https://artofproblemsolving.com/wiki/api.php?action=feedcontributions&user=Devenware&feedformat=atomAoPS Wiki - User contributions [en]2024-03-28T08:07:25ZUser contributionsMediaWiki 1.31.1https://artofproblemsolving.com/wiki/index.php?title=AoPS_Schoolhouse&diff=192485AoPS Schoolhouse2023-04-24T18:11:28Z<p>Devenware: /* Emojis */</p>
<hr />
<div>AoPS Schoolhouse is the system used for [[Math Jam|MathJams]] and online school. The main instructor (also a classroom admin) is responsible for posting the learning materials, while other classroom assistants serve as backup if the instructor is not present or answer questions posted by students. For example, in [[Mathcounts Week/Countdown Round|The World’s Largest Countdown Round]], there were 17 classroom assistants.<br />
<br />
Appearance of the new (beta) classroom design:<br />
[[File:The_new_aops_classroom.PNG|800px]]<br />
<br />
==Posting==<br />
If the chatroom is moderated, which is always the case during MathJams and usually the case during class, when students type things into the chatroom, the message goes to the instructors. From there, the instructors have the choice of sharing the question with the class, whispering an answer back, or doing nothing. Your post will show up in the classroom in your preferred color (you can change this in the settings) if it is chosen. Other people's chosen posts will show up in a blue color. If you get a reply back in a whisper, it will show up in a pink or purple color. If a moderator feels a question requires further discussion, they will open a mini window with you to more deeply discuss the question.<br />
<br />
==Character Limit==<br />
The character limit for posts in the AoPS classroom is <b>623 characters</b>. You can split your posts into two to bypass this limit. There is no limit for the number of posts you can send. However, you could get a warning for posting spam messages. (You can't post the same thing twice in a row.) You could also get a warning if you post messages too fast.<br />
<br />
==Emojis==<br />
Below is the list of emojis supported in the AoPS classroom and how to generate them. To adjust the emoji type you want, click on the arrow in the top right corner and select "Change Smiley Settings".<br />
<br />
{| class="wikitable"<br />
|- <!-- Start of a new row --><br />
| <center><b>Text</b></center> || <center><b>Classic</b></center> || <center><b>Alex</b></center> || <center><b>Grogg</b></center> || <center><b>Lizzie</b></center> || <center><b>Winnie</b></center><br />
|-<br />
| <pre>:)</pre> || <center>[[File:Smile-Classroom.gif]]</center> || <center>[[File:Smile-Classroom-Alex.png]]</center> || <center>[[File:Smile-Classroom-Grogg.png]]</center> || <center>[[File:Smile-Classroom-Lizzie.png]]</center> || <center>[[File:Smile-Classroom-Winnie.png]]</center><br />
|-<br />
| <pre>=)</pre> || <center>[[File:Smile-Classroom.gif]]</center> || <center>[[File:Smile-Classroom-Alex.png]]</center> || <center>[[File:Smile-Classroom-Grogg.png]]</center> || <center>[[File:Smile-Classroom-Lizzie.png]]</center> || <center>[[File:Smile-Classroom-Winnie.png]]</center><br />
|-<br />
| <pre>:-)</pre> || <center>[[File:Smile-Classroom.gif]]</center> || <center>[[File:Smile-Classroom-Alex.png]]</center> || <center>[[File:Smile-Classroom-Grogg.png]]</center> || <center>[[File:Smile-Classroom-Lizzie.png]]</center> || <center>[[File:Smile-Classroom-Winnie.png]]</center><br />
|-<br />
| <pre>=-)</pre> || <center>[[File:Smile-Classroom.gif]]</center> || <center>[[File:Smile-Classroom-Alex.png]]</center> || <center>[[File:Smile-Classroom-Grogg.png]]</center> || <center>[[File:Smile-Classroom-Lizzie.png]]</center> || <center>[[File:Smile-Classroom-Winnie.png]]</center><br />
|-<br />
| <pre>:(</pre> || <center>[[File:Frown-Classroom.gif]]</center> || <center>[[File:Frown-Classroom-Alex.png]]</center> || <center>[[File:Frown-Classroom-Grogg.png]]</center> || <center>[[File:Frown-Classroom-Lizzie.png]]</center> || <center>[[File:Frown-Classroom-Winnie.png]]</center><br />
|-<br />
| <pre>:-(</pre> || <center>[[File:Frown-Classroom.gif]]</center> || <center>[[File:Frown-Classroom-Alex.png]]</center> || <center>[[File:Frown-Classroom-Grogg.png]]</center> || <center>[[File:Frown-Classroom-Lizzie.png]]</center> || <center>[[File:Frown-Classroom-Winnie.png]]</center><br />
|-<br />
| <pre>;-)</pre> || <center>[[File:Wink-Classroom.gif]]</center> || <center>[[File:Wink-Classroom-Alex.png]]</center> || <center>[[File:Wink-Classroom-Grogg.png]]</center> || <center>[[File:Wink-Classroom-Lizzie.png]]</center> || <center>[[File:Wink-Classroom-Winnie.png]]</center><br />
|-<br />
| <pre>:D</pre> || <center>[[File:Bigsmile-Classroom.gif]]</center> || <center>[[File:Bigsmile-Classroom-Alex.png]]</center> || <center>[[File:Bigsmile-Classroom-Grogg.png]]</center> || <center>[[File:Bigsmile-Classroom-Lizzie.png]]</center> || <center>[[File:Bigsmile-Classroom-Winnie.png]]</center><br />
|-<br />
| <pre>D:</pre> || <center>[[File:Angry-Classroom.gif]]</center> || <center>[[File:Angry-Classroom-Alex.png]]</center> || <center>[[File:Angry-Classroom-Grogg.png]]</center> || <center>[[File:Angry-Classroom-Lizzie.png]]</center> || <center>[[File:Angry-Classroom-Winnie.png]]</center><br />
|-<br />
| <pre>:O</pre> || <center>[[File:Surprise-Classroom.gif]]</center> || <center>[[File:Surprise-Classroom-Alex.png]]</center> || <center>[[File:Surprise-Classroom-Grogg.png]]</center> || <center>[[File:Surprise-Classroom-Lizzie.png]]</center> || <center>[[File:Surprise-Classroom-Winnie.png]]</center><br />
|-<br />
| <pre>:P</pre> || <center>[[File:Tongue-Classroom.gif]]</center> || <center>[[File:Tongue-Classroom-Alex.png]]</center> || <center>[[File:Tongue-Classroom-Grogg.png]]</center> || <center>[[File:Tongue-Classroom-Lizzie.png]]</center> || <center>[[File:Tongue-Classroom-Winnie.png]]</center><br />
|-<br />
| <pre>:|</pre> || <center>[[File:Neutral-Classroom.gif]]</center> || <center>[[File:Neutral-Classroom-Alex.png]]</center> || <center>[[File:Neutral-Classroom-Grogg.png]]</center> || <center>[[File:Neutral-Classroom-Lizzie.png]]</center> || <center>[[File:Neutral-Classroom-Winnie.png]]</center><br />
|-<br />
| <pre>B-)</pre> || <center>[[File:Cool-Classroom.gif]]</center> || <center>[[File:Cool-Classroom-Alex.png]]</center> || <center>[[File:Cool-Classroom-Grogg.png]]</center> || <center>[[File:Cool-Classroom-Lizzie.png]]</center> || <center>[[File:Cool-Classroom-Winnie.png]]</center><br />
|-<br />
| <pre>O:)</pre> || <center>[[File:Saint-Classroom.gif]]</center> || <center>[[File:Saint-Classroom-Alex.png]]</center> || <center>[[File:Saint-Classroom-Grogg.png]]</center> || <center>[[File:Saint-Classroom-Lizzie.png]]</center> || <center>[[File:Saint-Classroom-Winnie.png]]</center><br />
|-<br />
| <pre>^_^</pre> || <center>[[File:Anime-Classroom.gif]]</center> || <center>[[File:Anime-Classroom-Alex.png]]</center> || <center>[[File:Anime-Classroom-Grogg.png]]</center> || <center>[[File:Anime-Classroom-Lizzie.png]]</center> || <center>[[File:Anime-Classroom-Winnie.png]]</center><br />
|-<br />
| <pre>^^</pre> || <center>[[File:Sweatdrop-Classroom.gif]]</center> || <center>[[File:Sweatdrop-Classroom-Alex.png]]</center> || <center>[[File:Sweatdrop-Classroom-Grogg.png]]</center> || <center>[[File:Sweatdrop-Classroom-Lizzie.png]]</center> || <center>[[File:Sweatdrop-Classroom-Winnie.png]]</center><br />
|-<br />
| <pre>:-3</pre> || <center>[[File:Smile3-Classroom.gif]]</center> || <center>[[File:Smile3-Classroom-Alex.png]]</center> || <center>[[File:Smile3-Classroom-Grogg.png]]</center> || <center>[[File:Smile3-Classroom-Lizzie.png]]</center> || <center>[[File:Smile3-Classroom-Winnie.png]]</center><br />
|-<br />
| <pre>;-3</pre> || <center>[[File:Wink3-Classroom.gif]]</center> || <center>[[File:Smile3-Classroom-Alex.png]]</center> || <center>[[File:Smile3-Classroom-Grogg.png]]</center> || <center>[[File:Smile3-Classroom-Lizzie.png]]</center> || <center>[[File:Smile3-Classroom-Winnie.png]]</center><br />
|-<br />
| <pre>o.O</pre> || <center>[[File:Boggle-Classroom.gif]]</center> || <center>[[File:Boggle-Classroom.gif]]</center> || <center>[[File:Boggle-Classroom.gif]]</center> || <center>[[File:Boggle-Classroom.gif]]</center> || <center>[[File:Boggle-Classroom.gif]]</center><br />
|-<br />
| <pre>O.o</pre> || <center>[[File:Boggle-Classroom.gif]]</center> || <center>[[File:Boggle-Classroom.gif]]</center> || <center>[[File:Boggle-Classroom.gif]]</center> || <center>[[File:Boggle-Classroom.gif]]</center> || <center>[[File:Boggle-Classroom.gif]]</center><br />
|-<br />
| <pre>:blue:</pre> || <center>[[File:Blue-Classroom.gif]]</center> || <center>[[File:Blue-Classroom-Alex.png]]</center> || <center>[[File:Blue-Classroom-Grogg.png]]</center> || <center>[[File:Blue-Classroom-Lizzie.png]]</center> || <center>[[File:Blue-Classroom-Winnie.png]]</center><br />
|-<br />
| <pre>:heart:</pre> || <center>[[File:Heart-Classroom.gif]]</center> || <center>[[File:Heart-Classroom-BA.png]]</center> || <center>[[File:Heart-Classroom-BA.png]]</center> || <center>[[File:Heart-Classroom-BA.png]]</center> || <center>[[File:Heart-Classroom-BA.png]]</center><br />
|-<br />
| <pre>:star:</pre> || <center>[[File:Star-Classroom.gif]]</center> || <center>[[File:Star-Classroom-BA.png]]</center> || <center>[[File:Star-Classroom-BA.png]]</center> || <center>[[File:Star-Classroom-BA.png]]</center> || <center>[[File:Star-Classroom-BA.png]]</center><br />
|-<br />
| <pre>:grin:</pre> || <center>[[File:Teeth-Classroom.gif]]</center> || <center>[[File:Teeth-Classroom-Alex.png]]</center> || <center>[[File:Teeth-Classroom-Grogg.png]]</center> || <center>[[File:Teeth-Classroom-Lizzie.png]]</center> || <center>[[File:Teeth-Classroom-Winnie.png]]</center><br />
|-<br />
| <pre>:sneaky:</pre> || <center>[[File:Sneaky-Classroom.gif]]</center> || <center>[[File:Sneaky-Classroom-Alex.png]]</center> || <center>[[File:Sneaky-Classroom-Grogg.png]]</center> || <center>[[File:Sneaky-Classroom-Lizzie.png]]</center> || <center>[[File:Sneaky-Classroom-Winnie.png]]</center><br />
|-<br />
| <pre>:zzz:</pre> || <center>[[File:Sleepy-Classroom.gif]]</center> || <center>[[File:Sleepy-Classroom-Alex.png]]</center> || <center>[[File:Sleepy-Classroom-Grogg.png]]</center> || <center>[[File:Sleepy-Classroom-Lizzie.png]]</center> || <center>[[File:Sleepy-Classroom-Winnie.png]]</center><br />
|-<br />
| <pre>:confused:</pre> || <center>[[File:Confuse-Classroom.gif]]</center> || <center>[[File:Confuse-Classroom-Alex.png]]</center> || <center>[[File:Confuse-Classroom-Grogg.png]]</center> || <center>[[File:Confuse-Classroom-Lizzie.png]]</center> || <center>[[File:Confuse-Classroom-Winnie.png]]</center><br />
|-<br />
| <pre>:laugh:</pre> || <center>[[File:Laugh-Classroom.gif]]</center> || <center>[[File:Laugh-Classroom-Alex.png]]</center> || <center>[[File:Laugh-Classroom-Grogg.png]]</center> || <center>[[File:Laugh-Classroom-Lizzie.png]]</center> || <center>[[File:Laugh-Classroom-Winnie.png]]</center><br />
|-<br />
| <pre>:grimace:</pre> || || <center>[[File:Grimace-Classroom-Alex.png]]</center> || <center>[[File:Grimace-Classroom-Grogg.png]]</center> || <center>[[File:Grimace-Classroom-Lizzie.png]]</center> || <center>[[File:Grimace-Classroom-Winnie.png]]</center><br />
|-<br />
| <pre>:smallgrin:</pre> || || <center>[[File:Smallgrin-Classroom-Alex.png]]</center> || <center>[[File:Smallgrin-Classroom-Grogg.png]]</center> || <center>[[File:Smallgrin-Classroom-Lizzie.png]]</center> || <center>[[File:Smallgrin-Classroom-Winnie.png]]</center><br />
|-<br />
| <pre>:ghost:</pre> || || <center>[[File:Ghost-Classroom-Alex.png]]</center> || <center>[[File:Ghost-Classroom-Grogg.png]]</center> || <center>[[File:Ghost-Classroom-Lizzie.png]]</center> || <center>[[File:Ghost-Classroom-Winnie.png]]</center><br />
|}</div>Devenwarehttps://artofproblemsolving.com/wiki/index.php?title=1987_AIME_Problems/Problem_14&diff=1571831987 AIME Problems/Problem 142021-07-01T23:28:26Z<p>Devenware: /* Solution 3 (Complex Numbers) */</p>
<hr />
<div>== Problem ==<br />
Compute<br />
<cmath>\frac{(10^4+324)(22^4+324)(34^4+324)(46^4+324)(58^4+324)}{(4^4+324)(16^4+324)(28^4+324)(40^4+324)(52^4+324)}.</cmath><br />
<br />
== Solution 1 (Sophie Germain Identity) ==<br />
The [[Sophie Germain Identity]] states that <math>a^4 + 4b^4</math> can be factored as <math>(a^2 + 2b^2 - 2ab)(a^2 + 2b^2 + 2ab)</math>. Each of the terms is in the form of <math>x^4 + 324</math>. Using Sophie Germain, we get that <cmath>x^4 + 4\cdot 3^4 = (x^2 + 2 \cdot 3^2 - 2\cdot 3\cdot x)(x^2 + 2 \cdot 3^2 + 2\cdot 3\cdot x) = (x(x-6) + 18)(x(x+6)+18),</cmath> so the original expression becomes<br />
<div style="text-align:center;"><math>\frac{[(10(10-6)+18)(10(10+6)+18)][(22(22-6)+18)(22(22+6)+18)]\cdots[(58(58-6)+18)(58(58+6)+18)]}{[(4(4-6)+18)(4(4+6)+18)][(16(16-6)+18)(16(16+6)+18)]\cdots[(52(52-6)+18)(52(52+6)+18)]}</math><br /><br /><br />
<br />
<math>= \frac{(10(4)+18)(10(16)+18)(22(16)+18)(22(28)+18)\cdots(58(52)+18)(58(64)+18)}{(4(-2)+18)(4(10)+18)(16(10)+18)(16(22)+18)\cdots(52(46)+18)(52(58)+18)}</math></div><br />
<br />
Almost all of the terms cancel out! We are left with <math>\frac{58(64)+18}{4(-2)+18} = \frac{3730}{10} = \boxed{373}</math>.<br />
<br />
== Solution 2 (Completing the Square and Difference of Squares) ==<br />
In both the numerator and the denominator, each factor is of the form <math>N^4+324=N^4+18^2</math> for some positive integer <math>N.</math> <br />
<br />
We factor <math>N^4+18^2</math> by completing the square, then applying the difference of squares:<br />
<cmath>\begin{align*}<br />
N^4+18^2&=\left(N^4+36N^2+18^2\right)-36N^2 \\<br />
&=\left(N^2+18\right)^2-(6N)^2 \\<br />
&=\left(N^2-6N+18\right)\left(N^2+6N+18\right) \\<br />
&=\left((N-3)^2+9\right)\left((N+3)^2+9\right).<br />
\end{align*}</cmath><br />
The original expression now becomes <cmath>\frac{\left[(7^2+9)(13^2+9)\right]\left[(19^2+9)(25^2+9)\right]\left[(31^2+9)(37^2+9)\right]\left[(43^2+9)(49^2+9)\right]\left[(55^2+9)(61^2+9)\right]}{\left[(1^2+9)(7^2+9)\right]\left[(13^2+9)(19^2+9)\right]\left[(25^2+9)(31^2+9)\right]\left[(37^2+9)(43^2+9)\right]\left[(49^2+9)(55^2+9)\right]}=\frac{61^2+9}{1^2+9}=\boxed{373}.</cmath><br />
~MRENTHUSIASM<br />
<br />
== Solution 3 (Complex Numbers) ==<br />
In both the numerator and the denominator, each factor is of the form <math>N^4+324=N^4+18^2</math> for some positive integer <math>N.</math><br />
<br />
We factor <math>N^4+18^2</math> by solving the equation <math>N^4+18^2</math> <math>=0</math>.<br />
<br />
<b>SOLUTION IN PROGRESS. WAITING FOR THE API ISSUE TO BE FIXED.</b><br />
<br />
<math>N^4+18^2=0^2</math> (code testing--this should be rendered correctly).<br />
<br />
TEST<br />
<br />
== Video Solution ==<br />
https://youtu.be/ZWqHxc0i7ro?t=1023<br />
<br />
~ pi_is_3.14<br />
<br />
== See also ==<br />
{{AIME box|year=1987|num-b=13|num-a=15}}<br />
<br />
[[Category:Intermediate Algebra Problems]]<br />
{{MAA Notice}}</div>Devenwarehttps://artofproblemsolving.com/wiki/index.php?title=1987_AIME_Problems/Problem_14&diff=1571821987 AIME Problems/Problem 142021-07-01T23:26:44Z<p>Devenware: /* Solution 3 (Complex Numbers) */</p>
<hr />
<div>== Problem ==<br />
Compute<br />
<cmath>\frac{(10^4+324)(22^4+324)(34^4+324)(46^4+324)(58^4+324)}{(4^4+324)(16^4+324)(28^4+324)(40^4+324)(52^4+324)}.</cmath><br />
<br />
== Solution 1 (Sophie Germain Identity) ==<br />
The [[Sophie Germain Identity]] states that <math>a^4 + 4b^4</math> can be factored as <math>(a^2 + 2b^2 - 2ab)(a^2 + 2b^2 + 2ab)</math>. Each of the terms is in the form of <math>x^4 + 324</math>. Using Sophie Germain, we get that <cmath>x^4 + 4\cdot 3^4 = (x^2 + 2 \cdot 3^2 - 2\cdot 3\cdot x)(x^2 + 2 \cdot 3^2 + 2\cdot 3\cdot x) = (x(x-6) + 18)(x(x+6)+18),</cmath> so the original expression becomes<br />
<div style="text-align:center;"><math>\frac{[(10(10-6)+18)(10(10+6)+18)][(22(22-6)+18)(22(22+6)+18)]\cdots[(58(58-6)+18)(58(58+6)+18)]}{[(4(4-6)+18)(4(4+6)+18)][(16(16-6)+18)(16(16+6)+18)]\cdots[(52(52-6)+18)(52(52+6)+18)]}</math><br /><br /><br />
<br />
<math>= \frac{(10(4)+18)(10(16)+18)(22(16)+18)(22(28)+18)\cdots(58(52)+18)(58(64)+18)}{(4(-2)+18)(4(10)+18)(16(10)+18)(16(22)+18)\cdots(52(46)+18)(52(58)+18)}</math></div><br />
<br />
Almost all of the terms cancel out! We are left with <math>\frac{58(64)+18}{4(-2)+18} = \frac{3730}{10} = \boxed{373}</math>.<br />
<br />
== Solution 2 (Completing the Square and Difference of Squares) ==<br />
In both the numerator and the denominator, each factor is of the form <math>N^4+324=N^4+18^2</math> for some positive integer <math>N.</math> <br />
<br />
We factor <math>N^4+18^2</math> by completing the square, then applying the difference of squares:<br />
<cmath>\begin{align*}<br />
N^4+18^2&=\left(N^4+36N^2+18^2\right)-36N^2 \\<br />
&=\left(N^2+18\right)^2-(6N)^2 \\<br />
&=\left(N^2-6N+18\right)\left(N^2+6N+18\right) \\<br />
&=\left((N-3)^2+9\right)\left((N+3)^2+9\right).<br />
\end{align*}</cmath><br />
The original expression now becomes <cmath>\frac{\left[(7^2+9)(13^2+9)\right]\left[(19^2+9)(25^2+9)\right]\left[(31^2+9)(37^2+9)\right]\left[(43^2+9)(49^2+9)\right]\left[(55^2+9)(61^2+9)\right]}{\left[(1^2+9)(7^2+9)\right]\left[(13^2+9)(19^2+9)\right]\left[(25^2+9)(31^2+9)\right]\left[(37^2+9)(43^2+9)\right]\left[(49^2+9)(55^2+9)\right]}=\frac{61^2+9}{1^2+9}=\boxed{373}.</cmath><br />
~MRENTHUSIASM<br />
<br />
== Solution 3 (Complex Numbers) ==<br />
In both the numerator and the denominator, each factor is of the form <math>N^4+324=N^4+18^2</math> for some positive integer <math>N.</math><br />
<br />
We factor <math>N^4+18^2</math> by solving the equation <math>N^4+18^2</math> <math>=0</math>.<br />
<br />
<b>SOLUTION IN PROGRESS. WAITING FOR THE API ISSUE TO BE FIXED.</b><br />
<br />
<math>N^4+18^2=0</math> (code testing--this should be rendered correctly).<br />
<br />
TEST<br />
<br />
== Video Solution ==<br />
https://youtu.be/ZWqHxc0i7ro?t=1023<br />
<br />
~ pi_is_3.14<br />
<br />
== See also ==<br />
{{AIME box|year=1987|num-b=13|num-a=15}}<br />
<br />
[[Category:Intermediate Algebra Problems]]<br />
{{MAA Notice}}</div>Devenwarehttps://artofproblemsolving.com/wiki/index.php?title=University_of_South_Carolina_High_School_Math_Contest/1993_Exam/Problems&diff=98301University of South Carolina High School Math Contest/1993 Exam/Problems2018-10-27T01:04:17Z<p>Devenware: /* Problem 29 */</p>
<hr />
<div>== Problem 1 ==<br />
If the width of a particular rectangle is doubled and the length is increased by 3, then the area is tripled. What is the length of the rectangle?<br />
<br />
<cmath> \mathrm{(A) \ } 1 \qquad \mathrm{(B) \ } 2 \qquad \mathrm{(C) \ } 3 \qquad \mathrm{(D) \ } 6 \qquad \mathrm{(E) \ } 9 </cmath><br />
<br />
[[University of South Carolina High School Math Contest/1993 Exam/Problem 1|Solution]]<br />
<br />
== Problem 2 ==<br />
Suppose the operation <math>\star</math> is defined by <math>a \star b = a+b+ab.</math> If <math>3\star x = 23,</math> then <math>x =</math><br />
<br />
<cmath> \mathrm{(A) \ } 2 \qquad \mathrm{(B) \ }3\qquad \mathrm{(C) \ }4 \qquad \mathrm{(D) \ }5 \qquad \mathrm{(E) \ }6 </cmath><br />
<br />
[[University of South Carolina High School Math Contest/1993 Exam/Problem 2|Solution]]<br />
<br />
== Problem 3 ==<br />
If 3 circles of radius 1 are mutually tangent as shown, what is the area of the gap they enclose?<br />
<br />
<center>[[Image:Usc93.3.PNG]]</center><br />
<br />
<cmath> \mathrm{(A) \ }\sqrt{3}-\frac{\pi}2 \qquad \mathrm{(B) \ } \frac 16 \qquad \mathrm{(C) \ }\frac 13 \qquad \mathrm{(D) \ } \frac{\sqrt{3}}2 - \frac{\pi}6 \qquad \mathrm{(E) \ } \frac{\pi}6 </cmath><br />
<br />
[[University of South Carolina High School Math Contest/1993 Exam/Problem 3|Solution]]<br />
<br />
== Problem 4 ==<br />
If <math>(1 + i)^{100}</math> is expanded and written in the form <math>a + bi</math> where <math>a</math> and <math>b</math> are real numbers, then <math>a =</math><br />
<br />
<cmath> \mathrm{(A) \ } -2^{50} \qquad \mathrm{(B) \ } 20^{50} - \frac{100!}{50!50!} \qquad \mathrm{(C) \ } \frac{100!}{(25!)^2 50!} \qquad \mathrm{(D) \ } 100! \left(-\frac 1{50!50!} + \frac 1{25!75!}\right) \qquad \mathrm{(E) \ } 0</cmath><br />
<br />
[[University of South Carolina High School Math Contest/1993 Exam/Problem 4|Solution]]<br />
<br />
== Problem 5 ==<br />
Suppose that <math>f</math> is a function with the property that for all <math>x</math> and <math>y, f(x + y) = f(x) + f(y) + 1</math> and <math>f(1) = 2.</math> What is the value of <math>f(3)</math>?<br />
<br />
<cmath> \mathrm{(A) \ }4 \qquad \mathrm{(B) \ }5 \qquad \mathrm{(C) \ }6 \qquad \mathrm{(D) \ }7 \qquad \mathrm{(E) \ }8 </cmath><br />
<br />
[[University of South Carolina High School Math Contest/1993 Exam/Problem 5|Solution]]<br />
<br />
== Problem 6 ==<br />
After a <math>p \%</math> price reduction, what increase does it take to restore the original price?<br />
<br />
<cmath> \mathrm{(A) \ }p\% \qquad \mathrm{(B) \ }\frac p{1-p}\% \qquad \mathrm{(C) \ } (100-p)\% \qquad \mathrm{(D) \ } \frac{100p}{100+p}\% \qquad \mathrm{(E) \ } \frac{100p}{100-p}\% </cmath><br />
<br />
[[University of South Carolina High School Math Contest/1993 Exam/Problem 6|Solution]]<br />
<br />
== Problem 7 ==<br />
Each card below covers up a number. The number written below each card is the sum of all the numbers covered by all of the other cards. What is the sum of all of the hidden numbers?<br />
<br />
<center>[[Image:Usc93.7.PNG]]</center><br />
<br />
<cmath> \mathrm{(A) \ }4.2 \qquad \mathrm{(B) \ }5 \qquad \mathrm{(C) \ }5.6 \qquad \mathrm{(D) \ }6.2 \qquad \mathrm{(E) \ }6.8 </cmath><br />
<br />
[[University of South Carolina High School Math Contest/1993 Exam/Problem 7|Solution]]<br />
<br />
== Problem 8 ==<br />
What is the coefficient of <math>x^3</math> in the expansion of <br />
<br />
<cmath>(1 + x + x^2 + x^3 + x^4 + x^5 )^6? </cmath><br />
<br />
<cmath> \mathrm{(A) \ } 40 \qquad \mathrm{(B) \ }48 \qquad \mathrm{(C) \ }56 \qquad \mathrm{(D) \ }62 \qquad \mathrm{(E) \ } 64 </cmath><br />
<br />
[[University of South Carolina High School Math Contest/1993 Exam/Problem 8|Solution]]<br />
<br />
== Problem 9 ==<br />
Suppose that <math>x</math> and <math>y</math> are integers such that <math>y > x > 1</math> and <math>y^2 - x^2 = 187</math>. Then one possible value of <math>xy</math> is<br />
<br />
<cmath> \mathrm{(A) \ }30 \qquad \mathrm{(B) \ }36 \qquad \mathrm{(C) \ }40 \qquad \mathrm{(D) \ }42 \qquad \mathrm{(E) \ }54 </cmath><br />
<br />
[[University of South Carolina High School Math Contest/1993 Exam/Problem 9|Solution]]<br />
<br />
== Problem 10 ==<br />
<math>\arcsin(1/3) + \arccos(1/3) + \arctan(1/3) + arccot(1/3) =</math><br />
<br />
<cmath> \mathrm{(A) \ }\pi \qquad \mathrm{(B) \ }\pi/2 \qquad \mathrm{(C) \ }\pi/3 \qquad \mathrm{(D) \ }2\pi/3 \qquad \mathrm{(E) \ }3/\pi/4 </cmath><br />
<br />
[[University of South Carolina High School Math Contest/1993 Exam/Problem 10|Solution]]<br />
<br />
== Problem 11 ==<br />
Suppose that 4 cards labeled 1 to 4 are placed randomly into 4 boxes also labeled 1 to 4, one card per box. What is the probability that no card gets placed into a box having the same label as the card?<br />
<br />
<cmath> \mathrm{(A) \ } 1/3 \qquad \mathrm{(B) \ }3/8 \qquad \mathrm{(C) \ }5/12 \qquad \mathrm{(D) \ } 1/2 \qquad \mathrm{(E) \ }9/16 </cmath><br />
<br />
[[University of South Carolina High School Math Contest/1993 Exam/Problem 11|Solution]]<br />
<br />
== Problem 12 ==<br />
If the equations <math> (1) x^2 + ax + b = 0</math> and <math> (2) x^2 + cx + d = 0 </math> have exactly one root in common, and <math> abcd\ne 0,</math> then the other root of equation <math> (2) </math> is <br />
<br />
<cmath> \mathrm{(A) \ }\frac{c-a}{b-d}d \qquad \mathrm{(B) \ }\frac{a+c}{b+d}d \qquad \mathrm{(C) \ }\frac{b+c}{a+d}c \qquad \mathrm{(D) \ }\frac{a-c}{b-d} \qquad \mathrm{(E) \ }\frac{a+c}{b-d}c </cmath><br />
<br />
[[University of South Carolina High School Math Contest/1993 Exam/Problem 12|Solution]]<br />
<br />
== Problem 13 ==<br />
Suppose that <math>x</math> and <math>y</math> are numbers such that <math>\sin(x+y) = 0.3</math> and <math>\sin(x-y) = 0.5</math>. Then <math> \sin (x)\cdot \cos (y) = </math><br />
<br />
<cmath> \mathrm{(A) \ }0.1 \qquad \mathrm{(B) \ }0.3 \qquad \mathrm{(C) \ }0.4 \qquad \mathrm{(D) \ }0.5 \qquad \mathrm{(E) \ }0.6 </cmath><br />
<br />
[[University of South Carolina High School Math Contest/1993 Exam/Problem 13|Solution]]<br />
<br />
== Problem 14 ==<br />
How many permutations of 1, 2, 3, 4, 5, 6, 7, 8, 9 have:<br />
* 1 appearing somewhere to the left of 2,<br />
* 3 somewhere to the left of 4, and<br />
* 5 somewhere to the left of 6?<br />
For example, 8 1 5 7 2 3 9 4 6 would be such a permutation.<br />
<br />
<cmath> \mathrm{(A) \ }9\cdot 7! \qquad \mathrm{(B) \ } 8! \qquad \mathrm{(C) \ }5!4! \qquad \mathrm{(D) \ }8!4! \qquad \mathrm{(E) \ }8!+6!+4! </cmath><br />
<br />
[[University of South Carolina High School Math Contest/1993 Exam/Problem 14|Solution]]<br />
<br />
== Problem 15 ==<br />
If we express the sum<br />
<br />
<cmath> \frac 1{3\cdot 5\cdot 7\cdot 11} + \frac 1{3\cdot 5\cdot 7\cdot 13} + \frac 1{3\cdot 5\cdot 11\cdot 13} + \frac 1{3\cdot 7\cdot 11\cdot 13} + \frac 1{5\cdot 7\cdot 11\cdot 13} </cmath><br />
<br />
as a rational number in reduced form, then the denominator will be<br />
<br />
<cmath> \mathrm{(A) \ }15015 \qquad \mathrm{(B) \ }5005 \qquad \mathrm{(C) \ }455 \qquad \mathrm{(D) \ }385 \qquad \mathrm{(E) \ }91 </cmath><br />
<br />
[[University of South Carolina High School Math Contest/1993 Exam/Problem 15|Solution]]<br />
<br />
== Problem 16 ==<br />
In the triangle below, <math>M, N, </math> and <math>P</math> are the midpoints of <math>BC, AB,</math> and <math>AC</math> respectively. <math>CN</math> and <math>AM</math> intersect at <math>O</math>. If the length of <math>CQ</math> is 4, then what is the length of <math>OQ</math>?<br />
<br />
<center>[[Image:Usc93.16.PNG]]</center><br />
<br />
<cmath> \mathrm{(A) \ }1 \qquad \mathrm{(B) \ }4/3 \qquad \mathrm{(C) \ }\sqrt{2} \qquad \mathrm{(D) \ }3/2 \qquad \mathrm{(E) \ }2 </cmath><br />
<br />
[[University of South Carolina High School Math Contest/1993 Exam/Problem 16|Solution]]<br />
<br />
== Problem 17 ==<br />
Let <math>[x]</math> represent the greatest integer that is less than or equal to <math>x</math>. For example, <math>[2.769]=2</math> and <math>[\pi]=3</math>. Then what is the value of <br />
<br />
<cmath> [\log_2 2] + [\log_2 3] + [\log_2 4] + \cdots + [\log_2 99] + [\log_2 100] ?</cmath><br />
<br />
<cmath> \mathrm{(A) \ } 480 \qquad \mathrm{(B) \ }481 \qquad \mathrm{(C) \ }482 \qquad \mathrm{(D) \ }483 \qquad \mathrm{(E) \ }484 </cmath><br />
<br />
[[University of South Carolina High School Math Contest/1993 Exam/Problem 17|Solution]]<br />
<br />
== Problem 18 ==<br />
The minimum value of the function<br />
<br />
<cmath>f(x) = \frac{\sin (x)}{\sqrt{1 - \cos^2 (x)}} + \frac{\cos(x)}{\sqrt{1 - \sin^2 (x) }} + \frac{\tan(x)}{\sqrt{\sec^2 (x) - 1}} + \frac{\cot (x)}{\sqrt{\csc^2 (x) - 1}}</cmath> <br />
<br />
as <math>x</math> varies over all numbers in the largest possible domain of <math>f</math>, is <br />
<br />
<cmath> \mathrm{(A) \ }-4 \qquad \mathrm{(B) \ }-2 \qquad \mathrm{(C) \ }0 \qquad \mathrm{(D) \ }2 \qquad \mathrm{(E) \ }4 </cmath><br />
<br />
[[University of South Carolina High School Math Contest/1993 Exam/Problem 18|Solution]]<br />
<br />
== Problem 19 ==<br />
In the figure below, there are 4 distinct dots <math>A, B, C,</math> and <math>D</math>, joined by edges. Each dot is to be colored either red, blue, green, or yellow. No two dots joined by an edge are to be colored with the same color. How many completed colorings are possible?<br />
<br />
<center>[[Image:Usc93.19.PNG]]</center><br />
<br />
<cmath> \mathrm{(A) \ }24 \qquad \mathrm{(B) \ }72 \qquad \mathrm{(C) \ }84 \qquad \mathrm{(D) \ }96 \qquad \mathrm{(E) \ }108 </cmath><br />
<br />
[[University of South Carolina High School Math Contest/1993 Exam/Problem 19|Solution]]<br />
<br />
== Problem 20 ==<br />
Let <math>A_1, A_2, \ldots , A_{63}</math> be the 63 nonempty subsets of <math>\{ 1,2,3,4,5,6 \}</math>. For each of these sets <math>A_i</math>, let <math>\pi(A_i)</math> denote the product of all the elements in <math>A_i</math>. Then what is the value of <math>\pi(A_1)+\pi(A_2)+\cdots+\pi(A_{63})</math>?<br />
<br />
<cmath> \mathrm{(A) \ }5003 \qquad \mathrm{(B) \ }5012 \qquad \mathrm{(C) \ }5039 \qquad \mathrm{(D) \ }5057 \qquad \mathrm{(E) \ }5093 </cmath><br />
<br />
[[University of South Carolina High School Math Contest/1993 Exam/Problem 20|Solution]]<br />
<br />
== Problem 21 ==<br />
Suppose that each pair of eight tennis players either played exactly one game last week or did not play at all. Each player participated in all but 12 games. How many games were played among the eight players?<br />
<br />
<cmath> \mathrm{(A) \ }10 \qquad \mathrm{(B) \ }12 \qquad \mathrm{(C) \ }14 \qquad \mathrm{(D) \ }16 \qquad \mathrm{(E) \ }18 </cmath><br />
<br />
[[University of South Carolina High School Math Contest/1993 Exam/Problem 21|Solution]]<br />
<br />
== Problem 22 ==<br />
Let <br />
<br />
<cmath> A = \left( 1 + \frac 12 + \frac 14 + \frac 18 + \frac 1{16} \right) \left( 1 + \frac 13 + \frac 19\right) \left( 1 + \frac 15\right) \left( 1 + \frac 17\right) \left( 1 + \frac 1{11} \right) \left( 1 + \frac 1{13}\right), </cmath><br />
<br />
<cmath>B = \left( 1 - \frac 12\right)^{-1} \left( 1 - \frac 13 \right)^{-1} \left(1 - \frac 15\right)^{-1} \left(1 - \frac 17\right)^{-1} \left(1-\frac 1{11}\right)^{-1} \left(1 - \frac 1{13}\right)^{-1}, </cmath><br />
<br />
and<br />
<br />
<cmath> C = 1 + \frac 12 + \frac 13 + \frac 14 + \frac 15 + \frac 16 + \frac 17 + \frac 18 + \frac 19 + \frac 1{10} + \frac 1{11} + \frac 1{12} + \frac 1{13} + \frac 1{14} + \frac 1{15} +\frac 1{16}. </cmath><br />
<br />
Then which of the following inequalities is true?<br />
<br />
<cmath> \mathrm{(A) \ } A > B > C \qquad \mathrm{(B) \ } B > A > C \qquad \mathrm{(C) \ } C > B > A \qquad \mathrm{(D) \ } C > A > B \qquad \mathrm{(E) \ } B > C > A </cmath><br />
<br />
[[University of South Carolina High School Math Contest/1993 Exam/Problem 22|Solution]]<br />
<br />
== Problem 23 ==<br />
The relation between the sets<br />
<br />
<cmath> M = \{ 12 m + 8 n + 4 l: m,n,l \rm{ \ are \ } \rm{integers}\} </cmath><br />
<br />
and<br />
<br />
<cmath> N= \{ 20 p + 16q + 12r: p,q,r \rm{ \ are \ } \rm{integers}\} </cmath><br />
<br />
is<br />
<cmath> \mathrm{(A) \ } M\subset N \qquad \mathrm{(B) \ } N\subset M \qquad \mathrm{(C) \ } M\cup N = \{0\} \qquad \mathrm{(D) \ }60244 \rm{ \ is \ } \rm{in \ } M \rm{ \ but \ } \rm{not \ } \rm{in \ } N \qquad \mathrm{(E) \ } M=N </cmath><br />
<br />
<br />
[[University of South Carolina High School Math Contest/1993 Exam/Problem 23|Solution]]<br />
<br />
== Problem 24 ==<br />
If <math>f(x) = \frac{1 + x}{1 - 3x}, f_1(x) = f(f(x)), f_2(x) = f(f_1(x)),</math> and in general <math>f_n(x) = f(f_{n-1}(x)),</math> then <math>f_{1993}(3)=</math><br />
<br />
<cmath> \mathrm{(A) \ }3 \qquad \mathrm{(B) \ }1993 \qquad \mathrm{(C) \ }\frac 12 \qquad \mathrm{(D) \ }\frac 15 \qquad \mathrm{(E) \ } -2^{-1993} </cmath><br />
<br />
[[University of South Carolina High School Math Contest/1993 Exam/Problem 24|Solution]]<br />
<br />
== Problem 25 ==<br />
What is the center of the circle passing through the point <math>(6,0)</math> and tangent to the circle <math>x^2 + y^2 = 4</math> at <math>(0,2)</math>? (Two circles are tangent at a point <math>P</math> if they intersect at <math>P</math> and at no other point.)<br />
<br />
<cmath> \mathrm{(A) \ }(0,-6) \qquad \mathrm{(B) \ } (1,-9) \qquad \mathrm{(C) \ } (-1,-9) \qquad \mathrm{(D) \ } (0,-9) \qquad \mathrm{(E) \ } \rm{none \ } \rm{of \ } \rm{these} </cmath><br />
<br />
[[University of South Carolina High School Math Contest/1993 Exam/Problem 25|Solution]]<br />
<br />
== Problem 26 ==<br />
Let <math>n=1667</math>. Then the first nonzero digit in the decimal expansion of <math>\sqrt{n^2 + 1} - n</math> is <br />
<br />
<cmath> \mathrm{(A) \ }1 \qquad \mathrm{(B) \ }2 \qquad \mathrm{(C) \ }3 \qquad \mathrm{(D) \ }4 \qquad \mathrm{(E) \ }5 </cmath><br />
<br />
[[University of South Carolina High School Math Contest/1993 Exam/Problem 26|Solution]]<br />
<br />
== Problem 27 ==<br />
Suppose <math>\triangle ABC</math> is a triangle with area 24 and that there is a point <math>P</math> inside <math>\triangle ABC</math> which is distance 2 from each of the sides of <math>\triangle ABC</math>. What is the perimeter of <math>\triangle ABC</math>?<br />
<br />
\[ <br />
\mathrm{(A) \ } 12 \qquad \mathrm{(B) \ }24 \qquad \mathrm{(C) \ }36 \qquad \mathrm{(D) \ }12\sqrt{2} \qquad \mathrm{(E) \ }12\sqrt{3} \]<br />
<br />
[[University of South Carolina High School Math Contest/1993 Exam/Problem 27|Solution]]<br />
<br />
== Problem 28 ==<br />
Suppose <math>\triangle ABC</math> is a triangle with 3 acute angles <math>A, B,</math> and <math>C</math>. Then the point <math>( \cos B - \sin A, \sin B - \cos A)</math><br />
<br />
(A) can be in the 1st quadrant and can be in the 2nd quadrant only<br />
<br />
(B) can be in the 3rd quadrant and can be in the 4th quadrant only<br />
<br />
(C) can be in the 2nd quadrant and can be in the 3rd quadrant only<br />
<br />
(D) can be in the 2nd quadrant only<br />
<br />
(E) can be in any of the 4 quadrants<br />
<br />
[[University of South Carolina High School Math Contest/1993 Exam/Problem 28|Solution]]<br />
<br />
== Problem 29 ==<br />
If the sides of a triangle have lengths 2, 3, and 4, what is the radius of the circle circumscribing the triangle?<br />
<br />
<cmath>\mathrm{(A)}\quad 2<br />
\quad \mathrm{(B) }\quad 8/\sqrt{15} <br />
\quad \mathrm{(C) }\quad 5/2<br />
\quad \mathrm{(D) }\quad \sqrt{6}<br />
\quad \mathrm{(E) }\quad (\sqrt{6} + 1)/2</cmath><br />
<br />
[[University of South Carolina High School Math Contest/1993 Exam/Problem 29|Solution]]<br />
<br />
== Problem 30 ==<br />
<cmath> \frac 1{1\cdot 2\cdot 3\cdot 4} + \frac 1{2\cdot 3\cdot 4\cdot 5} + \frac 1{3\cdot 4\cdot 5\cdot 6} + \cdots + \frac 1{28\cdot 29\cdot 30\cdot 31} = </cmath><br />
<br />
<br />
<cmath> \mathrm{(A) \ }1/18 \qquad \mathrm{(B) \ }1/21 \qquad \mathrm{(C) \ }4/93 \qquad \mathrm{(D) \ }128/2505 \qquad \mathrm{(E) \ } 749/13485</cmath><br />
<br />
[[University of South Carolina High School Math Contest/1993 Exam/Problem 30|Solution]]<br />
<br />
== See also ==<br />
* [[University of South Carolina High School Math Contest/1993 Exam]]</div>Devenwarehttps://artofproblemsolving.com/wiki/index.php?title=2016_AMC_10A_Problems/Problem_18&diff=980922016 AMC 10A Problems/Problem 182018-10-10T01:35:20Z<p>Devenware: /* Solution 2 */</p>
<hr />
<div>==Problem==<br />
<br />
Each vertex of a cube is to be labeled with an integer <math>1</math> through <math>8</math>, with each integer being used once, in such a way that the sum of the four numbers on the vertices of a face is the same for each face. Arrangements that can be obtained from each other through rotations of the cube are considered to be the same. How many different arrangements are possible?<br />
<br />
<math>\textbf{(A) } 1\qquad\textbf{(B) } 3\qquad\textbf{(C) }6 \qquad\textbf{(D) }12 \qquad\textbf{(E) }24</math><br />
<br />
==Solution 1==<br />
<br />
First of all, the adjacent faces have the same sum <math>(18</math>, because <math>1+2+3+4+5+6+7+8=36</math>, <math>36/2=18)</math>, <br />
so now consider the <math>\text{opposite sides}</math> (the two sides which are parallel but not on same face of the cube);<br />
they must have the same sum value too.<br />
Now think about the extreme condition 1 and 8, if they are not sharing the same side, which means they would become endpoints of <math>\text{opposite sides}</math>,<br />
we should have <math>1+X=8+Y</math>, but no solution for <math>[2,7]</math>, contradiction. <br />
<br />
Now we know <math>1</math> and <math>8</math> must share the same side, which sum is <math>9</math>, the <math>\text{opposite side}</math> also must have sum of <math>9</math>, same thing for the other two parallel sides.<br />
<br />
Now we have <math>4</math> parallel sides <math>1-8, 2-7, 3-6, 4-5</math>.<br />
thinking about <math>4</math> endpoints number need to have a sum of <math>18</math>.<br />
It is easy to notice only <math>1-7-6-4</math> and <math>8-2-3-5</math> would work.<br />
<br />
So if we fix one direction <math>1-8 (</math>or <math>8-1)</math> all other <math>3</math> parallel sides must lay in one particular direction. <math>(1-8,7-2,6-3,4-5)</math> or <math>(8-1,2-7,3-6,5-4)</math><br />
<br />
Now, the problem is same as the problem to arrange <math>4</math> points in a two-dimensional square. which is <math>\frac{4!}{4}</math>=<math>\boxed{\textbf{(C) }6.}</math><br />
<br />
== Solution 2 ==<br />
<br />
Again, all faces sum to <math>18.</math> If <math>x,y,z</math> are the vertices next to one, then the remaining vertices are <math>17-x-y, 17-y-z, 17-x-z, x+y+z-16.</math> Now it remains to test possibilities. Note that we must have <math>x+y+z>17.</math> Without loss of generality, let <math>x<y<z.</math><br />
<br />
<math>3,7,8:</math> Does not work.<br />
<math>4,6,8:</math> Works.<br />
<math>4,7,8:</math> Works.<br />
<math>5,6,7:</math> Does not work.<br />
<math>5,6,8:</math> Does not work.<br />
<math>5,7,8:</math> Does not work.<br />
<math>6,7,8:</math> Works.<br />
<br />
So our answer is <math>3\cdot 2=\boxed{\textbf{(C) }6.}</math><br />
<br />
== Solution 3 ==<br />
<br />
We know the sum of each face is <math>18.</math> If we look at an edge of the cube whose numbers sum to <math>x</math>, it must be possible to achieve the sum <math>18-x</math> in two distinct ways, looking at the two faces which contain the edge. If <math>8</math> and <math>6</math> were on the same face, it is possible to achieve the desired sum only with the numbers <math>1</math> and <math>3</math> since the values must be distinct. Similarly, if <math>8</math> and <math>7</math> were on the same face, the only way to get the sum is with <math>1</math> and <math>2</math>. This means that <math>6</math> and <math>7</math> are not on the same edge as <math>8</math>, or in other words they are diagonally across from it on the same face, or on the other end of the cube.<br />
<br />
Now we look at three cases, each yielding two solutions which are reflections of each other:<br />
<br />
1) <math>6</math> and <math>7</math> are diagonally opposite <math>8</math> on the same face.<br />
2) <math>6</math> is diagonally across the cube from <math>8</math>, while <math>7</math> is diagonally across from <math>8</math> on the same face.<br />
3) <math>7</math> is diagonally across the cube from <math>8</math>, while <math>6</math> is diagonally across from <math>8</math> on the same face.<br />
<br />
This means the answer is <math>3\cdot 2=\boxed{\textbf{(C) }6.}</math><br />
<br />
==See Also==<br />
{{AMC10 box|year=2016|ab=A|num-b=17|num-a=19}}<br />
{{AMC12 box|year=2016|ab=A|num-b=13|num-a=15}}<br />
{{MAA Notice}}</div>Devenwarehttps://artofproblemsolving.com/wiki/index.php?title=Separation_axioms&diff=94911Separation axioms2018-06-04T17:26:07Z<p>Devenware: /* Normal */</p>
<hr />
<div>The '''separation axioms''' are a series of definitions in [[topology]] that allow the classification of various topological spaces. The following [[axiom]]s are typically defined: <math>T_0 \subset T_1 \subset T_2 \subset T_{2 \frac 12} \subset T_3 \subset T_{3 \frac 12} \subset T_4 \subset T_5 \subset T_6</math>. Each axiom is a strictly stronger condition upon the topology than the previous axiom. <br />
<br />
== Accessible ==<br />
In a <math>T_1</math>, or an '''acessible''', space, every one-point set is closed. <br />
<br />
== Hausdorff ==<br />
<center><asy><br />
defaultpen(linewidth(1) + linetype("6 6"));<br />
pair A=(0,0),B=(1.6,0),C=(1.6,1),D=(0,1),shiftfactor=(2.5,0.2);<br />
<br />
/* draw an "open set" using Bezier" */<br />
picture neighborhood(){<br />
picture pic;<br />
path p = A{(0.2,-0.6)}..(2*A+B)/3{(0.4,-0.1)}..(A+2*B)/3{(0.4,-0.1)}..B{(0.3,0.7)}..(B+C)/2{(-0.5,0.7)}..C{(-0.1,0.6)}..(2*C+D)/3{(-0.7,0.5)}..(C+2*D)/3{(-0.5,0.5)}..D{(-0.1,-0.5)}..(A+D)/2{(0.6,-0.7)}..cycle;<br />
fill(pic,p,rgb(0.9,0.9,0.9));<br />
draw(pic,p);<br />
return pic;<br />
}<br />
<br />
/* actual drawing */ <br />
add(yscale(1.05)*neighborhood());<br />
add(shift(shiftfactor)*neighborhood());<br />
dot((A+C)/2);dot((A+C)/2 + shiftfactor); <br />
<br />
/* labels */ <br />
label("$x$",(A+C)/2,(1.2,-1.2)); label("$y$",(A+C)/2 + shiftfactor,(1.2,-1.2)); label("$U$",B,(1,-1)); label("$V$",B+shiftfactor,(1,-1)); <br />
</asy></center><br />
In a <math>T_2</math>, or an '''Hausdorff''', space, given any two distinct points <math>x,y</math> in a topological space <math>X</math>, there exists open sets <math>U, V</math> such that <math>x \in U, y \in V</math> and <math>U,V</math> are [[disjoint]]. <br />
<br />
An example of a space that is <math>T_1</math> but not <math>T_2</math> is the [[finite complement topology]] on any infinite space. <br />
<br />
== Regular ==<br />
<center><asy><br />
defaultpen(linewidth(1) + linetype("6 6"));<br />
pair A=(0,0),B=(1.6,0),C=(1.6,1),D=(0,1),shiftfactor=(2.5,0.2);<br />
<br />
/* draw an "open set" using Bezier" */<br />
picture neighborhood(){<br />
picture pic;<br />
path p = A{(0.2,-0.6)}..(2*A+B)/3{(0.4,-0.1)}..(A+2*B)/3{(0.4,-0.1)}..B{(0.3,0.7)}..(B+C)/2{(-0.5,0.7)}..C{(-0.1,0.6)}..(2*C+D)/3{(-0.7,0.5)}..(C+2*D)/3{(-0.5,0.5)}..D{(-0.1,-0.5)}..(A+D)/2{(0.6,-0.7)}..cycle;<br />
fill(pic,p,rgb(0.9,0.9,0.9));<br />
draw(pic,p);<br />
return pic;<br />
}<br />
<br />
pair SC = (A+C)/2+shiftfactor; path pSC = SC+(-0.25,-0.15)--SC+(-0.25,0.15)--SC+(0.25,0.15)--SC+(0.25,-0.15)--cycle;<br />
add(yscale(1.05)*neighborhood());<br />
add(shift(shiftfactor)*neighborhood());<br />
dot((A+C)/2); fill(pSC,rgb(0.7,0.7,0.7)); draw(pSC,linewidth(1)+linetype("6 4")); <br />
<br />
/* labels */<br />
label("$x$",(A+C)/2,(1.2,-1.2)); label("$B$",(A+C)/2 + shiftfactor,(2,-1)); label("$U$",B,(1,-1)); label("$V$",B+shiftfactor,(1,-1)); <br />
</asy></center><br />
In a <math>T_3</math>, or a '''regular''', space, given a point <math>x</math> and a closed set <math>A</math> in a topological space <math>X</math> that are disjoint, there exists open sets <math>U,V</math> such that <math>x \in U, A \subset V</math> and <math>U,V</math> are [[disjoint]]. <br />
<br />
An example of a Hausdorff space that is not regular is the space <math>\mathbb{R}_k</math>, the [[k-topology]] (or in more generality, a subspace of <math>\mathbb{R}</math> consisting of <math>\mathbb{R}</math> missing a [[countable]] number of elements). <br />
<br />
== Normal ==<br />
<center><asy><br />
defaultpen(linewidth(1) + linetype("6 6"));<br />
pair A=(0,0),B=(1.6,0),C=(1.6,1),D=(0,1),shiftfactor=(2.5,0.2);<br />
<br />
/* draw an "open set" using Bezier" */<br />
picture neighborhood(){<br />
picture pic;<br />
path p = A{(0.2,-0.6)}..(2*A+B)/3{(0.4,-0.1)}..(A+2*B)/3{(0.4,-0.1)}..B{(0.3,0.7)}..(B+C)/2{(-0.5,0.7)}..C{(-0.1,0.6)}..(2*C+D)/3{(-0.7,0.5)}..(C+2*D)/3{(-0.5,0.5)}..D{(-0.1,-0.5)}..(A+D)/2{(0.6,-0.7)}..cycle;<br />
fill(pic,p,rgb(0.9,0.9,0.9));<br />
draw(pic,p);<br />
return pic;<br />
}<br />
<br />
path oSC = (A+C)/2+(-0.25,-0.15)--(A+C)/2+(-0.25,0.15)--(A+C)/2+(0.25,0.15)--(A+C)/2+(0.25,-0.15)--cycle;<br />
pair SC = (A+C)/2+shiftfactor; path pSC = SC+(-0.25,-0.15)--SC+(-0.25,0.15)--SC+(0.25,0.15)--SC+(0.25,-0.15)--cycle;<br />
add(yscale(1.05)*neighborhood());<br />
add(shift(shiftfactor)*neighborhood());<br />
dot((A+C)/2); fill(pSC,rgb(0.7,0.7,0.7)); draw(pSC,linewidth(1)+linetype("6 4")); <br />
fill(oSC,rgb(0.7,0.7,0.7)); draw(oSC,linewidth(1)+linetype("6 4")); <br />
label("$A$",(A+C)/2,(2,-1)); label("$B$",(A+C)/2 + shiftfactor,(2,-1)); label("$U$",B,(1,-1)); label("$V$",B+shiftfactor,(1,-1)); <br />
</asy></center><br />
In a <math>T_4</math>, or a '''normal''', space, given any two disjoint closed sets <math>A,B</math> in a topological space <math>X</math>, there exists open sets <math>U,V</math> such that <math>A \subset U, B \in V</math> and <math>U,V</math> are [[disjoint]]. <br />
<br />
An example of a regular space that is not normal is the [[Sorgenfrey plane]]. <br />
<br />
{{stub}}<br />
<br />
[[Category:Topology]]</div>Devenwarehttps://artofproblemsolving.com/wiki/index.php?title=Gmaas&diff=86328Gmaas2017-07-10T23:38:53Z<p>Devenware: /* Known Facts About gmaas */</p>
<hr />
<div>Editing Gmaas (section)</div>Devenwarehttps://artofproblemsolving.com/wiki/index.php?title=Orbit-stabilizer_theorem&diff=80019Orbit-stabilizer theorem2016-08-18T16:41:28Z<p>Devenware: </p>
<hr />
<div>The '''orbit-stabilizer theorem''' is a [[combinatorics |combinatorial]] result in [[group theory]].<br />
<br />
Let <math>G</math> be a [[group]] [[group action|acting]] on a [[set]] <math>S</math>. For any <math>i \in S</math>, let <math>\text{stab}(i)</math> denote the [[stabilizer]] of <math>i</math>, and let <math>\text{orb}(i)</math> denote the [[orbit]] of <math>i</math>. The orbit-stabilizer theorem states that<br />
<cmath> \lvert G \rvert = \lvert \text{orb}(i) \rvert \cdot \lvert \text{stab}(i) \rvert . </cmath><br />
<br />
''Proof.'' Without loss of generality, let <math>G</math> operate on <math>S</math> from the left. We note that if <math>\alpha, \beta</math> are elements of <math>G</math> such that <math>\alpha(i) = \beta(i)</math>, then <math>\alpha^{-1} \beta \in \text{stab}(i)</math>. Hence for any <math>x \in \text{orb}(i)</math>, the set of elements <math>\alpha</math> of <math>G</math> for which <math>\alpha(i)= x</math> constitute a unique [[coset |left coset]] modulo <math>\text{stab}(i)</math>. Thus<br />
<cmath> \lvert \text{orb}(i) \rvert = \lvert G/\text{stab}(i) \rvert. </cmath><br />
The result then follows from [[Lagrange's Theorem]]. <math>\blacksquare</math><br />
<br />
== See also ==<br />
<br />
* [[Burnside's Lemma]]<br />
* [[Orbit]]<br />
* [[Stabilizer]]<br />
<br />
[[Category:Group theory]]</div>Devenwarehttps://artofproblemsolving.com/wiki/index.php?title=2004_AIME_I_Problems&diff=687722004 AIME I Problems2015-03-12T17:09:37Z<p>Devenware: /* Problem 15 */</p>
<hr />
<div>{{AIME Problems|year=2004|n=I}}<br />
<br />
== Problem 1 ==<br />
The digits of a positive integer <math> n </math> are four consecutive integers in decreasing order when read from left to right. What is the sum of the possible remainders when <math> n </math> is divided by 37?<br />
<br />
[[2004 AIME I Problems/Problem 1|Solution]]<br />
<br />
== Problem 2 ==<br />
Set <math> A </math> consists of <math> m </math> consecutive integers whose sum is <math> 2m, </math>and set <math> B </math> consists of <math> 2m </math> consecutive integers whose sum is <math> m. </math> The absolute value of the difference between the greatest element of <math> A </math> and the greatest element of <math> B </math> is 99. Find <math> m. </math><br />
<br />
[[2004 AIME I Problems/Problem 2|Solution]]<br />
<br />
== Problem 3 ==<br />
A convex polyhedron <math> P </math> has 26 vertices, 60 edges, and 36 faces, 24 of which are triangular, and 12 of which are quadrilaterals. A space diagonal is a line segment connecting two non-adjacent vertices that do not belong to the same face. How many space diagonals does <math> P </math> have?<br />
<br />
[[2004 AIME I Problems/Problem 3|Solution]]<br />
<br />
== Problem 4 ==<br />
A square has sides of length 2. Set <math> S </math> is the set of all line segments that have length 2 and whose endpoints are on adjacent sides of the square. The midpoints of the line segments in set <math> S </math> enclose a region whose area to the nearest hundredth is <math> k. </math> Find <math> 100k. </math><br />
<br />
[[2004 AIME I Problems/Problem 4|Solution]]<br />
<br />
== Problem 5 ==<br />
Alpha and Beta both took part in a two-day problem-solving competition. At the end of the second day, each had attempted questions worth a total of 500 points. Alpha scored 160 points out of 300 points attempted on the first day, and scored 140 points out of 200 points attempted on the second day. Beta who did not attempt 300 points on the first day, had a positive integer score on each of the two days, and Beta's daily success rate (points scored divided by points attempted) on each day was less than Alpha's on that day. Alpha's two-day success ratio was 300/500 = 3/5. The largest possible two-day success ratio that Beta could achieve is <math> m/n, </math> where <math> m </math> and <math> n </math> are relatively prime positive integers. What is <math> m+n </math>?<br />
<br />
[[2004 AIME I Problems/Problem 5|Solution]]<br />
<br />
== Problem 6 ==<br />
An integer is called snakelike if its decimal representation <math> a_1a_2a_3\cdots a_k </math> satisfies <math> a_i<a_{i+1} </math> if <math> i </math> is odd and <math> a_i>a_{i+1} </math> if <math> i </math> is even. How many snakelike integers between 1000 and 9999 have four distinct digits?<br />
<br />
[[2004 AIME I Problems/Problem 6|Solution]]<br />
<br />
== Problem 7 ==<br />
Let <math> C </math> be the coefficient of <math> x^2 </math> in the expansion of the product <math> (1 - x)(1 + 2x)(1 - 3x)\cdots(1 + 14x)(1 - 15x). </math> Find <math> |C|. </math><br />
<br />
[[2004 AIME I Problems/Problem 7|Solution]]<br />
<br />
== Problem 8 ==<br />
Define a regular <math> n </math>-pointed star to be the union of <math> n </math> line segments <math> P_1P_2, P_2P_3,\ldots, P_nP_1 </math> such that<br />
<br />
* the points <math> P_1, P_2,\ldots, P_n </math> are coplanar and no three of them are collinear,<br />
* each of the <math> n </math> line segments intersects at least one of the other line segments at a point other than an endpoint,<br />
* all of the angles at <math> P_1, P_2,\ldots, P_n </math> are congruent,<br />
* all of the <math> n </math> line segments <math> P_2P_3,\ldots, P_nP_1 </math> are congruent, and<br />
* the path <math> P_1P_2, P_2P_3,\ldots, P_nP_1 </math> turns counterclockwise at an angle of less than 180 degrees at each vertex.<br />
<br />
There are no regular 3-pointed, 4-pointed, or 6-pointed stars. All regular 5-pointed stars are similar, but there are two non-similar regular 7-pointed stars. How many non-similar regular 1000-pointed stars are there?<br />
<br />
[[2004 AIME I Problems/Problem 8|Solution]]<br />
<br />
== Problem 9 ==<br />
Let <math> ABC </math> be a triangle with sides 3, 4, and 5, and <math> DEFG </math> be a 6-by-7 rectangle. A segment is drawn to divide triangle <math> ABC </math> into a triangle <math> U_1 </math> and a trapezoid <math> V_1 </math> and another segment is drawn to divide rectangle <math> DEFG </math> into a triangle <math> U_2 </math> and a trapezoid <math> V_2 </math> such that <math> U_1 </math> is similar to <math> U_2 </math> and <math> V_1 </math> is similar to <math> V_2. </math> The minimum value of the area of <math> U_1 </math> can be written in the form <math> m/n, </math> where <math> m </math> and <math> n </math>are relatively prime positive integers. Find <math> m+n. </math><br />
<br />
[[2004 AIME I Problems/Problem 9|Solution]]<br />
<br />
== Problem 10 ==<br />
A circle of radius 1 is randomly placed in a 15-by-36 rectangle <math> ABCD </math> so that the circle lies completely within the rectangle. Given that the probability that the circle will not touch diagonal <math> AC </math> is <math> m/n, </math> where <math> m </math> and <math> n </math> are relatively prime positive integers, find <math> m + n. </math><br />
<br />
[[2004 AIME I Problems/Problem 10|Solution]]<br />
<br />
== Problem 11 ==<br />
A solid in the shape of a right circular cone is 4 inches tall and its base has a 3-inch radius. The entire surface of the cone, including its base, is painted. A plane parallel to the base of the cone divides the cone into two solids, a smaller cone-shaped solid <math> C </math> and a frustum-shaped solid <math> F, </math> in such a way that the ratio between the areas of the painted surfaces of <math> C </math> and <math> F </math> and the ratio between the volumes of <math> C </math> and <math> F </math> are both equal to <math> k. </math> Given that <math> k=m/n, </math> where <math> m </math> and <math> n </math> are relatively prime positive integers, find <math> m+n. </math><br />
<br />
[[2004 AIME I Problems/Problem 11|Solution]]<br />
<br />
== Problem 12 ==<br />
Let <math> S </math> be the set of ordered pairs <math> (x, y) </math> such that <math> 0 < x \le 1, 0<y\le 1, </math> and <math> \left[\log_2{\left(\frac 1x\right)}\right] </math> and <math> \left[\log_5{\left(\frac 1y\right)}\right] </math> are both even. Given that the area of the graph of <math> S </math> is <math> m/n, </math> where <math> m </math> and <math> n </math> are relatively prime positive integers, find <math> m+n. </math> The notation <math> [z] </math> denotes the greatest integer that is less than or equal to <math> z. </math><br />
<br />
[[2004 AIME I Problems/Problem 12|Solution]]<br />
<br />
== Problem 13 ==<br />
The polynomial <math> P(x)=(1+x+x^2+\cdots+x^{17})^2-x^{17} </math> has 34 complex roots of the form <math> z_k = r_k[\cos(2\pi a_k)+i\sin(2\pi a_k)], k=1, 2, 3,\ldots, 34, </math> with <math> 0 < a_1 \le a_2 \le a_3 \le \cdots \le a_{34} < 1 </math> and <math> r_k>0. </math> Given that <math> a_1 + a_2 + a_3 + a_4 + a_5 = m/n, </math> where <math> m </math> and <math> n </math> are relatively prime positive integers, find <math> m+n. </math><br />
<br />
[[2004 AIME I Problems/Problem 13|Solution]]<br />
<br />
== Problem 14 ==<br />
A unicorn is tethered by a 20-foot silver rope to the base of a magician's cylindrical tower whose radius is 8 feet. The rope is attached to the tower at ground level and to the unicorn at a height of 4 feet. The unicorn has pulled the rope taut, the end of the rope is 4 feet from the nearest point on the tower, and the length of the rope that is touching the tower is <math> \frac{a-\sqrt{b}}c </math> feet, where <math> a, b, </math> and <math> c </math> are positive integers, and <math> c </math> is prime. Find <math> a+b+c. </math><br />
<br />
[[2004 AIME I Problems/Problem 14|Solution]]<br />
<br />
== Problem 15 ==<br />
For all positive integers <math>x</math>, let<br />
<cmath><br />
f(x)=\begin{cases}1 &\mbox{if x = 1}\\ \frac x{10} &\mbox{if x is divisible by 10}\\ x+1 &\mbox{otherwise}\end{cases}<br />
</cmath><br />
and define a sequence as follows: <math>x_1=x</math> and <math>x_{n+1}=f(x_n)</math> for all positive integers <math>n</math>. Let <math>d(x)</math> be the smallest <math>n</math> such that <math>x_n=1</math>. (For example, <math>d(100)=3</math> and <math>d(87)=7</math>.) Let <math>m</math> be the number of positive integers <math>x</math> such that <math>d(x)=20</math>. Find the sum of the distinct prime factors of <math>m</math>.<br />
<br />
[[2004 AIME I Problems/Problem 15|Solution]]<br />
<br />
== See Also ==<br />
* [[2004 AIME I]]<br />
* [[American Invitational Mathematics Examination]]<br />
* [[AIME Problems and Solutions]]<br />
* [[Mathematics competition resources]]<br />
{{MAA Notice}}</div>Devenwarehttps://artofproblemsolving.com/wiki/index.php?title=2015_AMC_10A_Problems/Problem_9&diff=680412015 AMC 10A Problems/Problem 92015-03-01T23:49:44Z<p>Devenware: /* Problem */</p>
<hr />
<div>{{duplicate|[[2015 AMC 12A Problems|2015 AMC 12A #7]] and [[2015 AMC 10A Problems|2015 AMC 10A #9]]}}<br />
==Problem==<br />
Two right circular cylinders have the same volume. The radius of the second cylinder is <math>10\%</math> more than the radius of the first. What is the relationship between the heights of the two cylinders?<br />
<br />
<math>\textbf{(A)}\ \text{The second height is } 10\% \text{ less than the first.} \\ \textbf{(B)}\ \text{The first height is } 10\% \text{ more than the second.}\\ \textbf{(C)}\ \text{The second height is } 21\% \text{ less than the first.} \\ \textbf{(D)}\ \text{The first height is } 21\% \text{ more than the second.}\\ \textbf{(E)}\ \text{The second height is } 80\% \text{ of the first.}</math><br />
<br />
==Solution==<br />
Let the radius of the first cylinder be <math>r_1</math> and the radius of the second cylinder be <math>r_2</math>. Also, let the height of the first cylinder be <math>h_1</math> and the height of the second cylinder be <math>h_2</math>. We are told <cmath>r_2=\frac{11r_1}{10}</cmath> <cmath>\pi r_1^2h_1=\pi r_2^2h_2</cmath> Substituting the first equation into the second and dividing both sides by <math>\pi</math>, we get <cmath>r_1^2h_1=\frac{121r_1^2}{100}h_2\implies h_1=\frac{121h_2}{100}.</cmath> Therefore, <math>\boxed{\textbf{(D)}\ \text{The first height is } 21\% \text{ more than the second.}}</math><br />
<br />
==See Also==<br />
{{AMC10 box|year=2015|ab=A|num-b=8|num-a=10}}<br />
{{AMC12 box|year=2015|ab=A|num-b=6|num-a=8}}<br />
{{MAA Notice}}</div>Devenwarehttps://artofproblemsolving.com/wiki/index.php?title=2015_AMC_10A_Problems/Problem_6&diff=680402015 AMC 10A Problems/Problem 62015-03-01T23:46:57Z<p>Devenware: /* Problem */</p>
<hr />
<div>{{duplicate|[[2015 AMC 12A Problems|2015 AMC 12A #4]] and [[2015 AMC 10A Problems|2015 AMC 10A #6]]}}<br />
==Problem==<br />
<br />
The sum of two positive numbers is <math> 5 </math> times their difference. What is the ratio of the larger number to the smaller number?<br />
<br />
<math> \textbf{(A)}\ \frac{5}{4}\qquad\textbf{(B)}\ \frac{3}{2}\qquad\textbf{(C)}\ \frac{9}{5}\qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ \frac{5}{2} </math><br />
<br />
==Solution==<br />
<br />
Let <math>a</math> be the bigger number and <math>b</math> be the smaller.<br />
<br />
<math>a + b = 5(a - b)</math>.<br />
<br />
Solving gives <math>\frac{a}{b} = \frac32</math>, so the answer is <math>\boxed{\textbf{(B) }\frac32}</math>.<br />
<br />
==See Also==<br />
{{AMC10 box|year=2015|ab=A|num-b=5|num-a=7}}<br />
{{AMC12 box|year=2015|ab=A|num-b=3|num-a=5}}<br />
{{MAA Notice}}</div>Devenwarehttps://artofproblemsolving.com/wiki/index.php?title=2015_AMC_10A_Problems/Problem_5&diff=680392015 AMC 10A Problems/Problem 52015-03-01T23:46:35Z<p>Devenware: /* Problem */</p>
<hr />
<div>{{duplicate|[[2015 AMC 12A Problems|2015 AMC 12A #3]] and [[2015 AMC 10A Problems|2015 AMC 10A #5]]}}<br />
==Problem==<br />
<br />
Mr. Patrick teaches math to <math> 15 </math> students. He was grading tests and found that when he graded everyone's test except Payton's, the average grade for the class was <math> 80 </math>. After he graded Payton's test, the test average became <math> 81 </math>. What was Payton's score on the test?<br />
<br />
<math> \textbf{(A)}\ 81\qquad\textbf{(B)}\ 85\qquad\textbf{(C)}\ 91\qquad\textbf{(D)}\ 94\qquad\textbf{(E)}\ 95 </math><br />
<br />
==Solution==<br />
<br />
If the average of the first <math>14</math> peoples' scores was <math>80</math>, then the sum of all of their tests is <math>14*80 = 1120</math>. When Payton's score was added, the sum of all of the scores became <math>15*81 = 1215</math>. So, Payton's score must be <math>1215-1120 = \boxed{\textbf{(E) }95}</math><br />
<br />
<br />
==Alternate Solution==<br />
<br />
The average of a set of numbers is the value we get if we evenly distribute the total across all entries. So assume that the first <math>14</math> students each scored <math>80</math>. If Payton also scored an <math>80</math>, the average would still be <math>80</math>. In order to increase the overall average to <math>81</math>, we need to add one more point to all of the scores, including Payton's. This means we need to add a total of <math>15</math> more points, so Payton needs <math>80+15 = \boxed{\textbf{(E) }95}</math><br />
<br />
<br />
==See also==<br />
{{AMC10 box|year=2015|ab=A|num-b=4|num-a=6}}<br />
{{AMC12 box|year=2015|ab=A|num-b=2|num-a=4}}<br />
{{MAA Notice}}</div>Devenwarehttps://artofproblemsolving.com/wiki/index.php?title=2015_AMC_10A_Problems/Problem_4&diff=680382015 AMC 10A Problems/Problem 42015-03-01T23:46:16Z<p>Devenware: /* Problem */</p>
<hr />
<div>==Problem==<br />
<br />
Pablo, Sofia, and Mia got some candy eggs at a party. Pablo had three times as many eggs as Sofia, and Sofia had twice as many eggs as Mia. Pablo decides to give some of his eggs to Sofia and Mia so that all three will have the same number of eggs. What fraction of his eggs should Pablo give to Sofia?<br />
<br />
<math> \textbf{(A)}\ \frac{1}{12}\qquad\textbf{(B)}\ \frac{1}{6}\qquad\textbf{(C)}\ \frac{1}{4}\qquad\textbf{(D)}\ \frac{1}{3}\qquad\textbf{(E)}\ \frac{1}{2}</math><br />
<br />
==Solution==<br />
<br />
Assign a variable to the number of eggs Mia has, say <math>m</math>. Then, because we are given that Sofia has twice the number of eggs Mia has, Sofia has <math>2m</math> eggs, and Pablo, having three times the number of eggs as Sofia, has <math>6m</math> eggs.<br />
<br />
For them to all have the same number of eggs, they must each have <math>\frac{m+2m+6m}{3} = 3m</math> eggs. This means Pablo must give <math>2m</math> eggs to Mia and a <math>m</math> eggs to Sofia, so the answer is <math>\frac{m}{6m} = \boxed{\textbf{(B) }\frac{1}{6}}</math><br />
<br />
==See Also==<br />
{{AMC10 box|year=2015|ab=A|num-b=3|num-a=5}}<br />
{{MAA Notice}}</div>Devenwarehttps://artofproblemsolving.com/wiki/index.php?title=2015_AMC_10A_Problems/Problem_3&diff=680372015 AMC 10A Problems/Problem 32015-03-01T23:45:59Z<p>Devenware: /* Problem */</p>
<hr />
<div>==Problem==<br />
Ann made a <math>3</math>-step staircase using <math>18</math> toothpicks as shown in the figure. How many toothpicks does she need to add to complete a <math>5</math>-step staircase?<br />
<br />
<math>\textbf{(A)}\ 9\qquad\textbf{(B)}\ 18\qquad\textbf{(C)}\ 20\qquad\textbf{(D)}\ 22\qquad\textbf{(E)}\ 24</math><br />
<br />
==Solution==<br />
We can see that a <math>1</math>-step staircase requires <math>4</math> toothpicks and a <math>2</math>-step staircase requires <math>10</math> toothpicks. Thus, to go from a <math>1</math>-step to <math>2</math>-step staircase, <math>6</math> additional toothpicks are needed and to go from a <math>2</math>-step to <math>3</math>-step staircase, <math>8</math> additional toothpicks are needed. Applying this pattern, to go from a <math>3</math>-step to <math>4</math>-step staircase, <math>10</math> additional toothpicks are needed and to go from a <math>4</math>-step to <math>5</math>-step staircase, <math>12</math> additional toothpicks are needed. Our answer is <math>10+12=\boxed{\textbf{(D)}\ 22}</math><br />
<br />
==See Also==<br />
{{AMC10 box|year=2015|ab=A|num-b=2|num-a=4}}<br />
{{MAA Notice}}</div>Devenwarehttps://artofproblemsolving.com/wiki/index.php?title=2015_AMC_10A_Problems/Problem_2&diff=680362015 AMC 10A Problems/Problem 22015-03-01T23:45:42Z<p>Devenware: /* Problem */</p>
<hr />
<div>==Problem==<br />
<br />
A box contains a collection of triangular and square tiles. There are <math>25</math> tiles in the box, containing <math>84</math> edges total. How many square tiles are there in the box?<br />
<br />
<math> \textbf{(A)}\ 3\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 11</math><br />
<br />
==Solution==<br />
<br />
Let <math>a</math> be the amount of triangular tiles and <math>b</math> be the amount of square tiles.<br />
<br />
Triangles have <math>3</math> edges and squares have <math>4</math> edges, so we have a system of equations.<br />
<br />
We have <math>a + b</math> tiles total, so <math>a + b = 25</math>.<br />
<br />
We have <math>3a + 4b</math> edges total, so <math>3a + 4b = 84</math>.<br />
<br />
Solving gives, <math>a = 16</math> and <math>b = 9</math>, so the answer is <math>\boxed{\textbf{(D) }9}</math>.<br />
<br />
<br />
==Alternate Solution==<br />
<br />
If all of the tiles were triangles, there would be <math>75</math> edges. This is not enough, so there need to be some squares. Trading a triangle for a square results in one additional edge each time, so we must trade out <math>9</math> triangles for squares. Answer: <math>\boxed{\textbf{(D) }9}</math><br />
<br />
<br />
==See Also==<br />
{{AMC10 box|year=2015|ab=A|num-b=1|num-a=3}}<br />
{{MAA Notice}}</div>Devenwarehttps://artofproblemsolving.com/wiki/index.php?title=2015_AMC_10A_Problems/Problem_1&diff=680352015 AMC 10A Problems/Problem 12015-03-01T23:45:22Z<p>Devenware: /* Problem */</p>
<hr />
<div>{{duplicate|[[2015 AMC 12A Problems|2015 AMC 12A #1]] and [[2015 AMC 10A Problems|2015 AMC 10A #1]]}}<br />
==Problem==<br />
<br />
What is the value of <math>(2^0-1+5^2-0)^{-1}\times5?</math><br />
<br />
<math> \textbf{(A)}\ -125\qquad\textbf{(B)}\ -120\qquad\textbf{(C)}\ \frac{1}{5}\qquad\textbf{(D)}\ \frac{5}{24}\qquad\textbf{(E)}\ 25</math><br />
<br />
==Solution==<br />
<math>(2^0-1+5^2-0)^{-1}\times5 = (1-1+25-0)^{-1} \times 5 = 25^{-1} \times 5 = \frac{1}{25} \times 5 = \boxed{\textbf{(C) } \, \frac{1}{5}}</math>.<br />
<br />
==See Also==<br />
{{AMC10 box|year=2015|ab=A|before=First Problem|num-a=2}}<br />
{{AMC12 box|year=2015|ab=A|before=First Problem|num-a=2}}<br />
{{MAA Notice}}</div>Devenwarehttps://artofproblemsolving.com/wiki/index.php?title=2015_AMC_12A_Problems&diff=680342015 AMC 12A Problems2015-03-01T23:44:49Z<p>Devenware: /* Problem 14 */</p>
<hr />
<div>==Problem 1==<br />
<br />
What is the value of <math>(2^0-1+5^2-0)^{-1}\times5?</math><br />
<br />
<math> \textbf{(A)}\ -125\qquad\textbf{(B)}\ -120\qquad\textbf{(C)}\ \frac{1}{5}\qquad\textbf{(D)}\ \frac{5}{24}\qquad\textbf{(E)}\ 25 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
<br />
Two of the three sides of a triangle are 20 and 15. Which of the following numbers is not a possible perimeter of the triangle?<br />
<br />
<math> \textbf{(A)}\ 52\qquad\textbf{(B)}\ 57\qquad\textbf{(C)}\ 62\qquad\textbf{(D)}\ 67\qquad\textbf{(E)}\ 72 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
<br />
Mr. Patrick teaches math to 15 students. He was grading tests and found that when he graded everyone's test except Payton's, the average grade for the class was 80. after he graded Payton's test, the class average became 81. What was Payton's score on the test?<br />
<br />
<math> \textbf{(A)}\ 81\qquad\textbf{(B)}\ 85\qquad\textbf{(C)}\ 91\qquad\textbf{(D)}\ 94\qquad\textbf{(E)}\ 95 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
The sum of two positive numbers is 5 times their difference. What is the ratio of the larger number to the smaller?<br />
<br />
<math> \textbf{(A)}\ \frac54 \qquad\textbf{(B)}\ \frac32 \qquad\textbf{(C)}\ \frac95 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ \frac52 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
Amelia needs to estimate the quantity <math>\frac{a}{b} - c</math>, where <math>a, b,</math> and <math>c</math> are large positive integers. She rounds each of the integers so that the calculation will be easier to do mentally. In which of these situations will her answer necessarily be greater than the exact value of <math>\frac{a}{b} - c</math>?<br />
<br />
<math> \textbf{(A)}\ \text{She rounds all three numbers up.}\\<br />
\qquad\textbf{(B)}\ \text{She rounds } a \text{ and } b \text{ up, and she rounds } c \text{ down.}\\<br />
\qquad\textbf{(C)}\ \text{She rounds } a \text{ and } c \text{ up, and she rounds } b \text{ down.} \\<br />
\qquad\textbf{(D)}\ \text{She rounds } a \text{ up, and she rounds } b \text{ and } c \text{ down.}\\<br />
\qquad\textbf{(E)}\ \text{She rounds } c \text{ up, and she rounds } a \text{ and } b \text{ down.} </math><br />
<br />
[[2015 AMC 12A Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be <math>2 : 1</math>?<br />
<br />
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 8</math><br />
<br />
[[2015 AMC 12A Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
Two right circular cylinders have the same volume. The radius of the second cylinder is <math>10\%</math> more than the radius of the first. What is the relationship between the heights of the two cylinders?<br />
<br />
<math>\textbf{(A)}\ \text{The second height is } 10\% \text{ less than the first.} \\ \textbf{(B)}\ \text{The first height is } 10\% \text{ more than the second.}\\ \textbf{(C)}\ \text{The second height is } 21\% \text{ less than the first.} \\ \textbf{(D)}\ \text{The first height is } 21\% \text{ more than the second.}\\ \textbf{(E)}\ \text{The second height is } 80\% \text{ of the first.}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
The ratio of the length to the width of a rectangle is <math>4</math> : <math>3</math>. If the rectangle has diagonal of length <math>d</math>, then the area may be expressed as <math>kd^2</math> for some constant <math>k</math>. What is <math>k</math>?<br />
<br />
<math> \textbf{(A)}\ \frac27 \qquad\textbf{(B)}\ \frac37 \qquad\textbf{(C)}\ \frac{12}{25} \qquad\textbf{(D)}\ \frac{16}{26} \qquad\textbf{(E)}\ \frac34</math><br />
<br />
[[2015 AMC 12A Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
A box contains 2 red marbles, 2 green marbles, and 2 yellow marbles. Carol takes 2 marbles from the box at random; then Claudia takes 2 of the remaining marbles at random; and then Cheryl takes the last 2 marbles. What is the probability that Cheryl gets 2 marbles of the same color?<br />
<br />
<math> \textbf{(A)}\ \frac{1}{10} \qquad\textbf{(B)}\ \frac16 \qquad\textbf{(C)}\ \frac15 \qquad\textbf{(D)}\ \frac13 \qquad\textbf{(E)}\ \frac12</math><br />
<br />
[[2015 AMC 12A Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
Integers <math>x</math> and <math>y</math> with <math>x>y>0</math> satisfy <math>x+y+xy=80</math>. What is <math>x</math>?<br />
<br />
<math> \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 15 \qquad\textbf{(D)}\ 18 \qquad\textbf{(E)}\ 26</math><br />
<br />
[[2015 AMC 12A Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
On a sheet of paper, Isabella draws a circle of radius <math>2</math>, a circle of radius <math>3</math>, and all possible lines simultaneously tangent to both circles. Isabella notices that she has drawn exactly <math>k \ge 0</math> lines. How many different values of <math>k</math> are possible?<br />
<br />
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6</math><br />
<br />
[[2015 AMC 12A Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
The parabolas <math>y=ax^2 - 2</math> and <math>y=4 - bx^2</math> intersect the coordinate axes in exactly four points, and these four points are the vertices of a kite of area <math>12</math>. What is <math>a+b</math>?<br />
<br />
<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 1.5\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 2.5\qquad\textbf{(E)}\ 3</math><br />
<br />
[[2015 AMC 12A Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
<br />
A league with 12 teams holds a round-robin tournament, with each team playing every other team exactly once. Games either end with one team victorious or else end in a draw. A team scores 2 points for every game it wins and 1 point for every game it draws. Which of the following is NOT a true statement about the list of 12 scores?<br />
<br />
<math> \textbf{(A)}\ \text{There must be an even number of odd scores.}\\<br />
\qquad\textbf{(B)}\ \text{There must be an even number of even scores.}\\<br />
\qquad\textbf{(C)}\ \text{There cannot be two scores of }0\text{.}\\<br />
\qquad\textbf{(D)}\ \text{The sum of the scores must be at least }100\text{.}\\<br />
\qquad\textbf{(E)}\ \text{The highest score must be at least }12\text{.}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
What is the value of <math>a</math> for which <math>\frac{1}{\text{log}_2a} + \frac{1}{\text{log}_3a} + \frac{1}{\text{log}_4a} = 1</math>?<br />
<br />
<math>\textbf{(A)}\ 9\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 24\qquad\textbf{(E)}\ 36</math><br />
<br />
[[2015 AMC 12A Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
What is the minimum number of digits to the right of the decimal point needed to express the fraction <math>\frac{123456789}{2^{26}\cdot 5^4}</math> as a decimal?<br />
<br />
<math> \textbf{(A)}\ 4\qquad\textbf{(B)}\ 22\qquad\textbf{(C)}\ 26\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 104</math><br />
<br />
[[2015 AMC 12A Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
Tetrahedron <math>ABCD</math> has <math>AB=5,AC=3,BC=4,BD=4,AD=3,</math> and <math>CD=\frac{12}{5}\sqrt{2}</math>. What is the volume of the tetrahedron?<br />
<br />
<math> \textbf{(A)}\ 3\sqrt{2}\qquad\textbf{(B)}\ 2\sqrt{5}\qquad\textbf{(C)}\ \frac{24}{5}\qquad\textbf{(D)}\ 3\sqrt{3}\qquad\textbf{(E)}\ \frac{24}{5}\sqrt{2}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?<br />
<br />
<math> \textbf{(A)}\ \frac{47}{256} \qquad\textbf{(B)}\ \frac{3}{16} \qquad\textbf{(C)}\ \frac{49}{256} \qquad\textbf{(D)}\ \frac{25}{128} \qquad\textbf{(E)}\ \frac{51}{256}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
The zeros of the function <math>f(x) = x^2-ax+2a</math> are integers. What is the sum of the possible values of <math>a</math>?<br />
<br />
<math> \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 16 \qquad\textbf{(D)}\ 17 \qquad\textbf{(E)}\ 18</math><br />
<br />
[[2015 AMC 12A Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
For some positive integers <math>p</math>, there is a quadrilateral <math>ABCD</math> with positive integer side lengths, perimeter <math>p</math>, right angles at <math>B</math> and <math>C</math>, <math>AB=2</math>, and <math>CD=AD</math>. How many different values of <math>p<2015</math> are possible?<br />
<br />
<math> \textbf{(A)}\ 30 \qquad\textbf{(B)}\ 31 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}\ 62 \qquad\textbf{(E)}\ 63</math><br />
<br />
[[2015 AMC 12A Problems/Problem 19|Solution]]<br />
<br />
==Problem 20==<br />
<br />
Isosceles triangles <math>T</math> and <math>T'</math> are not congruent but have the same area and the same perimeter. The sides of <math>T</math> have lengths of <math>5,5,</math> and <math>8</math>, while those of <math>T'</math> have lengths of <math>a,a,</math> and <math>b</math>. Which of the following numbers is closest to <math>b</math>?<br />
<br />
<math> \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 8</math><br />
<br />
[[2015 AMC 12A Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
A circle of radius <math>r</math> passes through both foci of, and exactly four points on, the ellipse with equation <math>x^2+16y^2=16</math>. The set of all possible values of <math>r</math> is an interval <math>[a,b)</math>. What is <math>a+b</math>?<br />
<br />
<math> \textbf{(A)}\ 5\sqrt{2}+4 \qquad\textbf{(B)}\ \sqrt{17}+7 \qquad\textbf{(C)}\ 6\sqrt{2}+3 \qquad\textbf{(D)}\ \sqrt{15}+8 \qquad\textbf{(E)}\ 12</math><br />
<br />
[[2015 AMC 12A Problems/Problem 21|Solution]]<br />
<br />
==Problem 22==<br />
<br />
For each positive integer <math>n</math>, let <math>S(n)</math> be the number of sequences of length <math>n</math> consisting solely of the letters <math>A</math> and <math>B</math>, with no more than three <math>A</math>s in a row and no more than three <math>B</math>s in a row. What is the remainder when <math>S(2015)</math> is divided by 12?<br />
<br />
<math> \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 8 \qquad\textbf{(E)}\ 10</math><br />
<br />
[[2015 AMC 12A Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
Let <math>S</math> be a square of side length 1. Two points are chosen independently at random on the sides of <math>S</math>. The probability that the straight-line distance between the points is at least <math>\frac12</math> is <math>\frac{a-b\pi}{c}</math>, where <math>a,b,</math> and <math>c</math> are positive integers and <math>\text{gcd}(a,b,c) = 1</math>. What is <math>a+b+c</math>?<br />
<br />
<math> \textbf{(A)}\ 59 \qquad\textbf{(B)}\ 60 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}\ 62 \qquad\textbf{(E)}\ 63</math><br />
<br />
[[2015 AMC 12A Problems/Problem 23|Solution]]<br />
<br />
==Problem 24==<br />
<br />
Rational numbers <math>a</math> and <math>b</math> are chosen at random among all rational numbers in the interval <math>[0,2)</math> that can be written as fractions <math>\frac{n}{d}</math> where <math>n</math> and <math>d</math> are integers with <math>1 \le d \le 5</math>. What is the probability that<br />
<cmath>(\text{cos}(a\pi)+i\text{sin}(b\pi))^4</cmath><br />
is a real number?<br />
<br />
<math> \textbf{(A)}\ \frac{3}{5} \qquad\textbf{(B)}\ \frac{4}{25} \qquad\textbf{(C)}\ \frac{41}{200} \qquad\textbf{(D)}\ \frac{6}{25} \qquad\textbf{(E)}\ \frac{13}{50}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 24|Solution]]<br />
<br />
==Problem 25==<br />
<br />
A collection of circles in the upper half-plane, all tangent to the <math>x</math>-axis, is constructed in layers as follows. Layer <math>L_0</math> consists of two circles of radii <math>70^2</math> and <math>73^2</math> that are externally tangent. For <math>k\ge1</math>, the circles in <math>\bigcup_{j=0}^{k-1}L_j</math> are ordered according to their points of tangency with the <math>x</math>-axis. For every pair of consecutive circles in this order, a new circle is constructed externally tangent to each of the two circles in the pair. Layer <math>L_k</math> consists of the <math>2^{k-1}</math> circles constructed in this way. Let <math>S=\bigcup_{j=0}^{6}L_j</math>, and for every circle <math>C</math> denote by <math>r(C)</math> its radius. What is<br />
<cmath>\sum_{C\in S} \frac{1}{\sqrt{r(C)}}?</cmath><br />
<br />
<asy><br />
import olympiad;<br />
size(350);<br />
defaultpen(linewidth(0.7));<br />
// define a bunch of arrays and starting points<br />
pair[] coord = new pair[65];<br />
int[] trav = {32,16,8,4,2,1};<br />
coord[0] = (0,73^2); coord[64] = (2*73*70,70^2);<br />
// draw the big circles and the bottom line<br />
path arc1 = arc(coord[0],coord[0].y,260,360);<br />
path arc2 = arc(coord[64],coord[64].y,175,280);<br />
fill((coord[0].x-910,coord[0].y)--arc1--cycle,gray(0.75));<br />
fill((coord[64].x+870,coord[64].y+425)--arc2--cycle,gray(0.75));<br />
draw(arc1^^arc2);<br />
draw((-930,0)--(70^2+73^2+850,0));<br />
// We now apply the findCenter function 63 times to get<br />
// the location of the centers of all 63 constructed circles.<br />
// The complicated array setup ensures that all the circles<br />
// will be taken in the right order<br />
for(int i = 0;i<=5;i=i+1)<br />
{<br />
int skip = trav[i];<br />
for(int k=skip;k<=64 - skip; k = k + 2*skip)<br />
{<br />
pair cent1 = coord[k-skip], cent2 = coord[k+skip];<br />
real r1 = cent1.y, r2 = cent2.y, rn=r1*r2/((sqrt(r1)+sqrt(r2))^2);<br />
real shiftx = cent1.x + sqrt(4*r1*rn);<br />
coord[k] = (shiftx,rn);<br />
}<br />
// Draw the remaining 63 circles<br />
}<br />
for(int i=1;i<=63;i=i+1)<br />
{<br />
filldraw(circle(coord[i],coord[i].y),gray(0.75));<br />
}<br />
</asy><br />
<br />
<br />
$\textbf{(A)}\ \frac{286}{35} \qquad\textbf{(B)}\ \frac{583}{70} \qquad\textbf{(C)}\ \frac{715}{73}\qquad\textbf{(D)}\ \frac{143}{14} \qquad\textbf{(E)}\ \frac{1573}{146}$<br />
<br />
[[2015 AMC 12A Problems/Problem 25|Solution]]<br />
<br />
== See also ==<br />
* [[AMC Problems and Solutions]]<br />
{{MAA Notice}}</div>Devenwarehttps://artofproblemsolving.com/wiki/index.php?title=2015_AMC_12A_Problems&diff=680322015 AMC 12A Problems2015-03-01T23:35:46Z<p>Devenware: /* Problem 25 */</p>
<hr />
<div>==Problem 1==<br />
<br />
What is the value of <math>(2^0-1+5^2-0)^{-1}\times5?</math><br />
<br />
<math> \textbf{(A)}\ -125\qquad\textbf{(B)}\ -120\qquad\textbf{(C)}\ \frac{1}{5}\qquad\textbf{(D)}\ \frac{5}{24}\qquad\textbf{(E)}\ 25 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
<br />
Two of the three sides of a triangle are 20 and 15. Which of the following numbers is not a possible perimeter of the triangle?<br />
<br />
<math> \textbf{(A)}\ 52\qquad\textbf{(B)}\ 57\qquad\textbf{(C)}\ 62\qquad\textbf{(D)}\ 67\qquad\textbf{(E)}\ 72 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
<br />
Mr. Patrick teaches math to 15 students. He was grading tests and found that when he graded everyone's test except Payton's, the average grade for the class was 80. after he graded Payton's test, the class average became 81. What was Payton's score on the test?<br />
<br />
<math> \textbf{(A)}\ 81\qquad\textbf{(B)}\ 85\qquad\textbf{(C)}\ 91\qquad\textbf{(D)}\ 94\qquad\textbf{(E)}\ 95 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
The sum of two positive numbers is 5 times their difference. What is the ratio of the larger number to the smaller?<br />
<br />
<math> \textbf{(A)}\ \frac54 \qquad\textbf{(B)}\ \frac32 \qquad\textbf{(C)}\ \frac95 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ \frac52 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
Amelia needs to estimate the quantity <math>\frac{a}{b} - c</math>, where <math>a, b,</math> and <math>c</math> are large positive integers. She rounds each of the integers so that the calculation will be easier to do mentally. In which of these situations will her answer necessarily be greater than the exact value of <math>\frac{a}{b} - c</math>?<br />
<br />
<math> \textbf{(A)}\ \text{She rounds all three numbers up.}\\<br />
\qquad\textbf{(B)}\ \text{She rounds } a \text{ and } b \text{ up, and she rounds } c \text{ down.}\\<br />
\qquad\textbf{(C)}\ \text{She rounds } a \text{ and } c \text{ up, and she rounds } b \text{ down.} \\<br />
\qquad\textbf{(D)}\ \text{She rounds } a \text{ up, and she rounds } b \text{ and } c \text{ down.}\\<br />
\qquad\textbf{(E)}\ \text{She rounds } c \text{ up, and she rounds } a \text{ and } b \text{ down.} </math><br />
<br />
[[2015 AMC 12A Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be <math>2 : 1</math>?<br />
<br />
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 8</math><br />
<br />
[[2015 AMC 12A Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
Two right circular cylinders have the same volume. The radius of the second cylinder is <math>10\%</math> more than the radius of the first. What is the relationship between the heights of the two cylinders?<br />
<br />
<math>\textbf{(A)}\ \text{The second height is } 10\% \text{ less than the first.} \\ \textbf{(B)}\ \text{The first height is } 10\% \text{ more than the second.}\\ \textbf{(C)}\ \text{The second height is } 21\% \text{ less than the first.} \\ \textbf{(D)}\ \text{The first height is } 21\% \text{ more than the second.}\\ \textbf{(E)}\ \text{The second height is } 80\% \text{ of the first.}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
The ratio of the length to the width of a rectangle is <math>4</math> : <math>3</math>. If the rectangle has diagonal of length <math>d</math>, then the area may be expressed as <math>kd^2</math> for some constant <math>k</math>. What is <math>k</math>?<br />
<br />
<math> \textbf{(A)}\ \frac27 \qquad\textbf{(B)}\ \frac37 \qquad\textbf{(C)}\ \frac{12}{25} \qquad\textbf{(D)}\ \frac{16}{26} \qquad\textbf{(E)}\ \frac34</math><br />
<br />
[[2015 AMC 12A Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
A box contains 2 red marbles, 2 green marbles, and 2 yellow marbles. Carol takes 2 marbles from the box at random; then Claudia takes 2 of the remaining marbles at random; and then Cheryl takes the last 2 marbles. What is the probability that Cheryl gets 2 marbles of the same color?<br />
<br />
<math> \textbf{(A)}\ \frac{1}{10} \qquad\textbf{(B)}\ \frac16 \qquad\textbf{(C)}\ \frac15 \qquad\textbf{(D)}\ \frac13 \qquad\textbf{(E)}\ \frac12</math><br />
<br />
[[2015 AMC 12A Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
Integers <math>x</math> and <math>y</math> with <math>x>y>0</math> satisfy <math>x+y+xy=80</math>. What is <math>x</math>?<br />
<br />
<math> \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 15 \qquad\textbf{(D)}\ 18 \qquad\textbf{(E)}\ 26</math><br />
<br />
[[2015 AMC 12A Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
On a sheet of paper, Isabella draws a circle of radius <math>2</math>, a circle of radius <math>3</math>, and all possible lines simultaneously tangent to both circles. Isabella notices that she has drawn exactly <math>k \ge 0</math> lines. How many different values of <math>k</math> are possible?<br />
<br />
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6</math><br />
<br />
[[2015 AMC 12A Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
The parabolas <math>y=ax^2 - 2</math> and <math>y=4 - bx^2</math> intersect the coordinate axes in exactly four points, and these four points are the vertices of a kite of area <math>12</math>. What is <math>a+b</math>?<br />
<br />
<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 1.5\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 2.5\qquad\textbf{(E)}\ 3</math><br />
<br />
[[2015 AMC 12A Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
<br />
A league with 12 teams holds a round-robin tournament, with each team playing every other team exactly once. Games either end with one team victorious or else end in a draw. A team scores 2 points for every game it wins and 1 point for every game it draws. Which of the following is NOT a true statement about the list of 12 scores?<br />
<br />
<math> \textbf{(A)}\ \text{There must be an even number of odd scores.}\\<br />
\qquad\textbf{(B)}\ \text{There must be an even number of even scores.}\\<br />
\qquad\textbf{(C)}\ \text{There cannot be two scores of }0\text{.}\\<br />
\qquad\textbf{(D)}\ \text{The sum of the scores must be at least }100\text{.}\\<br />
\qquad\textbf{(E)}\ \text{The highest score must be at least }12\text{.}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
What is the value of <math>a</math> for which <math>\frac{1}{\text{log}_2a} + \frac{1}{\text{log}_3a} + \frac{1}{\text{log}_4a} = 1</math>?<br />
<br />
[[2015 AMC 12A Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
What is the minimum number of digits to the right of the decimal point needed to express the fraction <math>\frac{123456789}{2^{26}\cdot 5^4}</math> as a decimal?<br />
<br />
<math> \textbf{(A)}\ 4\qquad\textbf{(B)}\ 22\qquad\textbf{(C)}\ 26\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 104</math><br />
<br />
[[2015 AMC 12A Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
Tetrahedron <math>ABCD</math> has <math>AB=5,AC=3,BC=4,BD=4,AD=3,</math> and <math>CD=\frac{12}{5}\sqrt{2}</math>. What is the volume of the tetrahedron?<br />
<br />
<math> \textbf{(A)}\ 3\sqrt{2}\qquad\textbf{(B)}\ 2\sqrt{5}\qquad\textbf{(C)}\ \frac{24}{5}\qquad\textbf{(D)}\ 3\sqrt{3}\qquad\textbf{(E)}\ \frac{24}{5}\sqrt{2}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?<br />
<br />
<math> \textbf{(A)}\ \frac{47}{256} \qquad\textbf{(B)}\ \frac{3}{16} \qquad\textbf{(C)}\ \frac{49}{256} \qquad\textbf{(D)}\ \frac{25}{128} \qquad\textbf{(E)}\ \frac{51}{256}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
The zeros of the function <math>f(x) = x^2-ax+2a</math> are integers. What is the sum of the possible values of <math>a</math>?<br />
<br />
<math> \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 16 \qquad\textbf{(D)}\ 17 \qquad\textbf{(E)}\ 18</math><br />
<br />
[[2015 AMC 12A Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
For some positive integers <math>p</math>, there is a quadrilateral <math>ABCD</math> with positive integer side lengths, perimeter <math>p</math>, right angles at <math>B</math> and <math>C</math>, <math>AB=2</math>, and <math>CD=AD</math>. How many different values of <math>p<2015</math> are possible?<br />
<br />
<math> \textbf{(A)}\ 30 \qquad\textbf{(B)}\ 31 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}\ 62 \qquad\textbf{(E)}\ 63</math><br />
<br />
[[2015 AMC 12A Problems/Problem 19|Solution]]<br />
<br />
==Problem 20==<br />
<br />
Isosceles triangles <math>T</math> and <math>T'</math> are not congruent but have the same area and the same perimeter. The sides of <math>T</math> have lengths of <math>5,5,</math> and <math>8</math>, while those of <math>T'</math> have lengths of <math>a,a,</math> and <math>b</math>. Which of the following numbers is closest to <math>b</math>?<br />
<br />
<math> \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 8</math><br />
<br />
[[2015 AMC 12A Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
A circle of radius <math>r</math> passes through both foci of, and exactly four points on, the ellipse with equation <math>x^2+16y^2=16</math>. The set of all possible values of <math>r</math> is an interval <math>[a,b)</math>. What is <math>a+b</math>?<br />
<br />
<math> \textbf{(A)}\ 5\sqrt{2}+4 \qquad\textbf{(B)}\ \sqrt{17}+7 \qquad\textbf{(C)}\ 6\sqrt{2}+3 \qquad\textbf{(D)}\ \sqrt{15}+8 \qquad\textbf{(E)}\ 12</math><br />
<br />
[[2015 AMC 12A Problems/Problem 21|Solution]]<br />
<br />
==Problem 22==<br />
<br />
For each positive integer <math>n</math>, let <math>S(n)</math> be the number of sequences of length <math>n</math> consisting solely of the letters <math>A</math> and <math>B</math>, with no more than three <math>A</math>s in a row and no more than three <math>B</math>s in a row. What is the remainder when <math>S(2015)</math> is divided by 12?<br />
<br />
<math> \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 8 \qquad\textbf{(E)}\ 10</math><br />
<br />
[[2015 AMC 12A Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
Let <math>S</math> be a square of side length 1. Two points are chosen independently at random on the sides of <math>S</math>. The probability that the straight-line distance between the points is at least <math>\frac12</math> is <math>\frac{a-b\pi}{c}</math>, where <math>a,b,</math> and <math>c</math> are positive integers and <math>\text{gcd}(a,b,c) = 1</math>. What is <math>a+b+c</math>?<br />
<br />
<math> \textbf{(A)}\ 59 \qquad\textbf{(B)}\ 60 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}\ 62 \qquad\textbf{(E)}\ 63</math><br />
<br />
[[2015 AMC 12A Problems/Problem 23|Solution]]<br />
<br />
==Problem 24==<br />
<br />
Rational numbers <math>a</math> and <math>b</math> are chosen at random among all rational numbers in the interval <math>[0,2)</math> that can be written as fractions <math>\frac{n}{d}</math> where <math>n</math> and <math>d</math> are integers with <math>1 \le d \le 5</math>. What is the probability that<br />
<cmath>(\text{cos}(a\pi)+i\text{sin}(b\pi))^4</cmath><br />
is a real number?<br />
<br />
<math> \textbf{(A)}\ \frac{3}{5} \qquad\textbf{(B)}\ \frac{4}{25} \qquad\textbf{(C)}\ \frac{41}{200} \qquad\textbf{(D)}\ \frac{6}{25} \qquad\textbf{(E)}\ \frac{13}{50}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 24|Solution]]<br />
<br />
==Problem 25==<br />
<br />
A collection of circles in the upper half-plane, all tangent to the <math>x</math>-axis, is constructed in layers as follows. Layer <math>L_0</math> consists of two circles of radii <math>70^2</math> and <math>73^2</math> that are externally tangent. For <math>k\ge1</math>, the circles in <math>\bigcup_{j=0}^{k-1}L_j</math> are ordered according to their points of tangency with the <math>x</math>-axis. For every pair of consecutive circles in this order, a new circle is constructed externally tangent to each of the two circles in the pair. Layer <math>L_k</math> consists of the <math>2^{k-1}</math> circles constructed in this way. Let <math>S=\bigcup_{j=0}^{6}L_j</math>, and for every circle <math>C</math> denote by <math>r(C)</math> its radius. What is<br />
<cmath>\sum_{C\in S} \frac{1}{\sqrt{r(C)}}?</cmath><br />
<br />
<asy><br />
import olympiad;<br />
size(350);<br />
defaultpen(linewidth(0.7));<br />
// define a bunch of arrays and starting points<br />
pair[] coord = new pair[65];<br />
int[] trav = {32,16,8,4,2,1};<br />
coord[0] = (0,73^2); coord[64] = (2*73*70,70^2);<br />
// draw the big circles and the bottom line<br />
path arc1 = arc(coord[0],coord[0].y,260,360);<br />
path arc2 = arc(coord[64],coord[64].y,175,280);<br />
fill((coord[0].x-910,coord[0].y)--arc1--cycle,gray(0.75));<br />
fill((coord[64].x+870,coord[64].y+425)--arc2--cycle,gray(0.75));<br />
draw(arc1^^arc2);<br />
draw((-930,0)--(70^2+73^2+850,0));<br />
// We now apply the findCenter function 63 times to get<br />
// the location of the centers of all 63 constructed circles.<br />
// The complicated array setup ensures that all the circles<br />
// will be taken in the right order<br />
for(int i = 0;i<=5;i=i+1)<br />
{<br />
int skip = trav[i];<br />
for(int k=skip;k<=64 - skip; k = k + 2*skip)<br />
{<br />
pair cent1 = coord[k-skip], cent2 = coord[k+skip];<br />
real r1 = cent1.y, r2 = cent2.y, rn=r1*r2/((sqrt(r1)+sqrt(r2))^2);<br />
real shiftx = cent1.x + sqrt(4*r1*rn);<br />
coord[k] = (shiftx,rn);<br />
}<br />
// Draw the remaining 63 circles<br />
}<br />
for(int i=1;i<=63;i=i+1)<br />
{<br />
filldraw(circle(coord[i],coord[i].y),gray(0.75));<br />
}<br />
</asy><br />
<br />
<br />
$\textbf{(A)}\ \frac{286}{35} \qquad\textbf{(B)}\ \frac{583}{70} \qquad\textbf{(C)}\ \frac{715}{73}\qquad\textbf{(D)}\ \frac{143}{14} \qquad\textbf{(E)}\ \frac{1573}{146}$<br />
<br />
[[2015 AMC 12A Problems/Problem 25|Solution]]<br />
<br />
== See also ==<br />
* [[AMC Problems and Solutions]]<br />
{{MAA Notice}}</div>Devenwarehttps://artofproblemsolving.com/wiki/index.php?title=2015_AMC_12A_Problems&diff=680312015 AMC 12A Problems2015-03-01T23:33:30Z<p>Devenware: /* Problem 25 */</p>
<hr />
<div>==Problem 1==<br />
<br />
What is the value of <math>(2^0-1+5^2-0)^{-1}\times5?</math><br />
<br />
<math> \textbf{(A)}\ -125\qquad\textbf{(B)}\ -120\qquad\textbf{(C)}\ \frac{1}{5}\qquad\textbf{(D)}\ \frac{5}{24}\qquad\textbf{(E)}\ 25 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
<br />
Two of the three sides of a triangle are 20 and 15. Which of the following numbers is not a possible perimeter of the triangle?<br />
<br />
<math> \textbf{(A)}\ 52\qquad\textbf{(B)}\ 57\qquad\textbf{(C)}\ 62\qquad\textbf{(D)}\ 67\qquad\textbf{(E)}\ 72 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
<br />
Mr. Patrick teaches math to 15 students. He was grading tests and found that when he graded everyone's test except Payton's, the average grade for the class was 80. after he graded Payton's test, the class average became 81. What was Payton's score on the test?<br />
<br />
<math> \textbf{(A)}\ 81\qquad\textbf{(B)}\ 85\qquad\textbf{(C)}\ 91\qquad\textbf{(D)}\ 94\qquad\textbf{(E)}\ 95 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
The sum of two positive numbers is 5 times their difference. What is the ratio of the larger number to the smaller?<br />
<br />
<math> \textbf{(A)}\ \frac54 \qquad\textbf{(B)}\ \frac32 \qquad\textbf{(C)}\ \frac95 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ \frac52 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
Amelia needs to estimate the quantity <math>\frac{a}{b} - c</math>, where <math>a, b,</math> and <math>c</math> are large positive integers. She rounds each of the integers so that the calculation will be easier to do mentally. In which of these situations will her answer necessarily be greater than the exact value of <math>\frac{a}{b} - c</math>?<br />
<br />
<math> \textbf{(A)}\ \text{She rounds all three numbers up.}\\<br />
\qquad\textbf{(B)}\ \text{She rounds } a \text{ and } b \text{ up, and she rounds } c \text{ down.}\\<br />
\qquad\textbf{(C)}\ \text{She rounds } a \text{ and } c \text{ up, and she rounds } b \text{ down.} \\<br />
\qquad\textbf{(D)}\ \text{She rounds } a \text{ up, and she rounds } b \text{ and } c \text{ down.}\\<br />
\qquad\textbf{(E)}\ \text{She rounds } c \text{ up, and she rounds } a \text{ and } b \text{ down.} </math><br />
<br />
[[2015 AMC 12A Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be <math>2 : 1</math>?<br />
<br />
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 8</math><br />
<br />
[[2015 AMC 12A Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
Two right circular cylinders have the same volume. The radius of the second cylinder is <math>10\%</math> more than the radius of the first. What is the relationship between the heights of the two cylinders?<br />
<br />
<math>\textbf{(A)}\ \text{The second height is } 10\% \text{ less than the first.} \\ \textbf{(B)}\ \text{The first height is } 10\% \text{ more than the second.}\\ \textbf{(C)}\ \text{The second height is } 21\% \text{ less than the first.} \\ \textbf{(D)}\ \text{The first height is } 21\% \text{ more than the second.}\\ \textbf{(E)}\ \text{The second height is } 80\% \text{ of the first.}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
The ratio of the length to the width of a rectangle is <math>4</math> : <math>3</math>. If the rectangle has diagonal of length <math>d</math>, then the area may be expressed as <math>kd^2</math> for some constant <math>k</math>. What is <math>k</math>?<br />
<br />
<math> \textbf{(A)}\ \frac27 \qquad\textbf{(B)}\ \frac37 \qquad\textbf{(C)}\ \frac{12}{25} \qquad\textbf{(D)}\ \frac{16}{26} \qquad\textbf{(E)}\ \frac34</math><br />
<br />
[[2015 AMC 12A Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
A box contains 2 red marbles, 2 green marbles, and 2 yellow marbles. Carol takes 2 marbles from the box at random; then Claudia takes 2 of the remaining marbles at random; and then Cheryl takes the last 2 marbles. What is the probability that Cheryl gets 2 marbles of the same color?<br />
<br />
<math> \textbf{(A)}\ \frac{1}{10} \qquad\textbf{(B)}\ \frac16 \qquad\textbf{(C)}\ \frac15 \qquad\textbf{(D)}\ \frac13 \qquad\textbf{(E)}\ \frac12</math><br />
<br />
[[2015 AMC 12A Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
Integers <math>x</math> and <math>y</math> with <math>x>y>0</math> satisfy <math>x+y+xy=80</math>. What is <math>x</math>?<br />
<br />
<math> \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 15 \qquad\textbf{(D)}\ 18 \qquad\textbf{(E)}\ 26</math><br />
<br />
[[2015 AMC 12A Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
On a sheet of paper, Isabella draws a circle of radius <math>2</math>, a circle of radius <math>3</math>, and all possible lines simultaneously tangent to both circles. Isabella notices that she has drawn exactly <math>k \ge 0</math> lines. How many different values of <math>k</math> are possible?<br />
<br />
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6</math><br />
<br />
[[2015 AMC 12A Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
The parabolas <math>y=ax^2 - 2</math> and <math>y=4 - bx^2</math> intersect the coordinate axes in exactly four points, and these four points are the vertices of a kite of area <math>12</math>. What is <math>a+b</math>?<br />
<br />
<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 1.5\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 2.5\qquad\textbf{(E)}\ 3</math><br />
<br />
[[2015 AMC 12A Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
<br />
A league with 12 teams holds a round-robin tournament, with each team playing every other team exactly once. Games either end with one team victorious or else end in a draw. A team scores 2 points for every game it wins and 1 point for every game it draws. Which of the following is NOT a true statement about the list of 12 scores?<br />
<br />
<math> \textbf{(A)}\ \text{There must be an even number of odd scores.}\\<br />
\qquad\textbf{(B)}\ \text{There must be an even number of even scores.}\\<br />
\qquad\textbf{(C)}\ \text{There cannot be two scores of }0\text{.}\\<br />
\qquad\textbf{(D)}\ \text{The sum of the scores must be at least }100\text{.}\\<br />
\qquad\textbf{(E)}\ \text{The highest score must be at least }12\text{.}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
What is the value of <math>a</math> for which <math>\frac{1}{\text{log}_2a} + \frac{1}{\text{log}_3a} + \frac{1}{\text{log}_4a} = 1</math>?<br />
<br />
[[2015 AMC 12A Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
What is the minimum number of digits to the right of the decimal point needed to express the fraction <math>\frac{123456789}{2^{26}\cdot 5^4}</math> as a decimal?<br />
<br />
<math> \textbf{(A)}\ 4\qquad\textbf{(B)}\ 22\qquad\textbf{(C)}\ 26\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 104</math><br />
<br />
[[2015 AMC 12A Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
Tetrahedron <math>ABCD</math> has <math>AB=5,AC=3,BC=4,BD=4,AD=3,</math> and <math>CD=\frac{12}{5}\sqrt{2}</math>. What is the volume of the tetrahedron?<br />
<br />
<math> \textbf{(A)}\ 3\sqrt{2}\qquad\textbf{(B)}\ 2\sqrt{5}\qquad\textbf{(C)}\ \frac{24}{5}\qquad\textbf{(D)}\ 3\sqrt{3}\qquad\textbf{(E)}\ \frac{24}{5}\sqrt{2}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?<br />
<br />
<math> \textbf{(A)}\ \frac{47}{256} \qquad\textbf{(B)}\ \frac{3}{16} \qquad\textbf{(C)}\ \frac{49}{256} \qquad\textbf{(D)}\ \frac{25}{128} \qquad\textbf{(E)}\ \frac{51}{256}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
The zeros of the function <math>f(x) = x^2-ax+2a</math> are integers. What is the sum of the possible values of <math>a</math>?<br />
<br />
<math> \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 16 \qquad\textbf{(D)}\ 17 \qquad\textbf{(E)}\ 18</math><br />
<br />
[[2015 AMC 12A Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
For some positive integers <math>p</math>, there is a quadrilateral <math>ABCD</math> with positive integer side lengths, perimeter <math>p</math>, right angles at <math>B</math> and <math>C</math>, <math>AB=2</math>, and <math>CD=AD</math>. How many different values of <math>p<2015</math> are possible?<br />
<br />
<math> \textbf{(A)}\ 30 \qquad\textbf{(B)}\ 31 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}\ 62 \qquad\textbf{(E)}\ 63</math><br />
<br />
[[2015 AMC 12A Problems/Problem 19|Solution]]<br />
<br />
==Problem 20==<br />
<br />
Isosceles triangles <math>T</math> and <math>T'</math> are not congruent but have the same area and the same perimeter. The sides of <math>T</math> have lengths of <math>5,5,</math> and <math>8</math>, while those of <math>T'</math> have lengths of <math>a,a,</math> and <math>b</math>. Which of the following numbers is closest to <math>b</math>?<br />
<br />
<math> \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 8</math><br />
<br />
[[2015 AMC 12A Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
A circle of radius <math>r</math> passes through both foci of, and exactly four points on, the ellipse with equation <math>x^2+16y^2=16</math>. The set of all possible values of <math>r</math> is an interval <math>[a,b)</math>. What is <math>a+b</math>?<br />
<br />
<math> \textbf{(A)}\ 5\sqrt{2}+4 \qquad\textbf{(B)}\ \sqrt{17}+7 \qquad\textbf{(C)}\ 6\sqrt{2}+3 \qquad\textbf{(D)}\ \sqrt{15}+8 \qquad\textbf{(E)}\ 12</math><br />
<br />
[[2015 AMC 12A Problems/Problem 21|Solution]]<br />
<br />
==Problem 22==<br />
<br />
For each positive integer <math>n</math>, let <math>S(n)</math> be the number of sequences of length <math>n</math> consisting solely of the letters <math>A</math> and <math>B</math>, with no more than three <math>A</math>s in a row and no more than three <math>B</math>s in a row. What is the remainder when <math>S(2015)</math> is divided by 12?<br />
<br />
<math> \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 8 \qquad\textbf{(E)}\ 10</math><br />
<br />
[[2015 AMC 12A Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
Let <math>S</math> be a square of side length 1. Two points are chosen independently at random on the sides of <math>S</math>. The probability that the straight-line distance between the points is at least <math>\frac12</math> is <math>\frac{a-b\pi}{c}</math>, where <math>a,b,</math> and <math>c</math> are positive integers and <math>\text{gcd}(a,b,c) = 1</math>. What is <math>a+b+c</math>?<br />
<br />
<math> \textbf{(A)}\ 59 \qquad\textbf{(B)}\ 60 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}\ 62 \qquad\textbf{(E)}\ 63</math><br />
<br />
[[2015 AMC 12A Problems/Problem 23|Solution]]<br />
<br />
==Problem 24==<br />
<br />
Rational numbers <math>a</math> and <math>b</math> are chosen at random among all rational numbers in the interval <math>[0,2)</math> that can be written as fractions <math>\frac{n}{d}</math> where <math>n</math> and <math>d</math> are integers with <math>1 \le d \le 5</math>. What is the probability that<br />
<cmath>(\text{cos}(a\pi)+i\text{sin}(b\pi))^4</cmath><br />
is a real number?<br />
<br />
<math> \textbf{(A)}\ \frac{3}{5} \qquad\textbf{(B)}\ \frac{4}{25} \qquad\textbf{(C)}\ \frac{41}{200} \qquad\textbf{(D)}\ \frac{6}{25} \qquad\textbf{(E)}\ \frac{13}{50}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 24|Solution]]<br />
<br />
==Problem 25==<br />
<br />
A collection of circles in the upper half-plane, all tangent to the <math>x</math>-axis, is constructed in layers as follows. Layer <math>L_0</math> consists of two circles of radii <math>70^2</math> and <math>73^2</math> that are externally tangent. For <math>k\ge1</math>, the circles in <math>\bigcup_{j=0}^{k-1}L_j</math> are ordered according to their points of tangency with the <math>x</math>-axis. For every pair of consecutive circles in this order, a new circle is constructed externally tangent to each of the two circles in the pair. Layer <math>L_k</math> consists of the <math>2^{k-1}</math> circles constructed in this way. Let <math>S=\bigcup_{j=0}^{6}L_j</math>, and for every circle <math>C</math> denote by <math>r(C)</math> its radius. What is<br />
<cmath>\sum_{C\in S} \frac{1}{\sqrt{r(C)}}?</cmath><br />
<br />
<asy><br />
import olympiad;<br />
size(350);<br />
defaultpen(linewidth(0.7));<br />
// define a bunch of arrays and starting points<br />
pair[] coord = new pair[65];<br />
int[] trav = {32,16,8,4,2,1};<br />
coord[0] = (0,73^2); coord[64] = (2*73*70,70^2);<br />
// draw the big circles and the bottom line<br />
path arc1 = arc(coord[0],coord[0].y,260,360);<br />
path arc2 = arc(coord[64],coord[64].y,175,280);<br />
fill((coord[0].x-910,coord[0].y)--arc1--cycle,gray(0.75));<br />
fill((coord[64].x+870,coord[64].y+425)--arc2--cycle,gray(0.75));<br />
draw(arc1^^arc2);<br />
draw((-930,0)--(70^2+73^2+850,0));<br />
// We now apply the findCenter function 63 times to get<br />
// the location of the centers of all 63 constructed circles.<br />
// The complicated array setup ensures that all the circles<br />
// will be taken in the right order<br />
for(int i = 0;i<=5;i=i+1)<br />
{<br />
int skip = trav[i];<br />
for(int k=skip;k<=64 - skip; k = k + 2*skip)<br />
{<br />
pair cent1 = coord[k-skip], cent2 = coord[k+skip];<br />
real r1 = cent1.y, r2 = cent2.y, rn=r1*r2/((sqrt(r1)+sqrt(r2))^2);<br />
real shiftx = cent1.x + sqrt(4*r1*rn);<br />
coord[k] = (shiftx,rn);<br />
}<br />
// Draw the remaining 63 circles<br />
}<br />
for(int i=1;i<=63;i=i+1)<br />
{<br />
filldraw(circle(coord[i],coord[i].y),gray(0.75));<br />
}<br />
</asy><br />
<br />
<br />
<br />
$\textbf{(A)}\ \frac{286}{35} \qquad\textbf{(B)}\ \frac{583}{70} \qquad\textbf{(C)}\ \frac{715}{73}\qquad\textbf{(D)}\ \frac{143}{14} \qquad\textbf{(E)}\ \frac{1573}{146}$<br />
<br />
[[2015 AMC 12A Problems/Problem 25|Solution]]<br />
<br />
== See also ==<br />
* [[AMC Problems and Solutions]]<br />
{{MAA Notice}}</div>Devenwarehttps://artofproblemsolving.com/wiki/index.php?title=2015_AMC_12A_Problems&diff=680302015 AMC 12A Problems2015-03-01T23:32:31Z<p>Devenware: /* Problem 25 */</p>
<hr />
<div>==Problem 1==<br />
<br />
What is the value of <math>(2^0-1+5^2-0)^{-1}\times5?</math><br />
<br />
<math> \textbf{(A)}\ -125\qquad\textbf{(B)}\ -120\qquad\textbf{(C)}\ \frac{1}{5}\qquad\textbf{(D)}\ \frac{5}{24}\qquad\textbf{(E)}\ 25 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
<br />
Two of the three sides of a triangle are 20 and 15. Which of the following numbers is not a possible perimeter of the triangle?<br />
<br />
<math> \textbf{(A)}\ 52\qquad\textbf{(B)}\ 57\qquad\textbf{(C)}\ 62\qquad\textbf{(D)}\ 67\qquad\textbf{(E)}\ 72 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
<br />
Mr. Patrick teaches math to 15 students. He was grading tests and found that when he graded everyone's test except Payton's, the average grade for the class was 80. after he graded Payton's test, the class average became 81. What was Payton's score on the test?<br />
<br />
<math> \textbf{(A)}\ 81\qquad\textbf{(B)}\ 85\qquad\textbf{(C)}\ 91\qquad\textbf{(D)}\ 94\qquad\textbf{(E)}\ 95 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
The sum of two positive numbers is 5 times their difference. What is the ratio of the larger number to the smaller?<br />
<br />
<math> \textbf{(A)}\ \frac54 \qquad\textbf{(B)}\ \frac32 \qquad\textbf{(C)}\ \frac95 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ \frac52 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
Amelia needs to estimate the quantity <math>\frac{a}{b} - c</math>, where <math>a, b,</math> and <math>c</math> are large positive integers. She rounds each of the integers so that the calculation will be easier to do mentally. In which of these situations will her answer necessarily be greater than the exact value of <math>\frac{a}{b} - c</math>?<br />
<br />
<math> \textbf{(A)}\ \text{She rounds all three numbers up.}\\<br />
\qquad\textbf{(B)}\ \text{She rounds } a \text{ and } b \text{ up, and she rounds } c \text{ down.}\\<br />
\qquad\textbf{(C)}\ \text{She rounds } a \text{ and } c \text{ up, and she rounds } b \text{ down.} \\<br />
\qquad\textbf{(D)}\ \text{She rounds } a \text{ up, and she rounds } b \text{ and } c \text{ down.}\\<br />
\qquad\textbf{(E)}\ \text{She rounds } c \text{ up, and she rounds } a \text{ and } b \text{ down.} </math><br />
<br />
[[2015 AMC 12A Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be <math>2 : 1</math>?<br />
<br />
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 8</math><br />
<br />
[[2015 AMC 12A Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
Two right circular cylinders have the same volume. The radius of the second cylinder is <math>10\%</math> more than the radius of the first. What is the relationship between the heights of the two cylinders?<br />
<br />
<math>\textbf{(A)}\ \text{The second height is } 10\% \text{ less than the first.} \\ \textbf{(B)}\ \text{The first height is } 10\% \text{ more than the second.}\\ \textbf{(C)}\ \text{The second height is } 21\% \text{ less than the first.} \\ \textbf{(D)}\ \text{The first height is } 21\% \text{ more than the second.}\\ \textbf{(E)}\ \text{The second height is } 80\% \text{ of the first.}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
The ratio of the length to the width of a rectangle is <math>4</math> : <math>3</math>. If the rectangle has diagonal of length <math>d</math>, then the area may be expressed as <math>kd^2</math> for some constant <math>k</math>. What is <math>k</math>?<br />
<br />
<math> \textbf{(A)}\ \frac27 \qquad\textbf{(B)}\ \frac37 \qquad\textbf{(C)}\ \frac{12}{25} \qquad\textbf{(D)}\ \frac{16}{26} \qquad\textbf{(E)}\ \frac34</math><br />
<br />
[[2015 AMC 12A Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
A box contains 2 red marbles, 2 green marbles, and 2 yellow marbles. Carol takes 2 marbles from the box at random; then Claudia takes 2 of the remaining marbles at random; and then Cheryl takes the last 2 marbles. What is the probability that Cheryl gets 2 marbles of the same color?<br />
<br />
<math> \textbf{(A)}\ \frac{1}{10} \qquad\textbf{(B)}\ \frac16 \qquad\textbf{(C)}\ \frac15 \qquad\textbf{(D)}\ \frac13 \qquad\textbf{(E)}\ \frac12</math><br />
<br />
[[2015 AMC 12A Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
Integers <math>x</math> and <math>y</math> with <math>x>y>0</math> satisfy <math>x+y+xy=80</math>. What is <math>x</math>?<br />
<br />
<math> \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 15 \qquad\textbf{(D)}\ 18 \qquad\textbf{(E)}\ 26</math><br />
<br />
[[2015 AMC 12A Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
On a sheet of paper, Isabella draws a circle of radius <math>2</math>, a circle of radius <math>3</math>, and all possible lines simultaneously tangent to both circles. Isabella notices that she has drawn exactly <math>k \ge 0</math> lines. How many different values of <math>k</math> are possible?<br />
<br />
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6</math><br />
<br />
[[2015 AMC 12A Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
The parabolas <math>y=ax^2 - 2</math> and <math>y=4 - bx^2</math> intersect the coordinate axes in exactly four points, and these four points are the vertices of a kite of area <math>12</math>. What is <math>a+b</math>?<br />
<br />
<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 1.5\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 2.5\qquad\textbf{(E)}\ 3</math><br />
<br />
[[2015 AMC 12A Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
<br />
A league with 12 teams holds a round-robin tournament, with each team playing every other team exactly once. Games either end with one team victorious or else end in a draw. A team scores 2 points for every game it wins and 1 point for every game it draws. Which of the following is NOT a true statement about the list of 12 scores?<br />
<br />
<math> \textbf{(A)}\ \text{There must be an even number of odd scores.}\\<br />
\qquad\textbf{(B)}\ \text{There must be an even number of even scores.}\\<br />
\qquad\textbf{(C)}\ \text{There cannot be two scores of }0\text{.}\\<br />
\qquad\textbf{(D)}\ \text{The sum of the scores must be at least }100\text{.}\\<br />
\qquad\textbf{(E)}\ \text{The highest score must be at least }12\text{.}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
What is the value of <math>a</math> for which <math>\frac{1}{\text{log}_2a} + \frac{1}{\text{log}_3a} + \frac{1}{\text{log}_4a} = 1</math>?<br />
<br />
[[2015 AMC 12A Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
What is the minimum number of digits to the right of the decimal point needed to express the fraction <math>\frac{123456789}{2^{26}\cdot 5^4}</math> as a decimal?<br />
<br />
<math> \textbf{(A)}\ 4\qquad\textbf{(B)}\ 22\qquad\textbf{(C)}\ 26\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 104</math><br />
<br />
[[2015 AMC 12A Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
Tetrahedron <math>ABCD</math> has <math>AB=5,AC=3,BC=4,BD=4,AD=3,</math> and <math>CD=\frac{12}{5}\sqrt{2}</math>. What is the volume of the tetrahedron?<br />
<br />
<math> \textbf{(A)}\ 3\sqrt{2}\qquad\textbf{(B)}\ 2\sqrt{5}\qquad\textbf{(C)}\ \frac{24}{5}\qquad\textbf{(D)}\ 3\sqrt{3}\qquad\textbf{(E)}\ \frac{24}{5}\sqrt{2}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?<br />
<br />
<math> \textbf{(A)}\ \frac{47}{256} \qquad\textbf{(B)}\ \frac{3}{16} \qquad\textbf{(C)}\ \frac{49}{256} \qquad\textbf{(D)}\ \frac{25}{128} \qquad\textbf{(E)}\ \frac{51}{256}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
The zeros of the function <math>f(x) = x^2-ax+2a</math> are integers. What is the sum of the possible values of <math>a</math>?<br />
<br />
<math> \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 16 \qquad\textbf{(D)}\ 17 \qquad\textbf{(E)}\ 18</math><br />
<br />
[[2015 AMC 12A Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
For some positive integers <math>p</math>, there is a quadrilateral <math>ABCD</math> with positive integer side lengths, perimeter <math>p</math>, right angles at <math>B</math> and <math>C</math>, <math>AB=2</math>, and <math>CD=AD</math>. How many different values of <math>p<2015</math> are possible?<br />
<br />
<math> \textbf{(A)}\ 30 \qquad\textbf{(B)}\ 31 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}\ 62 \qquad\textbf{(E)}\ 63</math><br />
<br />
[[2015 AMC 12A Problems/Problem 19|Solution]]<br />
<br />
==Problem 20==<br />
<br />
Isosceles triangles <math>T</math> and <math>T'</math> are not congruent but have the same area and the same perimeter. The sides of <math>T</math> have lengths of <math>5,5,</math> and <math>8</math>, while those of <math>T'</math> have lengths of <math>a,a,</math> and <math>b</math>. Which of the following numbers is closest to <math>b</math>?<br />
<br />
<math> \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 8</math><br />
<br />
[[2015 AMC 12A Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
A circle of radius <math>r</math> passes through both foci of, and exactly four points on, the ellipse with equation <math>x^2+16y^2=16</math>. The set of all possible values of <math>r</math> is an interval <math>[a,b)</math>. What is <math>a+b</math>?<br />
<br />
<math> \textbf{(A)}\ 5\sqrt{2}+4 \qquad\textbf{(B)}\ \sqrt{17}+7 \qquad\textbf{(C)}\ 6\sqrt{2}+3 \qquad\textbf{(D)}\ \sqrt{15}+8 \qquad\textbf{(E)}\ 12</math><br />
<br />
[[2015 AMC 12A Problems/Problem 21|Solution]]<br />
<br />
==Problem 22==<br />
<br />
For each positive integer <math>n</math>, let <math>S(n)</math> be the number of sequences of length <math>n</math> consisting solely of the letters <math>A</math> and <math>B</math>, with no more than three <math>A</math>s in a row and no more than three <math>B</math>s in a row. What is the remainder when <math>S(2015)</math> is divided by 12?<br />
<br />
<math> \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 8 \qquad\textbf{(E)}\ 10</math><br />
<br />
[[2015 AMC 12A Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
Let <math>S</math> be a square of side length 1. Two points are chosen independently at random on the sides of <math>S</math>. The probability that the straight-line distance between the points is at least <math>\frac12</math> is <math>\frac{a-b\pi}{c}</math>, where <math>a,b,</math> and <math>c</math> are positive integers and <math>\text{gcd}(a,b,c) = 1</math>. What is <math>a+b+c</math>?<br />
<br />
<math> \textbf{(A)}\ 59 \qquad\textbf{(B)}\ 60 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}\ 62 \qquad\textbf{(E)}\ 63</math><br />
<br />
[[2015 AMC 12A Problems/Problem 23|Solution]]<br />
<br />
==Problem 24==<br />
<br />
Rational numbers <math>a</math> and <math>b</math> are chosen at random among all rational numbers in the interval <math>[0,2)</math> that can be written as fractions <math>\frac{n}{d}</math> where <math>n</math> and <math>d</math> are integers with <math>1 \le d \le 5</math>. What is the probability that<br />
<cmath>(\text{cos}(a\pi)+i\text{sin}(b\pi))^4</cmath><br />
is a real number?<br />
<br />
<math> \textbf{(A)}\ \frac{3}{5} \qquad\textbf{(B)}\ \frac{4}{25} \qquad\textbf{(C)}\ \frac{41}{200} \qquad\textbf{(D)}\ \frac{6}{25} \qquad\textbf{(E)}\ \frac{13}{50}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 24|Solution]]<br />
<br />
==Problem 25==<br />
<br />
A collection of circles in the upper half-plane, all tangent to the <math>x</math>-axis, is constructed in layers as follows. Layer <math>L_0</math> consists of two circles of radii <math>70^2</math> and <math>73^2</math> that are externally tangent. For <math>k\ge1</math>, the circles in <math>\bigcup_{j=0}^{k-1}L_j</math> are ordered according to their points of tangency with the <math>x</math>-axis. For every pair of consecutive circles in this order, a new circle is constructed externally tangent to each of the two circles in the pair. Layer <math>L_k</math> consists of the <math>2^{k-1}</math> circles constructed in this way. Let <math>S=\bigcup_{j=0}^{6}L_j</math>, and for every circle <math>C</math> denote by <math>r(C)</math> its radius. What is<br />
<cmath>\sum_{C\in S} \frac{1}{\sqrt{r(C)}}?</cmath><br />
<br />
<asy><br />
import olympiad;<br />
size(350);<br />
defaultpen(linewidth(0.7));<br />
// define a bunch of arrays and starting points<br />
pair[] coord = new pair[65];<br />
int[] trav = {32,16,8,4,2,1};<br />
coord[0] = (0,73^2); coord[64] = (2*73*70,70^2);<br />
// draw the big circles and the bottom line<br />
path arc1 = arc(coord[0],coord[0].y,260,360);<br />
path arc2 = arc(coord[64],coord[64].y,175,280);<br />
fill((coord[0].x-910,coord[0].y)--arc1--cycle,gray(0.75));<br />
fill((coord[64].x+870,coord[64].y+425)--arc2--cycle,gray(0.75));<br />
draw(arc1^^arc2);<br />
draw((-930,0)--(70^2+73^2+850,0));<br />
// We now apply the findCenter function 63 times to get<br />
// the location of the centers of all 63 constructed circles.<br />
// The complicated array setup ensures that all the circles<br />
// will be taken in the right order<br />
for(int i = 0;i<=5;i=i+1)<br />
{<br />
int skip = trav[i];<br />
for(int k=skip;k<=64 - skip; k = k + 2*skip)<br />
{<br />
pair cent1 = coord[k-skip], cent2 = coord[k+skip];<br />
real r1 = cent1.y, r2 = cent2.y, rn=r1*r2/((sqrt(r1)+sqrt(r2))^2);<br />
real shiftx = cent1.x + sqrt(4*r1*rn);<br />
coord[k] = (shiftx,rn);<br />
}<br />
// Draw the remaining 63 circles<br />
}<br />
for(int i=1;i<=63;i=i+1)<br />
{<br />
filldraw(circle(coord[i],coord[i].y),gray(0.75));<br />
}<br />
</asy><br />
<br />
<br />
<br />
$$ \textbf{(A)}\ \frac{286}{35} \qquad\textbf{(B)}\ \frac{583}{70} \qquad\textbf{(C)}\ \frac{715}{73} \qquad\textbf{(D)}\ \frac{143}{14} \qquad\textbf{(E)}\ \frac{1573}{146}$$<br />
<br />
[[2015 AMC 12A Problems/Problem 25|Solution]]<br />
<br />
== See also ==<br />
* [[AMC Problems and Solutions]]<br />
{{MAA Notice}}</div>Devenwarehttps://artofproblemsolving.com/wiki/index.php?title=2015_AMC_12A_Problems&diff=680292015 AMC 12A Problems2015-03-01T23:31:34Z<p>Devenware: /* Problem 25 */</p>
<hr />
<div>==Problem 1==<br />
<br />
What is the value of <math>(2^0-1+5^2-0)^{-1}\times5?</math><br />
<br />
<math> \textbf{(A)}\ -125\qquad\textbf{(B)}\ -120\qquad\textbf{(C)}\ \frac{1}{5}\qquad\textbf{(D)}\ \frac{5}{24}\qquad\textbf{(E)}\ 25 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
<br />
Two of the three sides of a triangle are 20 and 15. Which of the following numbers is not a possible perimeter of the triangle?<br />
<br />
<math> \textbf{(A)}\ 52\qquad\textbf{(B)}\ 57\qquad\textbf{(C)}\ 62\qquad\textbf{(D)}\ 67\qquad\textbf{(E)}\ 72 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
<br />
Mr. Patrick teaches math to 15 students. He was grading tests and found that when he graded everyone's test except Payton's, the average grade for the class was 80. after he graded Payton's test, the class average became 81. What was Payton's score on the test?<br />
<br />
<math> \textbf{(A)}\ 81\qquad\textbf{(B)}\ 85\qquad\textbf{(C)}\ 91\qquad\textbf{(D)}\ 94\qquad\textbf{(E)}\ 95 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
The sum of two positive numbers is 5 times their difference. What is the ratio of the larger number to the smaller?<br />
<br />
<math> \textbf{(A)}\ \frac54 \qquad\textbf{(B)}\ \frac32 \qquad\textbf{(C)}\ \frac95 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ \frac52 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
Amelia needs to estimate the quantity <math>\frac{a}{b} - c</math>, where <math>a, b,</math> and <math>c</math> are large positive integers. She rounds each of the integers so that the calculation will be easier to do mentally. In which of these situations will her answer necessarily be greater than the exact value of <math>\frac{a}{b} - c</math>?<br />
<br />
<math> \textbf{(A)}\ \text{She rounds all three numbers up.}\\<br />
\qquad\textbf{(B)}\ \text{She rounds } a \text{ and } b \text{ up, and she rounds } c \text{ down.}\\<br />
\qquad\textbf{(C)}\ \text{She rounds } a \text{ and } c \text{ up, and she rounds } b \text{ down.} \\<br />
\qquad\textbf{(D)}\ \text{She rounds } a \text{ up, and she rounds } b \text{ and } c \text{ down.}\\<br />
\qquad\textbf{(E)}\ \text{She rounds } c \text{ up, and she rounds } a \text{ and } b \text{ down.} </math><br />
<br />
[[2015 AMC 12A Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be <math>2 : 1</math>?<br />
<br />
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 8</math><br />
<br />
[[2015 AMC 12A Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
Two right circular cylinders have the same volume. The radius of the second cylinder is <math>10\%</math> more than the radius of the first. What is the relationship between the heights of the two cylinders?<br />
<br />
<math>\textbf{(A)}\ \text{The second height is } 10\% \text{ less than the first.} \\ \textbf{(B)}\ \text{The first height is } 10\% \text{ more than the second.}\\ \textbf{(C)}\ \text{The second height is } 21\% \text{ less than the first.} \\ \textbf{(D)}\ \text{The first height is } 21\% \text{ more than the second.}\\ \textbf{(E)}\ \text{The second height is } 80\% \text{ of the first.}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
The ratio of the length to the width of a rectangle is <math>4</math> : <math>3</math>. If the rectangle has diagonal of length <math>d</math>, then the area may be expressed as <math>kd^2</math> for some constant <math>k</math>. What is <math>k</math>?<br />
<br />
<math> \textbf{(A)}\ \frac27 \qquad\textbf{(B)}\ \frac37 \qquad\textbf{(C)}\ \frac{12}{25} \qquad\textbf{(D)}\ \frac{16}{26} \qquad\textbf{(E)}\ \frac34</math><br />
<br />
[[2015 AMC 12A Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
A box contains 2 red marbles, 2 green marbles, and 2 yellow marbles. Carol takes 2 marbles from the box at random; then Claudia takes 2 of the remaining marbles at random; and then Cheryl takes the last 2 marbles. What is the probability that Cheryl gets 2 marbles of the same color?<br />
<br />
<math> \textbf{(A)}\ \frac{1}{10} \qquad\textbf{(B)}\ \frac16 \qquad\textbf{(C)}\ \frac15 \qquad\textbf{(D)}\ \frac13 \qquad\textbf{(E)}\ \frac12</math><br />
<br />
[[2015 AMC 12A Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
Integers <math>x</math> and <math>y</math> with <math>x>y>0</math> satisfy <math>x+y+xy=80</math>. What is <math>x</math>?<br />
<br />
<math> \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 15 \qquad\textbf{(D)}\ 18 \qquad\textbf{(E)}\ 26</math><br />
<br />
[[2015 AMC 12A Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
On a sheet of paper, Isabella draws a circle of radius <math>2</math>, a circle of radius <math>3</math>, and all possible lines simultaneously tangent to both circles. Isabella notices that she has drawn exactly <math>k \ge 0</math> lines. How many different values of <math>k</math> are possible?<br />
<br />
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6</math><br />
<br />
[[2015 AMC 12A Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
The parabolas <math>y=ax^2 - 2</math> and <math>y=4 - bx^2</math> intersect the coordinate axes in exactly four points, and these four points are the vertices of a kite of area <math>12</math>. What is <math>a+b</math>?<br />
<br />
<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 1.5\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 2.5\qquad\textbf{(E)}\ 3</math><br />
<br />
[[2015 AMC 12A Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
<br />
A league with 12 teams holds a round-robin tournament, with each team playing every other team exactly once. Games either end with one team victorious or else end in a draw. A team scores 2 points for every game it wins and 1 point for every game it draws. Which of the following is NOT a true statement about the list of 12 scores?<br />
<br />
<math> \textbf{(A)}\ \text{There must be an even number of odd scores.}\\<br />
\qquad\textbf{(B)}\ \text{There must be an even number of even scores.}\\<br />
\qquad\textbf{(C)}\ \text{There cannot be two scores of }0\text{.}\\<br />
\qquad\textbf{(D)}\ \text{The sum of the scores must be at least }100\text{.}\\<br />
\qquad\textbf{(E)}\ \text{The highest score must be at least }12\text{.}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
What is the value of <math>a</math> for which <math>\frac{1}{\text{log}_2a} + \frac{1}{\text{log}_3a} + \frac{1}{\text{log}_4a} = 1</math>?<br />
<br />
[[2015 AMC 12A Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
What is the minimum number of digits to the right of the decimal point needed to express the fraction <math>\frac{123456789}{2^{26}\cdot 5^4}</math> as a decimal?<br />
<br />
<math> \textbf{(A)}\ 4\qquad\textbf{(B)}\ 22\qquad\textbf{(C)}\ 26\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 104</math><br />
<br />
[[2015 AMC 12A Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
Tetrahedron <math>ABCD</math> has <math>AB=5,AC=3,BC=4,BD=4,AD=3,</math> and <math>CD=\frac{12}{5}\sqrt{2}</math>. What is the volume of the tetrahedron?<br />
<br />
<math> \textbf{(A)}\ 3\sqrt{2}\qquad\textbf{(B)}\ 2\sqrt{5}\qquad\textbf{(C)}\ \frac{24}{5}\qquad\textbf{(D)}\ 3\sqrt{3}\qquad\textbf{(E)}\ \frac{24}{5}\sqrt{2}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?<br />
<br />
<math> \textbf{(A)}\ \frac{47}{256} \qquad\textbf{(B)}\ \frac{3}{16} \qquad\textbf{(C)}\ \frac{49}{256} \qquad\textbf{(D)}\ \frac{25}{128} \qquad\textbf{(E)}\ \frac{51}{256}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
The zeros of the function <math>f(x) = x^2-ax+2a</math> are integers. What is the sum of the possible values of <math>a</math>?<br />
<br />
<math> \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 16 \qquad\textbf{(D)}\ 17 \qquad\textbf{(E)}\ 18</math><br />
<br />
[[2015 AMC 12A Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
For some positive integers <math>p</math>, there is a quadrilateral <math>ABCD</math> with positive integer side lengths, perimeter <math>p</math>, right angles at <math>B</math> and <math>C</math>, <math>AB=2</math>, and <math>CD=AD</math>. How many different values of <math>p<2015</math> are possible?<br />
<br />
<math> \textbf{(A)}\ 30 \qquad\textbf{(B)}\ 31 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}\ 62 \qquad\textbf{(E)}\ 63</math><br />
<br />
[[2015 AMC 12A Problems/Problem 19|Solution]]<br />
<br />
==Problem 20==<br />
<br />
Isosceles triangles <math>T</math> and <math>T'</math> are not congruent but have the same area and the same perimeter. The sides of <math>T</math> have lengths of <math>5,5,</math> and <math>8</math>, while those of <math>T'</math> have lengths of <math>a,a,</math> and <math>b</math>. Which of the following numbers is closest to <math>b</math>?<br />
<br />
<math> \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 8</math><br />
<br />
[[2015 AMC 12A Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
A circle of radius <math>r</math> passes through both foci of, and exactly four points on, the ellipse with equation <math>x^2+16y^2=16</math>. The set of all possible values of <math>r</math> is an interval <math>[a,b)</math>. What is <math>a+b</math>?<br />
<br />
<math> \textbf{(A)}\ 5\sqrt{2}+4 \qquad\textbf{(B)}\ \sqrt{17}+7 \qquad\textbf{(C)}\ 6\sqrt{2}+3 \qquad\textbf{(D)}\ \sqrt{15}+8 \qquad\textbf{(E)}\ 12</math><br />
<br />
[[2015 AMC 12A Problems/Problem 21|Solution]]<br />
<br />
==Problem 22==<br />
<br />
For each positive integer <math>n</math>, let <math>S(n)</math> be the number of sequences of length <math>n</math> consisting solely of the letters <math>A</math> and <math>B</math>, with no more than three <math>A</math>s in a row and no more than three <math>B</math>s in a row. What is the remainder when <math>S(2015)</math> is divided by 12?<br />
<br />
<math> \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 8 \qquad\textbf{(E)}\ 10</math><br />
<br />
[[2015 AMC 12A Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
Let <math>S</math> be a square of side length 1. Two points are chosen independently at random on the sides of <math>S</math>. The probability that the straight-line distance between the points is at least <math>\frac12</math> is <math>\frac{a-b\pi}{c}</math>, where <math>a,b,</math> and <math>c</math> are positive integers and <math>\text{gcd}(a,b,c) = 1</math>. What is <math>a+b+c</math>?<br />
<br />
<math> \textbf{(A)}\ 59 \qquad\textbf{(B)}\ 60 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}\ 62 \qquad\textbf{(E)}\ 63</math><br />
<br />
[[2015 AMC 12A Problems/Problem 23|Solution]]<br />
<br />
==Problem 24==<br />
<br />
Rational numbers <math>a</math> and <math>b</math> are chosen at random among all rational numbers in the interval <math>[0,2)</math> that can be written as fractions <math>\frac{n}{d}</math> where <math>n</math> and <math>d</math> are integers with <math>1 \le d \le 5</math>. What is the probability that<br />
<cmath>(\text{cos}(a\pi)+i\text{sin}(b\pi))^4</cmath><br />
is a real number?<br />
<br />
<math> \textbf{(A)}\ \frac{3}{5} \qquad\textbf{(B)}\ \frac{4}{25} \qquad\textbf{(C)}\ \frac{41}{200} \qquad\textbf{(D)}\ \frac{6}{25} \qquad\textbf{(E)}\ \frac{13}{50}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 24|Solution]]<br />
<br />
==Problem 25==<br />
<br />
A collection of circles in the upper half-plane, all tangent to the <math>x</math>-axis, is constructed in layers as follows. Layer <math>L_0</math> consists of two circles of radii <math>70^2</math> and <math>73^2</math> that are externally tangent. For <math>k\ge1</math>, the circles in <math>\bigcup_{j=0}^{k-1}L_j</math> are ordered according to their points of tangency with the <math>x</math>-axis. For every pair of consecutive circles in this order, a new circle is constructed externally tangent to each of the two circles in the pair. Layer <math>L_k</math> consists of the <math>2^{k-1}</math> circles constructed in this way. Let <math>S=\bigcup_{j=0}^{6}L_j</math>, and for every circle <math>C</math> denote by <math>r(C)</math> its radius. What is<br />
<cmath>\sum_{C\in S} \frac{1}{\sqrt{r(C)}}?</cmath><br />
<br />
<asy><br />
import olympiad;<br />
size(350);<br />
defaultpen(linewidth(0.7));<br />
// define a bunch of arrays and starting points<br />
pair[] coord = new pair[65];<br />
int[] trav = {32,16,8,4,2,1};<br />
coord[0] = (0,73^2); coord[64] = (2*73*70,70^2);<br />
// draw the big circles and the bottom line<br />
path arc1 = arc(coord[0],coord[0].y,260,360);<br />
path arc2 = arc(coord[64],coord[64].y,175,280);<br />
fill((coord[0].x-910,coord[0].y)--arc1--cycle,gray(0.75));<br />
fill((coord[64].x+870,coord[64].y+425)--arc2--cycle,gray(0.75));<br />
draw(arc1^^arc2);<br />
draw((-930,0)--(70^2+73^2+850,0));<br />
// We now apply the findCenter function 63 times to get<br />
// the location of the centers of all 63 constructed circles.<br />
// The complicated array setup ensures that all the circles<br />
// will be taken in the right order<br />
for(int i = 0;i<=5;i=i+1)<br />
{<br />
int skip = trav[i];<br />
for(int k=skip;k<=64 - skip; k = k + 2*skip)<br />
{<br />
pair cent1 = coord[k-skip], cent2 = coord[k+skip];<br />
real r1 = cent1.y, r2 = cent2.y, rn=r1*r2/((sqrt(r1)+sqrt(r2))^2);<br />
real shiftx = cent1.x + sqrt(4*r1*rn);<br />
coord[k] = (shiftx,rn);<br />
}<br />
// Draw the remaining 63 circles<br />
}<br />
for(int i=1;i<=63;i=i+1)<br />
{<br />
filldraw(circle(coord[i],coord[i].y),gray(0.75));<br />
}</asy><br />
<br />
<br />
$\textbf{(A)}\ \frac{286}{35} \qquad\textbf{(B)}\ \frac{583}{70} \qquad\textbf{(C)}\ \frac{715}{73} \qquad\textbf{(D)}\ \frac{143}{14} \qquad\textbf{(E)}\ \frac{1573}{146}$<br />
<br />
[[2015 AMC 12A Problems/Problem 25|Solution]]<br />
<br />
== See also ==<br />
* [[AMC Problems and Solutions]]<br />
{{MAA Notice}}</div>Devenwarehttps://artofproblemsolving.com/wiki/index.php?title=2015_AMC_12A_Problems&diff=680282015 AMC 12A Problems2015-03-01T23:30:43Z<p>Devenware: /* Problem 25 */</p>
<hr />
<div>==Problem 1==<br />
<br />
What is the value of <math>(2^0-1+5^2-0)^{-1}\times5?</math><br />
<br />
<math> \textbf{(A)}\ -125\qquad\textbf{(B)}\ -120\qquad\textbf{(C)}\ \frac{1}{5}\qquad\textbf{(D)}\ \frac{5}{24}\qquad\textbf{(E)}\ 25 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
<br />
Two of the three sides of a triangle are 20 and 15. Which of the following numbers is not a possible perimeter of the triangle?<br />
<br />
<math> \textbf{(A)}\ 52\qquad\textbf{(B)}\ 57\qquad\textbf{(C)}\ 62\qquad\textbf{(D)}\ 67\qquad\textbf{(E)}\ 72 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
<br />
Mr. Patrick teaches math to 15 students. He was grading tests and found that when he graded everyone's test except Payton's, the average grade for the class was 80. after he graded Payton's test, the class average became 81. What was Payton's score on the test?<br />
<br />
<math> \textbf{(A)}\ 81\qquad\textbf{(B)}\ 85\qquad\textbf{(C)}\ 91\qquad\textbf{(D)}\ 94\qquad\textbf{(E)}\ 95 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
The sum of two positive numbers is 5 times their difference. What is the ratio of the larger number to the smaller?<br />
<br />
<math> \textbf{(A)}\ \frac54 \qquad\textbf{(B)}\ \frac32 \qquad\textbf{(C)}\ \frac95 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ \frac52 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
Amelia needs to estimate the quantity <math>\frac{a}{b} - c</math>, where <math>a, b,</math> and <math>c</math> are large positive integers. She rounds each of the integers so that the calculation will be easier to do mentally. In which of these situations will her answer necessarily be greater than the exact value of <math>\frac{a}{b} - c</math>?<br />
<br />
<math> \textbf{(A)}\ \text{She rounds all three numbers up.}\\<br />
\qquad\textbf{(B)}\ \text{She rounds } a \text{ and } b \text{ up, and she rounds } c \text{ down.}\\<br />
\qquad\textbf{(C)}\ \text{She rounds } a \text{ and } c \text{ up, and she rounds } b \text{ down.} \\<br />
\qquad\textbf{(D)}\ \text{She rounds } a \text{ up, and she rounds } b \text{ and } c \text{ down.}\\<br />
\qquad\textbf{(E)}\ \text{She rounds } c \text{ up, and she rounds } a \text{ and } b \text{ down.} </math><br />
<br />
[[2015 AMC 12A Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be <math>2 : 1</math>?<br />
<br />
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 8</math><br />
<br />
[[2015 AMC 12A Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
Two right circular cylinders have the same volume. The radius of the second cylinder is <math>10\%</math> more than the radius of the first. What is the relationship between the heights of the two cylinders?<br />
<br />
<math>\textbf{(A)}\ \text{The second height is } 10\% \text{ less than the first.} \\ \textbf{(B)}\ \text{The first height is } 10\% \text{ more than the second.}\\ \textbf{(C)}\ \text{The second height is } 21\% \text{ less than the first.} \\ \textbf{(D)}\ \text{The first height is } 21\% \text{ more than the second.}\\ \textbf{(E)}\ \text{The second height is } 80\% \text{ of the first.}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
The ratio of the length to the width of a rectangle is <math>4</math> : <math>3</math>. If the rectangle has diagonal of length <math>d</math>, then the area may be expressed as <math>kd^2</math> for some constant <math>k</math>. What is <math>k</math>?<br />
<br />
<math> \textbf{(A)}\ \frac27 \qquad\textbf{(B)}\ \frac37 \qquad\textbf{(C)}\ \frac{12}{25} \qquad\textbf{(D)}\ \frac{16}{26} \qquad\textbf{(E)}\ \frac34</math><br />
<br />
[[2015 AMC 12A Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
A box contains 2 red marbles, 2 green marbles, and 2 yellow marbles. Carol takes 2 marbles from the box at random; then Claudia takes 2 of the remaining marbles at random; and then Cheryl takes the last 2 marbles. What is the probability that Cheryl gets 2 marbles of the same color?<br />
<br />
<math> \textbf{(A)}\ \frac{1}{10} \qquad\textbf{(B)}\ \frac16 \qquad\textbf{(C)}\ \frac15 \qquad\textbf{(D)}\ \frac13 \qquad\textbf{(E)}\ \frac12</math><br />
<br />
[[2015 AMC 12A Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
Integers <math>x</math> and <math>y</math> with <math>x>y>0</math> satisfy <math>x+y+xy=80</math>. What is <math>x</math>?<br />
<br />
<math> \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 15 \qquad\textbf{(D)}\ 18 \qquad\textbf{(E)}\ 26</math><br />
<br />
[[2015 AMC 12A Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
On a sheet of paper, Isabella draws a circle of radius <math>2</math>, a circle of radius <math>3</math>, and all possible lines simultaneously tangent to both circles. Isabella notices that she has drawn exactly <math>k \ge 0</math> lines. How many different values of <math>k</math> are possible?<br />
<br />
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6</math><br />
<br />
[[2015 AMC 12A Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
The parabolas <math>y=ax^2 - 2</math> and <math>y=4 - bx^2</math> intersect the coordinate axes in exactly four points, and these four points are the vertices of a kite of area <math>12</math>. What is <math>a+b</math>?<br />
<br />
<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 1.5\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 2.5\qquad\textbf{(E)}\ 3</math><br />
<br />
[[2015 AMC 12A Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
<br />
A league with 12 teams holds a round-robin tournament, with each team playing every other team exactly once. Games either end with one team victorious or else end in a draw. A team scores 2 points for every game it wins and 1 point for every game it draws. Which of the following is NOT a true statement about the list of 12 scores?<br />
<br />
<math> \textbf{(A)}\ \text{There must be an even number of odd scores.}\\<br />
\qquad\textbf{(B)}\ \text{There must be an even number of even scores.}\\<br />
\qquad\textbf{(C)}\ \text{There cannot be two scores of }0\text{.}\\<br />
\qquad\textbf{(D)}\ \text{The sum of the scores must be at least }100\text{.}\\<br />
\qquad\textbf{(E)}\ \text{The highest score must be at least }12\text{.}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
What is the value of <math>a</math> for which <math>\frac{1}{\text{log}_2a} + \frac{1}{\text{log}_3a} + \frac{1}{\text{log}_4a} = 1</math>?<br />
<br />
[[2015 AMC 12A Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
What is the minimum number of digits to the right of the decimal point needed to express the fraction <math>\frac{123456789}{2^{26}\cdot 5^4}</math> as a decimal?<br />
<br />
<math> \textbf{(A)}\ 4\qquad\textbf{(B)}\ 22\qquad\textbf{(C)}\ 26\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 104</math><br />
<br />
[[2015 AMC 12A Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
Tetrahedron <math>ABCD</math> has <math>AB=5,AC=3,BC=4,BD=4,AD=3,</math> and <math>CD=\frac{12}{5}\sqrt{2}</math>. What is the volume of the tetrahedron?<br />
<br />
<math> \textbf{(A)}\ 3\sqrt{2}\qquad\textbf{(B)}\ 2\sqrt{5}\qquad\textbf{(C)}\ \frac{24}{5}\qquad\textbf{(D)}\ 3\sqrt{3}\qquad\textbf{(E)}\ \frac{24}{5}\sqrt{2}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?<br />
<br />
<math> \textbf{(A)}\ \frac{47}{256} \qquad\textbf{(B)}\ \frac{3}{16} \qquad\textbf{(C)}\ \frac{49}{256} \qquad\textbf{(D)}\ \frac{25}{128} \qquad\textbf{(E)}\ \frac{51}{256}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
The zeros of the function <math>f(x) = x^2-ax+2a</math> are integers. What is the sum of the possible values of <math>a</math>?<br />
<br />
<math> \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 16 \qquad\textbf{(D)}\ 17 \qquad\textbf{(E)}\ 18</math><br />
<br />
[[2015 AMC 12A Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
For some positive integers <math>p</math>, there is a quadrilateral <math>ABCD</math> with positive integer side lengths, perimeter <math>p</math>, right angles at <math>B</math> and <math>C</math>, <math>AB=2</math>, and <math>CD=AD</math>. How many different values of <math>p<2015</math> are possible?<br />
<br />
<math> \textbf{(A)}\ 30 \qquad\textbf{(B)}\ 31 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}\ 62 \qquad\textbf{(E)}\ 63</math><br />
<br />
[[2015 AMC 12A Problems/Problem 19|Solution]]<br />
<br />
==Problem 20==<br />
<br />
Isosceles triangles <math>T</math> and <math>T'</math> are not congruent but have the same area and the same perimeter. The sides of <math>T</math> have lengths of <math>5,5,</math> and <math>8</math>, while those of <math>T'</math> have lengths of <math>a,a,</math> and <math>b</math>. Which of the following numbers is closest to <math>b</math>?<br />
<br />
<math> \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 8</math><br />
<br />
[[2015 AMC 12A Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
A circle of radius <math>r</math> passes through both foci of, and exactly four points on, the ellipse with equation <math>x^2+16y^2=16</math>. The set of all possible values of <math>r</math> is an interval <math>[a,b)</math>. What is <math>a+b</math>?<br />
<br />
<math> \textbf{(A)}\ 5\sqrt{2}+4 \qquad\textbf{(B)}\ \sqrt{17}+7 \qquad\textbf{(C)}\ 6\sqrt{2}+3 \qquad\textbf{(D)}\ \sqrt{15}+8 \qquad\textbf{(E)}\ 12</math><br />
<br />
[[2015 AMC 12A Problems/Problem 21|Solution]]<br />
<br />
==Problem 22==<br />
<br />
For each positive integer <math>n</math>, let <math>S(n)</math> be the number of sequences of length <math>n</math> consisting solely of the letters <math>A</math> and <math>B</math>, with no more than three <math>A</math>s in a row and no more than three <math>B</math>s in a row. What is the remainder when <math>S(2015)</math> is divided by 12?<br />
<br />
<math> \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 8 \qquad\textbf{(E)}\ 10</math><br />
<br />
[[2015 AMC 12A Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
Let <math>S</math> be a square of side length 1. Two points are chosen independently at random on the sides of <math>S</math>. The probability that the straight-line distance between the points is at least <math>\frac12</math> is <math>\frac{a-b\pi}{c}</math>, where <math>a,b,</math> and <math>c</math> are positive integers and <math>\text{gcd}(a,b,c) = 1</math>. What is <math>a+b+c</math>?<br />
<br />
<math> \textbf{(A)}\ 59 \qquad\textbf{(B)}\ 60 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}\ 62 \qquad\textbf{(E)}\ 63</math><br />
<br />
[[2015 AMC 12A Problems/Problem 23|Solution]]<br />
<br />
==Problem 24==<br />
<br />
Rational numbers <math>a</math> and <math>b</math> are chosen at random among all rational numbers in the interval <math>[0,2)</math> that can be written as fractions <math>\frac{n}{d}</math> where <math>n</math> and <math>d</math> are integers with <math>1 \le d \le 5</math>. What is the probability that<br />
<cmath>(\text{cos}(a\pi)+i\text{sin}(b\pi))^4</cmath><br />
is a real number?<br />
<br />
<math> \textbf{(A)}\ \frac{3}{5} \qquad\textbf{(B)}\ \frac{4}{25} \qquad\textbf{(C)}\ \frac{41}{200} \qquad\textbf{(D)}\ \frac{6}{25} \qquad\textbf{(E)}\ \frac{13}{50}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 24|Solution]]<br />
<br />
==Problem 25==<br />
<br />
A collection of circles in the upper half-plane, all tangent to the <math>x</math>-axis, is constructed in layers as follows. Layer <math>L_0</math> consists of two circles of radii <math>70^2</math> and <math>73^2</math> that are externally tangent. For <math>k\ge1</math>, the circles in <math>\bigcup_{j=0}^{k-1}L_j</math> are ordered according to their points of tangency with the <math>x</math>-axis. For every pair of consecutive circles in this order, a new circle is constructed externally tangent to each of the two circles in the pair. Layer <math>L_k</math> consists of the <math>2^{k-1}</math> circles constructed in this way. Let <math>S=\bigcup_{j=0}^{6}L_j</math>, and for every circle <math>C</math> denote by <math>r(C)</math> its radius. What is<br />
<cmath>\sum_{C\in S} \frac{1}{\sqrt{r(C)}}?</cmath><br />
<br />
<asy><br />
import olympiad;<br />
size(350);<br />
defaultpen(linewidth(0.7));<br />
// define a bunch of arrays and starting points<br />
pair[] coord = new pair[65];<br />
int[] trav = {32,16,8,4,2,1};<br />
coord[0] = (0,73^2); coord[64] = (2*73*70,70^2);<br />
// draw the big circles and the bottom line<br />
path arc1 = arc(coord[0],coord[0].y,260,360);<br />
path arc2 = arc(coord[64],coord[64].y,175,280);<br />
fill((coord[0].x-910,coord[0].y)--arc1--cycle,gray(0.75));<br />
fill((coord[64].x+870,coord[64].y+425)--arc2--cycle,gray(0.75));<br />
draw(arc1^^arc2);<br />
draw((-930,0)--(70^2+73^2+850,0));<br />
// We now apply the findCenter function 63 times to get<br />
// the location of the centers of all 63 constructed circles.<br />
// The complicated array setup ensures that all the circles<br />
// will be taken in the right order<br />
for(int i = 0;i<=5;i=i+1)<br />
{<br />
int skip = trav[i];<br />
for(int k=skip;k<=64 - skip; k = k + 2*skip)<br />
{<br />
pair cent1 = coord[k-skip], cent2 = coord[k+skip];<br />
real r1 = cent1.y, r2 = cent2.y, rn=r1*r2/((sqrt(r1)+sqrt(r2))^2);<br />
real shiftx = cent1.x + sqrt(4*r1*rn);<br />
coord[k] = (shiftx,rn);<br />
}<br />
// Draw the remaining 63 circles<br />
}<br />
for(int i=1;i<=63;i=i+1)<br />
{<br />
filldraw(circle(coord[i],coord[i].y),gray(0.75));<br />
}</asy><br />
<br />
<br />
$ \textbf{(A)}\ \frac{286}{35} \qquad\textbf{(B)}\ \frac{583}{70} \qquad\textbf{(C)}\ \frac{715}{73} \qquad\textbf{(D)}\ \frac{143}{14} \qquad\textbf{(E)}\ \frac{1573}{146}$<br />
<br />
[[2015 AMC 12A Problems/Problem 25|Solution]]<br />
<br />
== See also ==<br />
* [[AMC Problems and Solutions]]<br />
{{MAA Notice}}</div>Devenwarehttps://artofproblemsolving.com/wiki/index.php?title=2015_AMC_12A_Problems&diff=680272015 AMC 12A Problems2015-03-01T23:30:26Z<p>Devenware: /* Problem 24 */</p>
<hr />
<div>==Problem 1==<br />
<br />
What is the value of <math>(2^0-1+5^2-0)^{-1}\times5?</math><br />
<br />
<math> \textbf{(A)}\ -125\qquad\textbf{(B)}\ -120\qquad\textbf{(C)}\ \frac{1}{5}\qquad\textbf{(D)}\ \frac{5}{24}\qquad\textbf{(E)}\ 25 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
<br />
Two of the three sides of a triangle are 20 and 15. Which of the following numbers is not a possible perimeter of the triangle?<br />
<br />
<math> \textbf{(A)}\ 52\qquad\textbf{(B)}\ 57\qquad\textbf{(C)}\ 62\qquad\textbf{(D)}\ 67\qquad\textbf{(E)}\ 72 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
<br />
Mr. Patrick teaches math to 15 students. He was grading tests and found that when he graded everyone's test except Payton's, the average grade for the class was 80. after he graded Payton's test, the class average became 81. What was Payton's score on the test?<br />
<br />
<math> \textbf{(A)}\ 81\qquad\textbf{(B)}\ 85\qquad\textbf{(C)}\ 91\qquad\textbf{(D)}\ 94\qquad\textbf{(E)}\ 95 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
The sum of two positive numbers is 5 times their difference. What is the ratio of the larger number to the smaller?<br />
<br />
<math> \textbf{(A)}\ \frac54 \qquad\textbf{(B)}\ \frac32 \qquad\textbf{(C)}\ \frac95 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ \frac52 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
Amelia needs to estimate the quantity <math>\frac{a}{b} - c</math>, where <math>a, b,</math> and <math>c</math> are large positive integers. She rounds each of the integers so that the calculation will be easier to do mentally. In which of these situations will her answer necessarily be greater than the exact value of <math>\frac{a}{b} - c</math>?<br />
<br />
<math> \textbf{(A)}\ \text{She rounds all three numbers up.}\\<br />
\qquad\textbf{(B)}\ \text{She rounds } a \text{ and } b \text{ up, and she rounds } c \text{ down.}\\<br />
\qquad\textbf{(C)}\ \text{She rounds } a \text{ and } c \text{ up, and she rounds } b \text{ down.} \\<br />
\qquad\textbf{(D)}\ \text{She rounds } a \text{ up, and she rounds } b \text{ and } c \text{ down.}\\<br />
\qquad\textbf{(E)}\ \text{She rounds } c \text{ up, and she rounds } a \text{ and } b \text{ down.} </math><br />
<br />
[[2015 AMC 12A Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be <math>2 : 1</math>?<br />
<br />
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 8</math><br />
<br />
[[2015 AMC 12A Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
Two right circular cylinders have the same volume. The radius of the second cylinder is <math>10\%</math> more than the radius of the first. What is the relationship between the heights of the two cylinders?<br />
<br />
<math>\textbf{(A)}\ \text{The second height is } 10\% \text{ less than the first.} \\ \textbf{(B)}\ \text{The first height is } 10\% \text{ more than the second.}\\ \textbf{(C)}\ \text{The second height is } 21\% \text{ less than the first.} \\ \textbf{(D)}\ \text{The first height is } 21\% \text{ more than the second.}\\ \textbf{(E)}\ \text{The second height is } 80\% \text{ of the first.}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
The ratio of the length to the width of a rectangle is <math>4</math> : <math>3</math>. If the rectangle has diagonal of length <math>d</math>, then the area may be expressed as <math>kd^2</math> for some constant <math>k</math>. What is <math>k</math>?<br />
<br />
<math> \textbf{(A)}\ \frac27 \qquad\textbf{(B)}\ \frac37 \qquad\textbf{(C)}\ \frac{12}{25} \qquad\textbf{(D)}\ \frac{16}{26} \qquad\textbf{(E)}\ \frac34</math><br />
<br />
[[2015 AMC 12A Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
A box contains 2 red marbles, 2 green marbles, and 2 yellow marbles. Carol takes 2 marbles from the box at random; then Claudia takes 2 of the remaining marbles at random; and then Cheryl takes the last 2 marbles. What is the probability that Cheryl gets 2 marbles of the same color?<br />
<br />
<math> \textbf{(A)}\ \frac{1}{10} \qquad\textbf{(B)}\ \frac16 \qquad\textbf{(C)}\ \frac15 \qquad\textbf{(D)}\ \frac13 \qquad\textbf{(E)}\ \frac12</math><br />
<br />
[[2015 AMC 12A Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
Integers <math>x</math> and <math>y</math> with <math>x>y>0</math> satisfy <math>x+y+xy=80</math>. What is <math>x</math>?<br />
<br />
<math> \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 15 \qquad\textbf{(D)}\ 18 \qquad\textbf{(E)}\ 26</math><br />
<br />
[[2015 AMC 12A Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
On a sheet of paper, Isabella draws a circle of radius <math>2</math>, a circle of radius <math>3</math>, and all possible lines simultaneously tangent to both circles. Isabella notices that she has drawn exactly <math>k \ge 0</math> lines. How many different values of <math>k</math> are possible?<br />
<br />
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6</math><br />
<br />
[[2015 AMC 12A Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
The parabolas <math>y=ax^2 - 2</math> and <math>y=4 - bx^2</math> intersect the coordinate axes in exactly four points, and these four points are the vertices of a kite of area <math>12</math>. What is <math>a+b</math>?<br />
<br />
<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 1.5\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 2.5\qquad\textbf{(E)}\ 3</math><br />
<br />
[[2015 AMC 12A Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
<br />
A league with 12 teams holds a round-robin tournament, with each team playing every other team exactly once. Games either end with one team victorious or else end in a draw. A team scores 2 points for every game it wins and 1 point for every game it draws. Which of the following is NOT a true statement about the list of 12 scores?<br />
<br />
<math> \textbf{(A)}\ \text{There must be an even number of odd scores.}\\<br />
\qquad\textbf{(B)}\ \text{There must be an even number of even scores.}\\<br />
\qquad\textbf{(C)}\ \text{There cannot be two scores of }0\text{.}\\<br />
\qquad\textbf{(D)}\ \text{The sum of the scores must be at least }100\text{.}\\<br />
\qquad\textbf{(E)}\ \text{The highest score must be at least }12\text{.}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
What is the value of <math>a</math> for which <math>\frac{1}{\text{log}_2a} + \frac{1}{\text{log}_3a} + \frac{1}{\text{log}_4a} = 1</math>?<br />
<br />
[[2015 AMC 12A Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
What is the minimum number of digits to the right of the decimal point needed to express the fraction <math>\frac{123456789}{2^{26}\cdot 5^4}</math> as a decimal?<br />
<br />
<math> \textbf{(A)}\ 4\qquad\textbf{(B)}\ 22\qquad\textbf{(C)}\ 26\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 104</math><br />
<br />
[[2015 AMC 12A Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
Tetrahedron <math>ABCD</math> has <math>AB=5,AC=3,BC=4,BD=4,AD=3,</math> and <math>CD=\frac{12}{5}\sqrt{2}</math>. What is the volume of the tetrahedron?<br />
<br />
<math> \textbf{(A)}\ 3\sqrt{2}\qquad\textbf{(B)}\ 2\sqrt{5}\qquad\textbf{(C)}\ \frac{24}{5}\qquad\textbf{(D)}\ 3\sqrt{3}\qquad\textbf{(E)}\ \frac{24}{5}\sqrt{2}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?<br />
<br />
<math> \textbf{(A)}\ \frac{47}{256} \qquad\textbf{(B)}\ \frac{3}{16} \qquad\textbf{(C)}\ \frac{49}{256} \qquad\textbf{(D)}\ \frac{25}{128} \qquad\textbf{(E)}\ \frac{51}{256}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
The zeros of the function <math>f(x) = x^2-ax+2a</math> are integers. What is the sum of the possible values of <math>a</math>?<br />
<br />
<math> \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 16 \qquad\textbf{(D)}\ 17 \qquad\textbf{(E)}\ 18</math><br />
<br />
[[2015 AMC 12A Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
For some positive integers <math>p</math>, there is a quadrilateral <math>ABCD</math> with positive integer side lengths, perimeter <math>p</math>, right angles at <math>B</math> and <math>C</math>, <math>AB=2</math>, and <math>CD=AD</math>. How many different values of <math>p<2015</math> are possible?<br />
<br />
<math> \textbf{(A)}\ 30 \qquad\textbf{(B)}\ 31 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}\ 62 \qquad\textbf{(E)}\ 63</math><br />
<br />
[[2015 AMC 12A Problems/Problem 19|Solution]]<br />
<br />
==Problem 20==<br />
<br />
Isosceles triangles <math>T</math> and <math>T'</math> are not congruent but have the same area and the same perimeter. The sides of <math>T</math> have lengths of <math>5,5,</math> and <math>8</math>, while those of <math>T'</math> have lengths of <math>a,a,</math> and <math>b</math>. Which of the following numbers is closest to <math>b</math>?<br />
<br />
<math> \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 8</math><br />
<br />
[[2015 AMC 12A Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
A circle of radius <math>r</math> passes through both foci of, and exactly four points on, the ellipse with equation <math>x^2+16y^2=16</math>. The set of all possible values of <math>r</math> is an interval <math>[a,b)</math>. What is <math>a+b</math>?<br />
<br />
<math> \textbf{(A)}\ 5\sqrt{2}+4 \qquad\textbf{(B)}\ \sqrt{17}+7 \qquad\textbf{(C)}\ 6\sqrt{2}+3 \qquad\textbf{(D)}\ \sqrt{15}+8 \qquad\textbf{(E)}\ 12</math><br />
<br />
[[2015 AMC 12A Problems/Problem 21|Solution]]<br />
<br />
==Problem 22==<br />
<br />
For each positive integer <math>n</math>, let <math>S(n)</math> be the number of sequences of length <math>n</math> consisting solely of the letters <math>A</math> and <math>B</math>, with no more than three <math>A</math>s in a row and no more than three <math>B</math>s in a row. What is the remainder when <math>S(2015)</math> is divided by 12?<br />
<br />
<math> \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 8 \qquad\textbf{(E)}\ 10</math><br />
<br />
[[2015 AMC 12A Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
Let <math>S</math> be a square of side length 1. Two points are chosen independently at random on the sides of <math>S</math>. The probability that the straight-line distance between the points is at least <math>\frac12</math> is <math>\frac{a-b\pi}{c}</math>, where <math>a,b,</math> and <math>c</math> are positive integers and <math>\text{gcd}(a,b,c) = 1</math>. What is <math>a+b+c</math>?<br />
<br />
<math> \textbf{(A)}\ 59 \qquad\textbf{(B)}\ 60 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}\ 62 \qquad\textbf{(E)}\ 63</math><br />
<br />
[[2015 AMC 12A Problems/Problem 23|Solution]]<br />
<br />
==Problem 24==<br />
<br />
Rational numbers <math>a</math> and <math>b</math> are chosen at random among all rational numbers in the interval <math>[0,2)</math> that can be written as fractions <math>\frac{n}{d}</math> where <math>n</math> and <math>d</math> are integers with <math>1 \le d \le 5</math>. What is the probability that<br />
<cmath>(\text{cos}(a\pi)+i\text{sin}(b\pi))^4</cmath><br />
is a real number?<br />
<br />
<math> \textbf{(A)}\ \frac{3}{5} \qquad\textbf{(B)}\ \frac{4}{25} \qquad\textbf{(C)}\ \frac{41}{200} \qquad\textbf{(D)}\ \frac{6}{25} \qquad\textbf{(E)}\ \frac{13}{50}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 24|Solution]]<br />
<br />
==Problem 25==<br />
<br />
A collection of circles in the upper half-plane, all tangent to the <math>x</math>-axis, is constructed in layers as follows. Layer <math>L_0</math> consists of two circles of radii <math>70^2</math> and <math>73^2</math> that are externally tangent. For <math>k\ge1</math>, the circles in <math>\bigcup_{j=0}^{k-1}L_j</math> are ordered according to their points of tangency with the <math>x</math>-axis. For every pair of consecutive circles in this order, a new circle is constructed externally tangent to each of the two circles in the pair. Layer <math>L_k</math> consists of the <math>2^{k-1}</math> circles constructed in this way. Let <math>S=\bigcup_{j=0}^{6}L_j</math>, and for every circle <math>C</math> denote by <math>r(C)</math> its radius. What is<br />
<cmath>\sum_{C\in S} \frac{1}{\sqrt{r(C)}}?</cmath><br />
<br />
<asy><br />
import olympiad;<br />
size(350);<br />
defaultpen(linewidth(0.7));<br />
// define a bunch of arrays and starting points<br />
pair[] coord = new pair[65];<br />
int[] trav = {32,16,8,4,2,1};<br />
coord[0] = (0,73^2); coord[64] = (2*73*70,70^2);<br />
// draw the big circles and the bottom line<br />
path arc1 = arc(coord[0],coord[0].y,260,360);<br />
path arc2 = arc(coord[64],coord[64].y,175,280);<br />
fill((coord[0].x-910,coord[0].y)--arc1--cycle,gray(0.75));<br />
fill((coord[64].x+870,coord[64].y+425)--arc2--cycle,gray(0.75));<br />
draw(arc1^^arc2);<br />
draw((-930,0)--(70^2+73^2+850,0));<br />
// We now apply the findCenter function 63 times to get<br />
// the location of the centers of all 63 constructed circles.<br />
// The complicated array setup ensures that all the circles<br />
// will be taken in the right order<br />
for(int i = 0;i<=5;i=i+1)<br />
{<br />
int skip = trav[i];<br />
for(int k=skip;k<=64 - skip; k = k + 2*skip)<br />
{<br />
pair cent1 = coord[k-skip], cent2 = coord[k+skip];<br />
real r1 = cent1.y, r2 = cent2.y, rn=r1*r2/((sqrt(r1)+sqrt(r2))^2);<br />
real shiftx = cent1.x + sqrt(4*r1*rn);<br />
coord[k] = (shiftx,rn);<br />
}<br />
// Draw the remaining 63 circles<br />
}<br />
for(int i=1;i<=63;i=i+1)<br />
{<br />
filldraw(circle(coord[i],coord[i].y),gray(0.75));<br />
}</asy><br />
<br />
<br />
$ \textbf{(A)}\ \frac{286}{35} \qquad\textbf{(B)}\ \frac{583}{70} \qquad\textbf{(C)}\ \frac{715}{73} \qquad\textbf{(D)}}\ \frac{143}{14} \qquad\textbf{(E)}\ \frac{1573}{146}$<br />
<br />
[[2015 AMC 12A Problems/Problem 25|Solution]]<br />
<br />
== See also ==<br />
* [[AMC Problems and Solutions]]<br />
{{MAA Notice}}</div>Devenwarehttps://artofproblemsolving.com/wiki/index.php?title=2015_AMC_12A_Problems&diff=680262015 AMC 12A Problems2015-03-01T23:30:06Z<p>Devenware: /* Problem 23 */</p>
<hr />
<div>==Problem 1==<br />
<br />
What is the value of <math>(2^0-1+5^2-0)^{-1}\times5?</math><br />
<br />
<math> \textbf{(A)}\ -125\qquad\textbf{(B)}\ -120\qquad\textbf{(C)}\ \frac{1}{5}\qquad\textbf{(D)}\ \frac{5}{24}\qquad\textbf{(E)}\ 25 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
<br />
Two of the three sides of a triangle are 20 and 15. Which of the following numbers is not a possible perimeter of the triangle?<br />
<br />
<math> \textbf{(A)}\ 52\qquad\textbf{(B)}\ 57\qquad\textbf{(C)}\ 62\qquad\textbf{(D)}\ 67\qquad\textbf{(E)}\ 72 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
<br />
Mr. Patrick teaches math to 15 students. He was grading tests and found that when he graded everyone's test except Payton's, the average grade for the class was 80. after he graded Payton's test, the class average became 81. What was Payton's score on the test?<br />
<br />
<math> \textbf{(A)}\ 81\qquad\textbf{(B)}\ 85\qquad\textbf{(C)}\ 91\qquad\textbf{(D)}\ 94\qquad\textbf{(E)}\ 95 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
The sum of two positive numbers is 5 times their difference. What is the ratio of the larger number to the smaller?<br />
<br />
<math> \textbf{(A)}\ \frac54 \qquad\textbf{(B)}\ \frac32 \qquad\textbf{(C)}\ \frac95 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ \frac52 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
Amelia needs to estimate the quantity <math>\frac{a}{b} - c</math>, where <math>a, b,</math> and <math>c</math> are large positive integers. She rounds each of the integers so that the calculation will be easier to do mentally. In which of these situations will her answer necessarily be greater than the exact value of <math>\frac{a}{b} - c</math>?<br />
<br />
<math> \textbf{(A)}\ \text{She rounds all three numbers up.}\\<br />
\qquad\textbf{(B)}\ \text{She rounds } a \text{ and } b \text{ up, and she rounds } c \text{ down.}\\<br />
\qquad\textbf{(C)}\ \text{She rounds } a \text{ and } c \text{ up, and she rounds } b \text{ down.} \\<br />
\qquad\textbf{(D)}\ \text{She rounds } a \text{ up, and she rounds } b \text{ and } c \text{ down.}\\<br />
\qquad\textbf{(E)}\ \text{She rounds } c \text{ up, and she rounds } a \text{ and } b \text{ down.} </math><br />
<br />
[[2015 AMC 12A Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be <math>2 : 1</math>?<br />
<br />
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 8</math><br />
<br />
[[2015 AMC 12A Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
Two right circular cylinders have the same volume. The radius of the second cylinder is <math>10\%</math> more than the radius of the first. What is the relationship between the heights of the two cylinders?<br />
<br />
<math>\textbf{(A)}\ \text{The second height is } 10\% \text{ less than the first.} \\ \textbf{(B)}\ \text{The first height is } 10\% \text{ more than the second.}\\ \textbf{(C)}\ \text{The second height is } 21\% \text{ less than the first.} \\ \textbf{(D)}\ \text{The first height is } 21\% \text{ more than the second.}\\ \textbf{(E)}\ \text{The second height is } 80\% \text{ of the first.}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
The ratio of the length to the width of a rectangle is <math>4</math> : <math>3</math>. If the rectangle has diagonal of length <math>d</math>, then the area may be expressed as <math>kd^2</math> for some constant <math>k</math>. What is <math>k</math>?<br />
<br />
<math> \textbf{(A)}\ \frac27 \qquad\textbf{(B)}\ \frac37 \qquad\textbf{(C)}\ \frac{12}{25} \qquad\textbf{(D)}\ \frac{16}{26} \qquad\textbf{(E)}\ \frac34</math><br />
<br />
[[2015 AMC 12A Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
A box contains 2 red marbles, 2 green marbles, and 2 yellow marbles. Carol takes 2 marbles from the box at random; then Claudia takes 2 of the remaining marbles at random; and then Cheryl takes the last 2 marbles. What is the probability that Cheryl gets 2 marbles of the same color?<br />
<br />
<math> \textbf{(A)}\ \frac{1}{10} \qquad\textbf{(B)}\ \frac16 \qquad\textbf{(C)}\ \frac15 \qquad\textbf{(D)}\ \frac13 \qquad\textbf{(E)}\ \frac12</math><br />
<br />
[[2015 AMC 12A Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
Integers <math>x</math> and <math>y</math> with <math>x>y>0</math> satisfy <math>x+y+xy=80</math>. What is <math>x</math>?<br />
<br />
<math> \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 15 \qquad\textbf{(D)}\ 18 \qquad\textbf{(E)}\ 26</math><br />
<br />
[[2015 AMC 12A Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
On a sheet of paper, Isabella draws a circle of radius <math>2</math>, a circle of radius <math>3</math>, and all possible lines simultaneously tangent to both circles. Isabella notices that she has drawn exactly <math>k \ge 0</math> lines. How many different values of <math>k</math> are possible?<br />
<br />
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6</math><br />
<br />
[[2015 AMC 12A Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
The parabolas <math>y=ax^2 - 2</math> and <math>y=4 - bx^2</math> intersect the coordinate axes in exactly four points, and these four points are the vertices of a kite of area <math>12</math>. What is <math>a+b</math>?<br />
<br />
<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 1.5\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 2.5\qquad\textbf{(E)}\ 3</math><br />
<br />
[[2015 AMC 12A Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
<br />
A league with 12 teams holds a round-robin tournament, with each team playing every other team exactly once. Games either end with one team victorious or else end in a draw. A team scores 2 points for every game it wins and 1 point for every game it draws. Which of the following is NOT a true statement about the list of 12 scores?<br />
<br />
<math> \textbf{(A)}\ \text{There must be an even number of odd scores.}\\<br />
\qquad\textbf{(B)}\ \text{There must be an even number of even scores.}\\<br />
\qquad\textbf{(C)}\ \text{There cannot be two scores of }0\text{.}\\<br />
\qquad\textbf{(D)}\ \text{The sum of the scores must be at least }100\text{.}\\<br />
\qquad\textbf{(E)}\ \text{The highest score must be at least }12\text{.}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
What is the value of <math>a</math> for which <math>\frac{1}{\text{log}_2a} + \frac{1}{\text{log}_3a} + \frac{1}{\text{log}_4a} = 1</math>?<br />
<br />
[[2015 AMC 12A Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
What is the minimum number of digits to the right of the decimal point needed to express the fraction <math>\frac{123456789}{2^{26}\cdot 5^4}</math> as a decimal?<br />
<br />
<math> \textbf{(A)}\ 4\qquad\textbf{(B)}\ 22\qquad\textbf{(C)}\ 26\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 104</math><br />
<br />
[[2015 AMC 12A Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
Tetrahedron <math>ABCD</math> has <math>AB=5,AC=3,BC=4,BD=4,AD=3,</math> and <math>CD=\frac{12}{5}\sqrt{2}</math>. What is the volume of the tetrahedron?<br />
<br />
<math> \textbf{(A)}\ 3\sqrt{2}\qquad\textbf{(B)}\ 2\sqrt{5}\qquad\textbf{(C)}\ \frac{24}{5}\qquad\textbf{(D)}\ 3\sqrt{3}\qquad\textbf{(E)}\ \frac{24}{5}\sqrt{2}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?<br />
<br />
<math> \textbf{(A)}\ \frac{47}{256} \qquad\textbf{(B)}\ \frac{3}{16} \qquad\textbf{(C)}\ \frac{49}{256} \qquad\textbf{(D)}\ \frac{25}{128} \qquad\textbf{(E)}\ \frac{51}{256}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
The zeros of the function <math>f(x) = x^2-ax+2a</math> are integers. What is the sum of the possible values of <math>a</math>?<br />
<br />
<math> \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 16 \qquad\textbf{(D)}\ 17 \qquad\textbf{(E)}\ 18</math><br />
<br />
[[2015 AMC 12A Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
For some positive integers <math>p</math>, there is a quadrilateral <math>ABCD</math> with positive integer side lengths, perimeter <math>p</math>, right angles at <math>B</math> and <math>C</math>, <math>AB=2</math>, and <math>CD=AD</math>. How many different values of <math>p<2015</math> are possible?<br />
<br />
<math> \textbf{(A)}\ 30 \qquad\textbf{(B)}\ 31 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}\ 62 \qquad\textbf{(E)}\ 63</math><br />
<br />
[[2015 AMC 12A Problems/Problem 19|Solution]]<br />
<br />
==Problem 20==<br />
<br />
Isosceles triangles <math>T</math> and <math>T'</math> are not congruent but have the same area and the same perimeter. The sides of <math>T</math> have lengths of <math>5,5,</math> and <math>8</math>, while those of <math>T'</math> have lengths of <math>a,a,</math> and <math>b</math>. Which of the following numbers is closest to <math>b</math>?<br />
<br />
<math> \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 8</math><br />
<br />
[[2015 AMC 12A Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
A circle of radius <math>r</math> passes through both foci of, and exactly four points on, the ellipse with equation <math>x^2+16y^2=16</math>. The set of all possible values of <math>r</math> is an interval <math>[a,b)</math>. What is <math>a+b</math>?<br />
<br />
<math> \textbf{(A)}\ 5\sqrt{2}+4 \qquad\textbf{(B)}\ \sqrt{17}+7 \qquad\textbf{(C)}\ 6\sqrt{2}+3 \qquad\textbf{(D)}\ \sqrt{15}+8 \qquad\textbf{(E)}\ 12</math><br />
<br />
[[2015 AMC 12A Problems/Problem 21|Solution]]<br />
<br />
==Problem 22==<br />
<br />
For each positive integer <math>n</math>, let <math>S(n)</math> be the number of sequences of length <math>n</math> consisting solely of the letters <math>A</math> and <math>B</math>, with no more than three <math>A</math>s in a row and no more than three <math>B</math>s in a row. What is the remainder when <math>S(2015)</math> is divided by 12?<br />
<br />
<math> \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 8 \qquad\textbf{(E)}\ 10</math><br />
<br />
[[2015 AMC 12A Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
Let <math>S</math> be a square of side length 1. Two points are chosen independently at random on the sides of <math>S</math>. The probability that the straight-line distance between the points is at least <math>\frac12</math> is <math>\frac{a-b\pi}{c}</math>, where <math>a,b,</math> and <math>c</math> are positive integers and <math>\text{gcd}(a,b,c) = 1</math>. What is <math>a+b+c</math>?<br />
<br />
<math> \textbf{(A)}\ 59 \qquad\textbf{(B)}\ 60 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}\ 62 \qquad\textbf{(E)}\ 63</math><br />
<br />
[[2015 AMC 12A Problems/Problem 23|Solution]]<br />
<br />
==Problem 24==<br />
<br />
Rational numbers <math>a</math> and <math>b</math> are chosen at random among all rational numbers in the interval <math>[0,2)</math> that can be written as fractions <math>\frac{n}{d}</math> where <math>n</math> and <math>d</math> are integers with <math>1 \le d \le 5</math>. What is the probability that<br />
<cmath>(\text{cos}(a\pi)+i\text{sin}(b\pi))^4</cmath><br />
is a real number?<br />
<br />
<math> \textbf{(A)}\ \frac{3}{5} \qquad\textbf{(B)}\ \frac{4}{25} \qquad\textbf{(C)}\ \frac{41}{200} \qquad\textbf{(D)}}\ \frac{6}{25} \qquad\textbf{(E)}\ \frac{13}{50}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 24|Solution]]<br />
<br />
==Problem 25==<br />
<br />
A collection of circles in the upper half-plane, all tangent to the <math>x</math>-axis, is constructed in layers as follows. Layer <math>L_0</math> consists of two circles of radii <math>70^2</math> and <math>73^2</math> that are externally tangent. For <math>k\ge1</math>, the circles in <math>\bigcup_{j=0}^{k-1}L_j</math> are ordered according to their points of tangency with the <math>x</math>-axis. For every pair of consecutive circles in this order, a new circle is constructed externally tangent to each of the two circles in the pair. Layer <math>L_k</math> consists of the <math>2^{k-1}</math> circles constructed in this way. Let <math>S=\bigcup_{j=0}^{6}L_j</math>, and for every circle <math>C</math> denote by <math>r(C)</math> its radius. What is<br />
<cmath>\sum_{C\in S} \frac{1}{\sqrt{r(C)}}?</cmath><br />
<br />
<asy><br />
import olympiad;<br />
size(350);<br />
defaultpen(linewidth(0.7));<br />
// define a bunch of arrays and starting points<br />
pair[] coord = new pair[65];<br />
int[] trav = {32,16,8,4,2,1};<br />
coord[0] = (0,73^2); coord[64] = (2*73*70,70^2);<br />
// draw the big circles and the bottom line<br />
path arc1 = arc(coord[0],coord[0].y,260,360);<br />
path arc2 = arc(coord[64],coord[64].y,175,280);<br />
fill((coord[0].x-910,coord[0].y)--arc1--cycle,gray(0.75));<br />
fill((coord[64].x+870,coord[64].y+425)--arc2--cycle,gray(0.75));<br />
draw(arc1^^arc2);<br />
draw((-930,0)--(70^2+73^2+850,0));<br />
// We now apply the findCenter function 63 times to get<br />
// the location of the centers of all 63 constructed circles.<br />
// The complicated array setup ensures that all the circles<br />
// will be taken in the right order<br />
for(int i = 0;i<=5;i=i+1)<br />
{<br />
int skip = trav[i];<br />
for(int k=skip;k<=64 - skip; k = k + 2*skip)<br />
{<br />
pair cent1 = coord[k-skip], cent2 = coord[k+skip];<br />
real r1 = cent1.y, r2 = cent2.y, rn=r1*r2/((sqrt(r1)+sqrt(r2))^2);<br />
real shiftx = cent1.x + sqrt(4*r1*rn);<br />
coord[k] = (shiftx,rn);<br />
}<br />
// Draw the remaining 63 circles<br />
}<br />
for(int i=1;i<=63;i=i+1)<br />
{<br />
filldraw(circle(coord[i],coord[i].y),gray(0.75));<br />
}</asy><br />
<br />
<br />
$ \textbf{(A)}\ \frac{286}{35} \qquad\textbf{(B)}\ \frac{583}{70} \qquad\textbf{(C)}\ \frac{715}{73} \qquad\textbf{(D)}}\ \frac{143}{14} \qquad\textbf{(E)}\ \frac{1573}{146}$<br />
<br />
[[2015 AMC 12A Problems/Problem 25|Solution]]<br />
<br />
== See also ==<br />
* [[AMC Problems and Solutions]]<br />
{{MAA Notice}}</div>Devenwarehttps://artofproblemsolving.com/wiki/index.php?title=2015_AMC_12A_Problems&diff=680252015 AMC 12A Problems2015-03-01T23:29:51Z<p>Devenware: /* Problem 22 */</p>
<hr />
<div>==Problem 1==<br />
<br />
What is the value of <math>(2^0-1+5^2-0)^{-1}\times5?</math><br />
<br />
<math> \textbf{(A)}\ -125\qquad\textbf{(B)}\ -120\qquad\textbf{(C)}\ \frac{1}{5}\qquad\textbf{(D)}\ \frac{5}{24}\qquad\textbf{(E)}\ 25 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
<br />
Two of the three sides of a triangle are 20 and 15. Which of the following numbers is not a possible perimeter of the triangle?<br />
<br />
<math> \textbf{(A)}\ 52\qquad\textbf{(B)}\ 57\qquad\textbf{(C)}\ 62\qquad\textbf{(D)}\ 67\qquad\textbf{(E)}\ 72 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
<br />
Mr. Patrick teaches math to 15 students. He was grading tests and found that when he graded everyone's test except Payton's, the average grade for the class was 80. after he graded Payton's test, the class average became 81. What was Payton's score on the test?<br />
<br />
<math> \textbf{(A)}\ 81\qquad\textbf{(B)}\ 85\qquad\textbf{(C)}\ 91\qquad\textbf{(D)}\ 94\qquad\textbf{(E)}\ 95 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
The sum of two positive numbers is 5 times their difference. What is the ratio of the larger number to the smaller?<br />
<br />
<math> \textbf{(A)}\ \frac54 \qquad\textbf{(B)}\ \frac32 \qquad\textbf{(C)}\ \frac95 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ \frac52 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
Amelia needs to estimate the quantity <math>\frac{a}{b} - c</math>, where <math>a, b,</math> and <math>c</math> are large positive integers. She rounds each of the integers so that the calculation will be easier to do mentally. In which of these situations will her answer necessarily be greater than the exact value of <math>\frac{a}{b} - c</math>?<br />
<br />
<math> \textbf{(A)}\ \text{She rounds all three numbers up.}\\<br />
\qquad\textbf{(B)}\ \text{She rounds } a \text{ and } b \text{ up, and she rounds } c \text{ down.}\\<br />
\qquad\textbf{(C)}\ \text{She rounds } a \text{ and } c \text{ up, and she rounds } b \text{ down.} \\<br />
\qquad\textbf{(D)}\ \text{She rounds } a \text{ up, and she rounds } b \text{ and } c \text{ down.}\\<br />
\qquad\textbf{(E)}\ \text{She rounds } c \text{ up, and she rounds } a \text{ and } b \text{ down.} </math><br />
<br />
[[2015 AMC 12A Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be <math>2 : 1</math>?<br />
<br />
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 8</math><br />
<br />
[[2015 AMC 12A Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
Two right circular cylinders have the same volume. The radius of the second cylinder is <math>10\%</math> more than the radius of the first. What is the relationship between the heights of the two cylinders?<br />
<br />
<math>\textbf{(A)}\ \text{The second height is } 10\% \text{ less than the first.} \\ \textbf{(B)}\ \text{The first height is } 10\% \text{ more than the second.}\\ \textbf{(C)}\ \text{The second height is } 21\% \text{ less than the first.} \\ \textbf{(D)}\ \text{The first height is } 21\% \text{ more than the second.}\\ \textbf{(E)}\ \text{The second height is } 80\% \text{ of the first.}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
The ratio of the length to the width of a rectangle is <math>4</math> : <math>3</math>. If the rectangle has diagonal of length <math>d</math>, then the area may be expressed as <math>kd^2</math> for some constant <math>k</math>. What is <math>k</math>?<br />
<br />
<math> \textbf{(A)}\ \frac27 \qquad\textbf{(B)}\ \frac37 \qquad\textbf{(C)}\ \frac{12}{25} \qquad\textbf{(D)}\ \frac{16}{26} \qquad\textbf{(E)}\ \frac34</math><br />
<br />
[[2015 AMC 12A Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
A box contains 2 red marbles, 2 green marbles, and 2 yellow marbles. Carol takes 2 marbles from the box at random; then Claudia takes 2 of the remaining marbles at random; and then Cheryl takes the last 2 marbles. What is the probability that Cheryl gets 2 marbles of the same color?<br />
<br />
<math> \textbf{(A)}\ \frac{1}{10} \qquad\textbf{(B)}\ \frac16 \qquad\textbf{(C)}\ \frac15 \qquad\textbf{(D)}\ \frac13 \qquad\textbf{(E)}\ \frac12</math><br />
<br />
[[2015 AMC 12A Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
Integers <math>x</math> and <math>y</math> with <math>x>y>0</math> satisfy <math>x+y+xy=80</math>. What is <math>x</math>?<br />
<br />
<math> \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 15 \qquad\textbf{(D)}\ 18 \qquad\textbf{(E)}\ 26</math><br />
<br />
[[2015 AMC 12A Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
On a sheet of paper, Isabella draws a circle of radius <math>2</math>, a circle of radius <math>3</math>, and all possible lines simultaneously tangent to both circles. Isabella notices that she has drawn exactly <math>k \ge 0</math> lines. How many different values of <math>k</math> are possible?<br />
<br />
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6</math><br />
<br />
[[2015 AMC 12A Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
The parabolas <math>y=ax^2 - 2</math> and <math>y=4 - bx^2</math> intersect the coordinate axes in exactly four points, and these four points are the vertices of a kite of area <math>12</math>. What is <math>a+b</math>?<br />
<br />
<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 1.5\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 2.5\qquad\textbf{(E)}\ 3</math><br />
<br />
[[2015 AMC 12A Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
<br />
A league with 12 teams holds a round-robin tournament, with each team playing every other team exactly once. Games either end with one team victorious or else end in a draw. A team scores 2 points for every game it wins and 1 point for every game it draws. Which of the following is NOT a true statement about the list of 12 scores?<br />
<br />
<math> \textbf{(A)}\ \text{There must be an even number of odd scores.}\\<br />
\qquad\textbf{(B)}\ \text{There must be an even number of even scores.}\\<br />
\qquad\textbf{(C)}\ \text{There cannot be two scores of }0\text{.}\\<br />
\qquad\textbf{(D)}\ \text{The sum of the scores must be at least }100\text{.}\\<br />
\qquad\textbf{(E)}\ \text{The highest score must be at least }12\text{.}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
What is the value of <math>a</math> for which <math>\frac{1}{\text{log}_2a} + \frac{1}{\text{log}_3a} + \frac{1}{\text{log}_4a} = 1</math>?<br />
<br />
[[2015 AMC 12A Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
What is the minimum number of digits to the right of the decimal point needed to express the fraction <math>\frac{123456789}{2^{26}\cdot 5^4}</math> as a decimal?<br />
<br />
<math> \textbf{(A)}\ 4\qquad\textbf{(B)}\ 22\qquad\textbf{(C)}\ 26\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 104</math><br />
<br />
[[2015 AMC 12A Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
Tetrahedron <math>ABCD</math> has <math>AB=5,AC=3,BC=4,BD=4,AD=3,</math> and <math>CD=\frac{12}{5}\sqrt{2}</math>. What is the volume of the tetrahedron?<br />
<br />
<math> \textbf{(A)}\ 3\sqrt{2}\qquad\textbf{(B)}\ 2\sqrt{5}\qquad\textbf{(C)}\ \frac{24}{5}\qquad\textbf{(D)}\ 3\sqrt{3}\qquad\textbf{(E)}\ \frac{24}{5}\sqrt{2}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?<br />
<br />
<math> \textbf{(A)}\ \frac{47}{256} \qquad\textbf{(B)}\ \frac{3}{16} \qquad\textbf{(C)}\ \frac{49}{256} \qquad\textbf{(D)}\ \frac{25}{128} \qquad\textbf{(E)}\ \frac{51}{256}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
The zeros of the function <math>f(x) = x^2-ax+2a</math> are integers. What is the sum of the possible values of <math>a</math>?<br />
<br />
<math> \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 16 \qquad\textbf{(D)}\ 17 \qquad\textbf{(E)}\ 18</math><br />
<br />
[[2015 AMC 12A Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
For some positive integers <math>p</math>, there is a quadrilateral <math>ABCD</math> with positive integer side lengths, perimeter <math>p</math>, right angles at <math>B</math> and <math>C</math>, <math>AB=2</math>, and <math>CD=AD</math>. How many different values of <math>p<2015</math> are possible?<br />
<br />
<math> \textbf{(A)}\ 30 \qquad\textbf{(B)}\ 31 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}\ 62 \qquad\textbf{(E)}\ 63</math><br />
<br />
[[2015 AMC 12A Problems/Problem 19|Solution]]<br />
<br />
==Problem 20==<br />
<br />
Isosceles triangles <math>T</math> and <math>T'</math> are not congruent but have the same area and the same perimeter. The sides of <math>T</math> have lengths of <math>5,5,</math> and <math>8</math>, while those of <math>T'</math> have lengths of <math>a,a,</math> and <math>b</math>. Which of the following numbers is closest to <math>b</math>?<br />
<br />
<math> \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 8</math><br />
<br />
[[2015 AMC 12A Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
A circle of radius <math>r</math> passes through both foci of, and exactly four points on, the ellipse with equation <math>x^2+16y^2=16</math>. The set of all possible values of <math>r</math> is an interval <math>[a,b)</math>. What is <math>a+b</math>?<br />
<br />
<math> \textbf{(A)}\ 5\sqrt{2}+4 \qquad\textbf{(B)}\ \sqrt{17}+7 \qquad\textbf{(C)}\ 6\sqrt{2}+3 \qquad\textbf{(D)}\ \sqrt{15}+8 \qquad\textbf{(E)}\ 12</math><br />
<br />
[[2015 AMC 12A Problems/Problem 21|Solution]]<br />
<br />
==Problem 22==<br />
<br />
For each positive integer <math>n</math>, let <math>S(n)</math> be the number of sequences of length <math>n</math> consisting solely of the letters <math>A</math> and <math>B</math>, with no more than three <math>A</math>s in a row and no more than three <math>B</math>s in a row. What is the remainder when <math>S(2015)</math> is divided by 12?<br />
<br />
<math> \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 8 \qquad\textbf{(E)}\ 10</math><br />
<br />
[[2015 AMC 12A Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
Let <math>S</math> be a square of side length 1. Two points are chosen independently at random on the sides of <math>S</math>. The probability that the straight-line distance between the points is at least <math>\frac12</math> is <math>\frac{a-b\pi}{c}</math>, where <math>a,b,</math> and <math>c</math> are positive integers and <math>\text{gcd}(a,b,c) = 1</math>. What is <math>a+b+c</math>?<br />
<br />
<math> \textbf{(A)}\ 59 \qquad\textbf{(B)}\ 60 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}}\ 62 \qquad\textbf{(E)}\ 63</math><br />
<br />
[[2015 AMC 12A Problems/Problem 23|Solution]]<br />
<br />
==Problem 24==<br />
<br />
Rational numbers <math>a</math> and <math>b</math> are chosen at random among all rational numbers in the interval <math>[0,2)</math> that can be written as fractions <math>\frac{n}{d}</math> where <math>n</math> and <math>d</math> are integers with <math>1 \le d \le 5</math>. What is the probability that<br />
<cmath>(\text{cos}(a\pi)+i\text{sin}(b\pi))^4</cmath><br />
is a real number?<br />
<br />
<math> \textbf{(A)}\ \frac{3}{5} \qquad\textbf{(B)}\ \frac{4}{25} \qquad\textbf{(C)}\ \frac{41}{200} \qquad\textbf{(D)}}\ \frac{6}{25} \qquad\textbf{(E)}\ \frac{13}{50}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 24|Solution]]<br />
<br />
==Problem 25==<br />
<br />
A collection of circles in the upper half-plane, all tangent to the <math>x</math>-axis, is constructed in layers as follows. Layer <math>L_0</math> consists of two circles of radii <math>70^2</math> and <math>73^2</math> that are externally tangent. For <math>k\ge1</math>, the circles in <math>\bigcup_{j=0}^{k-1}L_j</math> are ordered according to their points of tangency with the <math>x</math>-axis. For every pair of consecutive circles in this order, a new circle is constructed externally tangent to each of the two circles in the pair. Layer <math>L_k</math> consists of the <math>2^{k-1}</math> circles constructed in this way. Let <math>S=\bigcup_{j=0}^{6}L_j</math>, and for every circle <math>C</math> denote by <math>r(C)</math> its radius. What is<br />
<cmath>\sum_{C\in S} \frac{1}{\sqrt{r(C)}}?</cmath><br />
<br />
<asy><br />
import olympiad;<br />
size(350);<br />
defaultpen(linewidth(0.7));<br />
// define a bunch of arrays and starting points<br />
pair[] coord = new pair[65];<br />
int[] trav = {32,16,8,4,2,1};<br />
coord[0] = (0,73^2); coord[64] = (2*73*70,70^2);<br />
// draw the big circles and the bottom line<br />
path arc1 = arc(coord[0],coord[0].y,260,360);<br />
path arc2 = arc(coord[64],coord[64].y,175,280);<br />
fill((coord[0].x-910,coord[0].y)--arc1--cycle,gray(0.75));<br />
fill((coord[64].x+870,coord[64].y+425)--arc2--cycle,gray(0.75));<br />
draw(arc1^^arc2);<br />
draw((-930,0)--(70^2+73^2+850,0));<br />
// We now apply the findCenter function 63 times to get<br />
// the location of the centers of all 63 constructed circles.<br />
// The complicated array setup ensures that all the circles<br />
// will be taken in the right order<br />
for(int i = 0;i<=5;i=i+1)<br />
{<br />
int skip = trav[i];<br />
for(int k=skip;k<=64 - skip; k = k + 2*skip)<br />
{<br />
pair cent1 = coord[k-skip], cent2 = coord[k+skip];<br />
real r1 = cent1.y, r2 = cent2.y, rn=r1*r2/((sqrt(r1)+sqrt(r2))^2);<br />
real shiftx = cent1.x + sqrt(4*r1*rn);<br />
coord[k] = (shiftx,rn);<br />
}<br />
// Draw the remaining 63 circles<br />
}<br />
for(int i=1;i<=63;i=i+1)<br />
{<br />
filldraw(circle(coord[i],coord[i].y),gray(0.75));<br />
}</asy><br />
<br />
<br />
$ \textbf{(A)}\ \frac{286}{35} \qquad\textbf{(B)}\ \frac{583}{70} \qquad\textbf{(C)}\ \frac{715}{73} \qquad\textbf{(D)}}\ \frac{143}{14} \qquad\textbf{(E)}\ \frac{1573}{146}$<br />
<br />
[[2015 AMC 12A Problems/Problem 25|Solution]]<br />
<br />
== See also ==<br />
* [[AMC Problems and Solutions]]<br />
{{MAA Notice}}</div>Devenwarehttps://artofproblemsolving.com/wiki/index.php?title=2015_AMC_12A_Problems&diff=680242015 AMC 12A Problems2015-03-01T23:29:29Z<p>Devenware: /* Problem 21 */</p>
<hr />
<div>==Problem 1==<br />
<br />
What is the value of <math>(2^0-1+5^2-0)^{-1}\times5?</math><br />
<br />
<math> \textbf{(A)}\ -125\qquad\textbf{(B)}\ -120\qquad\textbf{(C)}\ \frac{1}{5}\qquad\textbf{(D)}\ \frac{5}{24}\qquad\textbf{(E)}\ 25 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
<br />
Two of the three sides of a triangle are 20 and 15. Which of the following numbers is not a possible perimeter of the triangle?<br />
<br />
<math> \textbf{(A)}\ 52\qquad\textbf{(B)}\ 57\qquad\textbf{(C)}\ 62\qquad\textbf{(D)}\ 67\qquad\textbf{(E)}\ 72 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
<br />
Mr. Patrick teaches math to 15 students. He was grading tests and found that when he graded everyone's test except Payton's, the average grade for the class was 80. after he graded Payton's test, the class average became 81. What was Payton's score on the test?<br />
<br />
<math> \textbf{(A)}\ 81\qquad\textbf{(B)}\ 85\qquad\textbf{(C)}\ 91\qquad\textbf{(D)}\ 94\qquad\textbf{(E)}\ 95 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
The sum of two positive numbers is 5 times their difference. What is the ratio of the larger number to the smaller?<br />
<br />
<math> \textbf{(A)}\ \frac54 \qquad\textbf{(B)}\ \frac32 \qquad\textbf{(C)}\ \frac95 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ \frac52 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
Amelia needs to estimate the quantity <math>\frac{a}{b} - c</math>, where <math>a, b,</math> and <math>c</math> are large positive integers. She rounds each of the integers so that the calculation will be easier to do mentally. In which of these situations will her answer necessarily be greater than the exact value of <math>\frac{a}{b} - c</math>?<br />
<br />
<math> \textbf{(A)}\ \text{She rounds all three numbers up.}\\<br />
\qquad\textbf{(B)}\ \text{She rounds } a \text{ and } b \text{ up, and she rounds } c \text{ down.}\\<br />
\qquad\textbf{(C)}\ \text{She rounds } a \text{ and } c \text{ up, and she rounds } b \text{ down.} \\<br />
\qquad\textbf{(D)}\ \text{She rounds } a \text{ up, and she rounds } b \text{ and } c \text{ down.}\\<br />
\qquad\textbf{(E)}\ \text{She rounds } c \text{ up, and she rounds } a \text{ and } b \text{ down.} </math><br />
<br />
[[2015 AMC 12A Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be <math>2 : 1</math>?<br />
<br />
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 8</math><br />
<br />
[[2015 AMC 12A Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
Two right circular cylinders have the same volume. The radius of the second cylinder is <math>10\%</math> more than the radius of the first. What is the relationship between the heights of the two cylinders?<br />
<br />
<math>\textbf{(A)}\ \text{The second height is } 10\% \text{ less than the first.} \\ \textbf{(B)}\ \text{The first height is } 10\% \text{ more than the second.}\\ \textbf{(C)}\ \text{The second height is } 21\% \text{ less than the first.} \\ \textbf{(D)}\ \text{The first height is } 21\% \text{ more than the second.}\\ \textbf{(E)}\ \text{The second height is } 80\% \text{ of the first.}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
The ratio of the length to the width of a rectangle is <math>4</math> : <math>3</math>. If the rectangle has diagonal of length <math>d</math>, then the area may be expressed as <math>kd^2</math> for some constant <math>k</math>. What is <math>k</math>?<br />
<br />
<math> \textbf{(A)}\ \frac27 \qquad\textbf{(B)}\ \frac37 \qquad\textbf{(C)}\ \frac{12}{25} \qquad\textbf{(D)}\ \frac{16}{26} \qquad\textbf{(E)}\ \frac34</math><br />
<br />
[[2015 AMC 12A Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
A box contains 2 red marbles, 2 green marbles, and 2 yellow marbles. Carol takes 2 marbles from the box at random; then Claudia takes 2 of the remaining marbles at random; and then Cheryl takes the last 2 marbles. What is the probability that Cheryl gets 2 marbles of the same color?<br />
<br />
<math> \textbf{(A)}\ \frac{1}{10} \qquad\textbf{(B)}\ \frac16 \qquad\textbf{(C)}\ \frac15 \qquad\textbf{(D)}\ \frac13 \qquad\textbf{(E)}\ \frac12</math><br />
<br />
[[2015 AMC 12A Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
Integers <math>x</math> and <math>y</math> with <math>x>y>0</math> satisfy <math>x+y+xy=80</math>. What is <math>x</math>?<br />
<br />
<math> \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 15 \qquad\textbf{(D)}\ 18 \qquad\textbf{(E)}\ 26</math><br />
<br />
[[2015 AMC 12A Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
On a sheet of paper, Isabella draws a circle of radius <math>2</math>, a circle of radius <math>3</math>, and all possible lines simultaneously tangent to both circles. Isabella notices that she has drawn exactly <math>k \ge 0</math> lines. How many different values of <math>k</math> are possible?<br />
<br />
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6</math><br />
<br />
[[2015 AMC 12A Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
The parabolas <math>y=ax^2 - 2</math> and <math>y=4 - bx^2</math> intersect the coordinate axes in exactly four points, and these four points are the vertices of a kite of area <math>12</math>. What is <math>a+b</math>?<br />
<br />
<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 1.5\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 2.5\qquad\textbf{(E)}\ 3</math><br />
<br />
[[2015 AMC 12A Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
<br />
A league with 12 teams holds a round-robin tournament, with each team playing every other team exactly once. Games either end with one team victorious or else end in a draw. A team scores 2 points for every game it wins and 1 point for every game it draws. Which of the following is NOT a true statement about the list of 12 scores?<br />
<br />
<math> \textbf{(A)}\ \text{There must be an even number of odd scores.}\\<br />
\qquad\textbf{(B)}\ \text{There must be an even number of even scores.}\\<br />
\qquad\textbf{(C)}\ \text{There cannot be two scores of }0\text{.}\\<br />
\qquad\textbf{(D)}\ \text{The sum of the scores must be at least }100\text{.}\\<br />
\qquad\textbf{(E)}\ \text{The highest score must be at least }12\text{.}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
What is the value of <math>a</math> for which <math>\frac{1}{\text{log}_2a} + \frac{1}{\text{log}_3a} + \frac{1}{\text{log}_4a} = 1</math>?<br />
<br />
[[2015 AMC 12A Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
What is the minimum number of digits to the right of the decimal point needed to express the fraction <math>\frac{123456789}{2^{26}\cdot 5^4}</math> as a decimal?<br />
<br />
<math> \textbf{(A)}\ 4\qquad\textbf{(B)}\ 22\qquad\textbf{(C)}\ 26\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 104</math><br />
<br />
[[2015 AMC 12A Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
Tetrahedron <math>ABCD</math> has <math>AB=5,AC=3,BC=4,BD=4,AD=3,</math> and <math>CD=\frac{12}{5}\sqrt{2}</math>. What is the volume of the tetrahedron?<br />
<br />
<math> \textbf{(A)}\ 3\sqrt{2}\qquad\textbf{(B)}\ 2\sqrt{5}\qquad\textbf{(C)}\ \frac{24}{5}\qquad\textbf{(D)}\ 3\sqrt{3}\qquad\textbf{(E)}\ \frac{24}{5}\sqrt{2}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?<br />
<br />
<math> \textbf{(A)}\ \frac{47}{256} \qquad\textbf{(B)}\ \frac{3}{16} \qquad\textbf{(C)}\ \frac{49}{256} \qquad\textbf{(D)}\ \frac{25}{128} \qquad\textbf{(E)}\ \frac{51}{256}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
The zeros of the function <math>f(x) = x^2-ax+2a</math> are integers. What is the sum of the possible values of <math>a</math>?<br />
<br />
<math> \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 16 \qquad\textbf{(D)}\ 17 \qquad\textbf{(E)}\ 18</math><br />
<br />
[[2015 AMC 12A Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
For some positive integers <math>p</math>, there is a quadrilateral <math>ABCD</math> with positive integer side lengths, perimeter <math>p</math>, right angles at <math>B</math> and <math>C</math>, <math>AB=2</math>, and <math>CD=AD</math>. How many different values of <math>p<2015</math> are possible?<br />
<br />
<math> \textbf{(A)}\ 30 \qquad\textbf{(B)}\ 31 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}\ 62 \qquad\textbf{(E)}\ 63</math><br />
<br />
[[2015 AMC 12A Problems/Problem 19|Solution]]<br />
<br />
==Problem 20==<br />
<br />
Isosceles triangles <math>T</math> and <math>T'</math> are not congruent but have the same area and the same perimeter. The sides of <math>T</math> have lengths of <math>5,5,</math> and <math>8</math>, while those of <math>T'</math> have lengths of <math>a,a,</math> and <math>b</math>. Which of the following numbers is closest to <math>b</math>?<br />
<br />
<math> \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 8</math><br />
<br />
[[2015 AMC 12A Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
A circle of radius <math>r</math> passes through both foci of, and exactly four points on, the ellipse with equation <math>x^2+16y^2=16</math>. The set of all possible values of <math>r</math> is an interval <math>[a,b)</math>. What is <math>a+b</math>?<br />
<br />
<math> \textbf{(A)}\ 5\sqrt{2}+4 \qquad\textbf{(B)}\ \sqrt{17}+7 \qquad\textbf{(C)}\ 6\sqrt{2}+3 \qquad\textbf{(D)}\ \sqrt{15}+8 \qquad\textbf{(E)}\ 12</math><br />
<br />
[[2015 AMC 12A Problems/Problem 21|Solution]]<br />
<br />
==Problem 22==<br />
<br />
For each positive integer <math>n</math>, let <math>S(n)</math> be the number of sequences of length <math>n</math> consisting solely of the letters <math>A</math> and <math>B</math>, with no more than three <math>A</math>s in a row and no more than three <math>B</math>s in a row. What is the remainder when <math>S(2015)</math> is divided by 12?<br />
<br />
<math> \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}}\ 8 \qquad\textbf{(E)}\ 10</math><br />
<br />
[[2015 AMC 12A Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
Let <math>S</math> be a square of side length 1. Two points are chosen independently at random on the sides of <math>S</math>. The probability that the straight-line distance between the points is at least <math>\frac12</math> is <math>\frac{a-b\pi}{c}</math>, where <math>a,b,</math> and <math>c</math> are positive integers and <math>\text{gcd}(a,b,c) = 1</math>. What is <math>a+b+c</math>?<br />
<br />
<math> \textbf{(A)}\ 59 \qquad\textbf{(B)}\ 60 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}}\ 62 \qquad\textbf{(E)}\ 63</math><br />
<br />
[[2015 AMC 12A Problems/Problem 23|Solution]]<br />
<br />
==Problem 24==<br />
<br />
Rational numbers <math>a</math> and <math>b</math> are chosen at random among all rational numbers in the interval <math>[0,2)</math> that can be written as fractions <math>\frac{n}{d}</math> where <math>n</math> and <math>d</math> are integers with <math>1 \le d \le 5</math>. What is the probability that<br />
<cmath>(\text{cos}(a\pi)+i\text{sin}(b\pi))^4</cmath><br />
is a real number?<br />
<br />
<math> \textbf{(A)}\ \frac{3}{5} \qquad\textbf{(B)}\ \frac{4}{25} \qquad\textbf{(C)}\ \frac{41}{200} \qquad\textbf{(D)}}\ \frac{6}{25} \qquad\textbf{(E)}\ \frac{13}{50}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 24|Solution]]<br />
<br />
==Problem 25==<br />
<br />
A collection of circles in the upper half-plane, all tangent to the <math>x</math>-axis, is constructed in layers as follows. Layer <math>L_0</math> consists of two circles of radii <math>70^2</math> and <math>73^2</math> that are externally tangent. For <math>k\ge1</math>, the circles in <math>\bigcup_{j=0}^{k-1}L_j</math> are ordered according to their points of tangency with the <math>x</math>-axis. For every pair of consecutive circles in this order, a new circle is constructed externally tangent to each of the two circles in the pair. Layer <math>L_k</math> consists of the <math>2^{k-1}</math> circles constructed in this way. Let <math>S=\bigcup_{j=0}^{6}L_j</math>, and for every circle <math>C</math> denote by <math>r(C)</math> its radius. What is<br />
<cmath>\sum_{C\in S} \frac{1}{\sqrt{r(C)}}?</cmath><br />
<br />
<asy><br />
import olympiad;<br />
size(350);<br />
defaultpen(linewidth(0.7));<br />
// define a bunch of arrays and starting points<br />
pair[] coord = new pair[65];<br />
int[] trav = {32,16,8,4,2,1};<br />
coord[0] = (0,73^2); coord[64] = (2*73*70,70^2);<br />
// draw the big circles and the bottom line<br />
path arc1 = arc(coord[0],coord[0].y,260,360);<br />
path arc2 = arc(coord[64],coord[64].y,175,280);<br />
fill((coord[0].x-910,coord[0].y)--arc1--cycle,gray(0.75));<br />
fill((coord[64].x+870,coord[64].y+425)--arc2--cycle,gray(0.75));<br />
draw(arc1^^arc2);<br />
draw((-930,0)--(70^2+73^2+850,0));<br />
// We now apply the findCenter function 63 times to get<br />
// the location of the centers of all 63 constructed circles.<br />
// The complicated array setup ensures that all the circles<br />
// will be taken in the right order<br />
for(int i = 0;i<=5;i=i+1)<br />
{<br />
int skip = trav[i];<br />
for(int k=skip;k<=64 - skip; k = k + 2*skip)<br />
{<br />
pair cent1 = coord[k-skip], cent2 = coord[k+skip];<br />
real r1 = cent1.y, r2 = cent2.y, rn=r1*r2/((sqrt(r1)+sqrt(r2))^2);<br />
real shiftx = cent1.x + sqrt(4*r1*rn);<br />
coord[k] = (shiftx,rn);<br />
}<br />
// Draw the remaining 63 circles<br />
}<br />
for(int i=1;i<=63;i=i+1)<br />
{<br />
filldraw(circle(coord[i],coord[i].y),gray(0.75));<br />
}</asy><br />
<br />
<br />
$ \textbf{(A)}\ \frac{286}{35} \qquad\textbf{(B)}\ \frac{583}{70} \qquad\textbf{(C)}\ \frac{715}{73} \qquad\textbf{(D)}}\ \frac{143}{14} \qquad\textbf{(E)}\ \frac{1573}{146}$<br />
<br />
[[2015 AMC 12A Problems/Problem 25|Solution]]<br />
<br />
== See also ==<br />
* [[AMC Problems and Solutions]]<br />
{{MAA Notice}}</div>Devenwarehttps://artofproblemsolving.com/wiki/index.php?title=2015_AMC_12A_Problems&diff=680232015 AMC 12A Problems2015-03-01T23:29:13Z<p>Devenware: /* Problem 20 */</p>
<hr />
<div>==Problem 1==<br />
<br />
What is the value of <math>(2^0-1+5^2-0)^{-1}\times5?</math><br />
<br />
<math> \textbf{(A)}\ -125\qquad\textbf{(B)}\ -120\qquad\textbf{(C)}\ \frac{1}{5}\qquad\textbf{(D)}\ \frac{5}{24}\qquad\textbf{(E)}\ 25 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
<br />
Two of the three sides of a triangle are 20 and 15. Which of the following numbers is not a possible perimeter of the triangle?<br />
<br />
<math> \textbf{(A)}\ 52\qquad\textbf{(B)}\ 57\qquad\textbf{(C)}\ 62\qquad\textbf{(D)}\ 67\qquad\textbf{(E)}\ 72 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
<br />
Mr. Patrick teaches math to 15 students. He was grading tests and found that when he graded everyone's test except Payton's, the average grade for the class was 80. after he graded Payton's test, the class average became 81. What was Payton's score on the test?<br />
<br />
<math> \textbf{(A)}\ 81\qquad\textbf{(B)}\ 85\qquad\textbf{(C)}\ 91\qquad\textbf{(D)}\ 94\qquad\textbf{(E)}\ 95 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
The sum of two positive numbers is 5 times their difference. What is the ratio of the larger number to the smaller?<br />
<br />
<math> \textbf{(A)}\ \frac54 \qquad\textbf{(B)}\ \frac32 \qquad\textbf{(C)}\ \frac95 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ \frac52 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
Amelia needs to estimate the quantity <math>\frac{a}{b} - c</math>, where <math>a, b,</math> and <math>c</math> are large positive integers. She rounds each of the integers so that the calculation will be easier to do mentally. In which of these situations will her answer necessarily be greater than the exact value of <math>\frac{a}{b} - c</math>?<br />
<br />
<math> \textbf{(A)}\ \text{She rounds all three numbers up.}\\<br />
\qquad\textbf{(B)}\ \text{She rounds } a \text{ and } b \text{ up, and she rounds } c \text{ down.}\\<br />
\qquad\textbf{(C)}\ \text{She rounds } a \text{ and } c \text{ up, and she rounds } b \text{ down.} \\<br />
\qquad\textbf{(D)}\ \text{She rounds } a \text{ up, and she rounds } b \text{ and } c \text{ down.}\\<br />
\qquad\textbf{(E)}\ \text{She rounds } c \text{ up, and she rounds } a \text{ and } b \text{ down.} </math><br />
<br />
[[2015 AMC 12A Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be <math>2 : 1</math>?<br />
<br />
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 8</math><br />
<br />
[[2015 AMC 12A Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
Two right circular cylinders have the same volume. The radius of the second cylinder is <math>10\%</math> more than the radius of the first. What is the relationship between the heights of the two cylinders?<br />
<br />
<math>\textbf{(A)}\ \text{The second height is } 10\% \text{ less than the first.} \\ \textbf{(B)}\ \text{The first height is } 10\% \text{ more than the second.}\\ \textbf{(C)}\ \text{The second height is } 21\% \text{ less than the first.} \\ \textbf{(D)}\ \text{The first height is } 21\% \text{ more than the second.}\\ \textbf{(E)}\ \text{The second height is } 80\% \text{ of the first.}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
The ratio of the length to the width of a rectangle is <math>4</math> : <math>3</math>. If the rectangle has diagonal of length <math>d</math>, then the area may be expressed as <math>kd^2</math> for some constant <math>k</math>. What is <math>k</math>?<br />
<br />
<math> \textbf{(A)}\ \frac27 \qquad\textbf{(B)}\ \frac37 \qquad\textbf{(C)}\ \frac{12}{25} \qquad\textbf{(D)}\ \frac{16}{26} \qquad\textbf{(E)}\ \frac34</math><br />
<br />
[[2015 AMC 12A Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
A box contains 2 red marbles, 2 green marbles, and 2 yellow marbles. Carol takes 2 marbles from the box at random; then Claudia takes 2 of the remaining marbles at random; and then Cheryl takes the last 2 marbles. What is the probability that Cheryl gets 2 marbles of the same color?<br />
<br />
<math> \textbf{(A)}\ \frac{1}{10} \qquad\textbf{(B)}\ \frac16 \qquad\textbf{(C)}\ \frac15 \qquad\textbf{(D)}\ \frac13 \qquad\textbf{(E)}\ \frac12</math><br />
<br />
[[2015 AMC 12A Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
Integers <math>x</math> and <math>y</math> with <math>x>y>0</math> satisfy <math>x+y+xy=80</math>. What is <math>x</math>?<br />
<br />
<math> \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 15 \qquad\textbf{(D)}\ 18 \qquad\textbf{(E)}\ 26</math><br />
<br />
[[2015 AMC 12A Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
On a sheet of paper, Isabella draws a circle of radius <math>2</math>, a circle of radius <math>3</math>, and all possible lines simultaneously tangent to both circles. Isabella notices that she has drawn exactly <math>k \ge 0</math> lines. How many different values of <math>k</math> are possible?<br />
<br />
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6</math><br />
<br />
[[2015 AMC 12A Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
The parabolas <math>y=ax^2 - 2</math> and <math>y=4 - bx^2</math> intersect the coordinate axes in exactly four points, and these four points are the vertices of a kite of area <math>12</math>. What is <math>a+b</math>?<br />
<br />
<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 1.5\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 2.5\qquad\textbf{(E)}\ 3</math><br />
<br />
[[2015 AMC 12A Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
<br />
A league with 12 teams holds a round-robin tournament, with each team playing every other team exactly once. Games either end with one team victorious or else end in a draw. A team scores 2 points for every game it wins and 1 point for every game it draws. Which of the following is NOT a true statement about the list of 12 scores?<br />
<br />
<math> \textbf{(A)}\ \text{There must be an even number of odd scores.}\\<br />
\qquad\textbf{(B)}\ \text{There must be an even number of even scores.}\\<br />
\qquad\textbf{(C)}\ \text{There cannot be two scores of }0\text{.}\\<br />
\qquad\textbf{(D)}\ \text{The sum of the scores must be at least }100\text{.}\\<br />
\qquad\textbf{(E)}\ \text{The highest score must be at least }12\text{.}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
What is the value of <math>a</math> for which <math>\frac{1}{\text{log}_2a} + \frac{1}{\text{log}_3a} + \frac{1}{\text{log}_4a} = 1</math>?<br />
<br />
[[2015 AMC 12A Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
What is the minimum number of digits to the right of the decimal point needed to express the fraction <math>\frac{123456789}{2^{26}\cdot 5^4}</math> as a decimal?<br />
<br />
<math> \textbf{(A)}\ 4\qquad\textbf{(B)}\ 22\qquad\textbf{(C)}\ 26\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 104</math><br />
<br />
[[2015 AMC 12A Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
Tetrahedron <math>ABCD</math> has <math>AB=5,AC=3,BC=4,BD=4,AD=3,</math> and <math>CD=\frac{12}{5}\sqrt{2}</math>. What is the volume of the tetrahedron?<br />
<br />
<math> \textbf{(A)}\ 3\sqrt{2}\qquad\textbf{(B)}\ 2\sqrt{5}\qquad\textbf{(C)}\ \frac{24}{5}\qquad\textbf{(D)}\ 3\sqrt{3}\qquad\textbf{(E)}\ \frac{24}{5}\sqrt{2}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?<br />
<br />
<math> \textbf{(A)}\ \frac{47}{256} \qquad\textbf{(B)}\ \frac{3}{16} \qquad\textbf{(C)}\ \frac{49}{256} \qquad\textbf{(D)}\ \frac{25}{128} \qquad\textbf{(E)}\ \frac{51}{256}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
The zeros of the function <math>f(x) = x^2-ax+2a</math> are integers. What is the sum of the possible values of <math>a</math>?<br />
<br />
<math> \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 16 \qquad\textbf{(D)}\ 17 \qquad\textbf{(E)}\ 18</math><br />
<br />
[[2015 AMC 12A Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
For some positive integers <math>p</math>, there is a quadrilateral <math>ABCD</math> with positive integer side lengths, perimeter <math>p</math>, right angles at <math>B</math> and <math>C</math>, <math>AB=2</math>, and <math>CD=AD</math>. How many different values of <math>p<2015</math> are possible?<br />
<br />
<math> \textbf{(A)}\ 30 \qquad\textbf{(B)}\ 31 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}\ 62 \qquad\textbf{(E)}\ 63</math><br />
<br />
[[2015 AMC 12A Problems/Problem 19|Solution]]<br />
<br />
==Problem 20==<br />
<br />
Isosceles triangles <math>T</math> and <math>T'</math> are not congruent but have the same area and the same perimeter. The sides of <math>T</math> have lengths of <math>5,5,</math> and <math>8</math>, while those of <math>T'</math> have lengths of <math>a,a,</math> and <math>b</math>. Which of the following numbers is closest to <math>b</math>?<br />
<br />
<math> \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 8</math><br />
<br />
[[2015 AMC 12A Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
A circle of radius <math>r</math> passes through both foci of, and exactly four points on, the ellipse with equation <math>x^2+16y^2=16</math>. The set of all possible values of <math>r</math> is an interval <math>[a,b)</math>. What is <math>a+b</math>?<br />
<br />
<math> \textbf{(A)}\ 5\sqrt{2}+4 \qquad\textbf{(B)}\ \sqrt{17}+7 \qquad\textbf{(C)}\ 6\sqrt{2}+3 \qquad\textbf{(D)}}\ \sqrt{15}+8 \qquad\textbf{(E)}\ 12</math><br />
<br />
[[2015 AMC 12A Problems/Problem 21|Solution]]<br />
<br />
==Problem 22==<br />
<br />
For each positive integer <math>n</math>, let <math>S(n)</math> be the number of sequences of length <math>n</math> consisting solely of the letters <math>A</math> and <math>B</math>, with no more than three <math>A</math>s in a row and no more than three <math>B</math>s in a row. What is the remainder when <math>S(2015)</math> is divided by 12?<br />
<br />
<math> \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}}\ 8 \qquad\textbf{(E)}\ 10</math><br />
<br />
[[2015 AMC 12A Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
Let <math>S</math> be a square of side length 1. Two points are chosen independently at random on the sides of <math>S</math>. The probability that the straight-line distance between the points is at least <math>\frac12</math> is <math>\frac{a-b\pi}{c}</math>, where <math>a,b,</math> and <math>c</math> are positive integers and <math>\text{gcd}(a,b,c) = 1</math>. What is <math>a+b+c</math>?<br />
<br />
<math> \textbf{(A)}\ 59 \qquad\textbf{(B)}\ 60 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}}\ 62 \qquad\textbf{(E)}\ 63</math><br />
<br />
[[2015 AMC 12A Problems/Problem 23|Solution]]<br />
<br />
==Problem 24==<br />
<br />
Rational numbers <math>a</math> and <math>b</math> are chosen at random among all rational numbers in the interval <math>[0,2)</math> that can be written as fractions <math>\frac{n}{d}</math> where <math>n</math> and <math>d</math> are integers with <math>1 \le d \le 5</math>. What is the probability that<br />
<cmath>(\text{cos}(a\pi)+i\text{sin}(b\pi))^4</cmath><br />
is a real number?<br />
<br />
<math> \textbf{(A)}\ \frac{3}{5} \qquad\textbf{(B)}\ \frac{4}{25} \qquad\textbf{(C)}\ \frac{41}{200} \qquad\textbf{(D)}}\ \frac{6}{25} \qquad\textbf{(E)}\ \frac{13}{50}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 24|Solution]]<br />
<br />
==Problem 25==<br />
<br />
A collection of circles in the upper half-plane, all tangent to the <math>x</math>-axis, is constructed in layers as follows. Layer <math>L_0</math> consists of two circles of radii <math>70^2</math> and <math>73^2</math> that are externally tangent. For <math>k\ge1</math>, the circles in <math>\bigcup_{j=0}^{k-1}L_j</math> are ordered according to their points of tangency with the <math>x</math>-axis. For every pair of consecutive circles in this order, a new circle is constructed externally tangent to each of the two circles in the pair. Layer <math>L_k</math> consists of the <math>2^{k-1}</math> circles constructed in this way. Let <math>S=\bigcup_{j=0}^{6}L_j</math>, and for every circle <math>C</math> denote by <math>r(C)</math> its radius. What is<br />
<cmath>\sum_{C\in S} \frac{1}{\sqrt{r(C)}}?</cmath><br />
<br />
<asy><br />
import olympiad;<br />
size(350);<br />
defaultpen(linewidth(0.7));<br />
// define a bunch of arrays and starting points<br />
pair[] coord = new pair[65];<br />
int[] trav = {32,16,8,4,2,1};<br />
coord[0] = (0,73^2); coord[64] = (2*73*70,70^2);<br />
// draw the big circles and the bottom line<br />
path arc1 = arc(coord[0],coord[0].y,260,360);<br />
path arc2 = arc(coord[64],coord[64].y,175,280);<br />
fill((coord[0].x-910,coord[0].y)--arc1--cycle,gray(0.75));<br />
fill((coord[64].x+870,coord[64].y+425)--arc2--cycle,gray(0.75));<br />
draw(arc1^^arc2);<br />
draw((-930,0)--(70^2+73^2+850,0));<br />
// We now apply the findCenter function 63 times to get<br />
// the location of the centers of all 63 constructed circles.<br />
// The complicated array setup ensures that all the circles<br />
// will be taken in the right order<br />
for(int i = 0;i<=5;i=i+1)<br />
{<br />
int skip = trav[i];<br />
for(int k=skip;k<=64 - skip; k = k + 2*skip)<br />
{<br />
pair cent1 = coord[k-skip], cent2 = coord[k+skip];<br />
real r1 = cent1.y, r2 = cent2.y, rn=r1*r2/((sqrt(r1)+sqrt(r2))^2);<br />
real shiftx = cent1.x + sqrt(4*r1*rn);<br />
coord[k] = (shiftx,rn);<br />
}<br />
// Draw the remaining 63 circles<br />
}<br />
for(int i=1;i<=63;i=i+1)<br />
{<br />
filldraw(circle(coord[i],coord[i].y),gray(0.75));<br />
}</asy><br />
<br />
<br />
$ \textbf{(A)}\ \frac{286}{35} \qquad\textbf{(B)}\ \frac{583}{70} \qquad\textbf{(C)}\ \frac{715}{73} \qquad\textbf{(D)}}\ \frac{143}{14} \qquad\textbf{(E)}\ \frac{1573}{146}$<br />
<br />
[[2015 AMC 12A Problems/Problem 25|Solution]]<br />
<br />
== See also ==<br />
* [[AMC Problems and Solutions]]<br />
{{MAA Notice}}</div>Devenwarehttps://artofproblemsolving.com/wiki/index.php?title=2015_AMC_12A_Problems&diff=680212015 AMC 12A Problems2015-03-01T23:28:57Z<p>Devenware: /* Problem 19 */</p>
<hr />
<div>==Problem 1==<br />
<br />
What is the value of <math>(2^0-1+5^2-0)^{-1}\times5?</math><br />
<br />
<math> \textbf{(A)}\ -125\qquad\textbf{(B)}\ -120\qquad\textbf{(C)}\ \frac{1}{5}\qquad\textbf{(D)}\ \frac{5}{24}\qquad\textbf{(E)}\ 25 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
<br />
Two of the three sides of a triangle are 20 and 15. Which of the following numbers is not a possible perimeter of the triangle?<br />
<br />
<math> \textbf{(A)}\ 52\qquad\textbf{(B)}\ 57\qquad\textbf{(C)}\ 62\qquad\textbf{(D)}\ 67\qquad\textbf{(E)}\ 72 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
<br />
Mr. Patrick teaches math to 15 students. He was grading tests and found that when he graded everyone's test except Payton's, the average grade for the class was 80. after he graded Payton's test, the class average became 81. What was Payton's score on the test?<br />
<br />
<math> \textbf{(A)}\ 81\qquad\textbf{(B)}\ 85\qquad\textbf{(C)}\ 91\qquad\textbf{(D)}\ 94\qquad\textbf{(E)}\ 95 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
The sum of two positive numbers is 5 times their difference. What is the ratio of the larger number to the smaller?<br />
<br />
<math> \textbf{(A)}\ \frac54 \qquad\textbf{(B)}\ \frac32 \qquad\textbf{(C)}\ \frac95 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ \frac52 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
Amelia needs to estimate the quantity <math>\frac{a}{b} - c</math>, where <math>a, b,</math> and <math>c</math> are large positive integers. She rounds each of the integers so that the calculation will be easier to do mentally. In which of these situations will her answer necessarily be greater than the exact value of <math>\frac{a}{b} - c</math>?<br />
<br />
<math> \textbf{(A)}\ \text{She rounds all three numbers up.}\\<br />
\qquad\textbf{(B)}\ \text{She rounds } a \text{ and } b \text{ up, and she rounds } c \text{ down.}\\<br />
\qquad\textbf{(C)}\ \text{She rounds } a \text{ and } c \text{ up, and she rounds } b \text{ down.} \\<br />
\qquad\textbf{(D)}\ \text{She rounds } a \text{ up, and she rounds } b \text{ and } c \text{ down.}\\<br />
\qquad\textbf{(E)}\ \text{She rounds } c \text{ up, and she rounds } a \text{ and } b \text{ down.} </math><br />
<br />
[[2015 AMC 12A Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be <math>2 : 1</math>?<br />
<br />
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 8</math><br />
<br />
[[2015 AMC 12A Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
Two right circular cylinders have the same volume. The radius of the second cylinder is <math>10\%</math> more than the radius of the first. What is the relationship between the heights of the two cylinders?<br />
<br />
<math>\textbf{(A)}\ \text{The second height is } 10\% \text{ less than the first.} \\ \textbf{(B)}\ \text{The first height is } 10\% \text{ more than the second.}\\ \textbf{(C)}\ \text{The second height is } 21\% \text{ less than the first.} \\ \textbf{(D)}\ \text{The first height is } 21\% \text{ more than the second.}\\ \textbf{(E)}\ \text{The second height is } 80\% \text{ of the first.}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
The ratio of the length to the width of a rectangle is <math>4</math> : <math>3</math>. If the rectangle has diagonal of length <math>d</math>, then the area may be expressed as <math>kd^2</math> for some constant <math>k</math>. What is <math>k</math>?<br />
<br />
<math> \textbf{(A)}\ \frac27 \qquad\textbf{(B)}\ \frac37 \qquad\textbf{(C)}\ \frac{12}{25} \qquad\textbf{(D)}\ \frac{16}{26} \qquad\textbf{(E)}\ \frac34</math><br />
<br />
[[2015 AMC 12A Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
A box contains 2 red marbles, 2 green marbles, and 2 yellow marbles. Carol takes 2 marbles from the box at random; then Claudia takes 2 of the remaining marbles at random; and then Cheryl takes the last 2 marbles. What is the probability that Cheryl gets 2 marbles of the same color?<br />
<br />
<math> \textbf{(A)}\ \frac{1}{10} \qquad\textbf{(B)}\ \frac16 \qquad\textbf{(C)}\ \frac15 \qquad\textbf{(D)}\ \frac13 \qquad\textbf{(E)}\ \frac12</math><br />
<br />
[[2015 AMC 12A Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
Integers <math>x</math> and <math>y</math> with <math>x>y>0</math> satisfy <math>x+y+xy=80</math>. What is <math>x</math>?<br />
<br />
<math> \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 15 \qquad\textbf{(D)}\ 18 \qquad\textbf{(E)}\ 26</math><br />
<br />
[[2015 AMC 12A Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
On a sheet of paper, Isabella draws a circle of radius <math>2</math>, a circle of radius <math>3</math>, and all possible lines simultaneously tangent to both circles. Isabella notices that she has drawn exactly <math>k \ge 0</math> lines. How many different values of <math>k</math> are possible?<br />
<br />
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6</math><br />
<br />
[[2015 AMC 12A Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
The parabolas <math>y=ax^2 - 2</math> and <math>y=4 - bx^2</math> intersect the coordinate axes in exactly four points, and these four points are the vertices of a kite of area <math>12</math>. What is <math>a+b</math>?<br />
<br />
<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 1.5\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 2.5\qquad\textbf{(E)}\ 3</math><br />
<br />
[[2015 AMC 12A Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
<br />
A league with 12 teams holds a round-robin tournament, with each team playing every other team exactly once. Games either end with one team victorious or else end in a draw. A team scores 2 points for every game it wins and 1 point for every game it draws. Which of the following is NOT a true statement about the list of 12 scores?<br />
<br />
<math> \textbf{(A)}\ \text{There must be an even number of odd scores.}\\<br />
\qquad\textbf{(B)}\ \text{There must be an even number of even scores.}\\<br />
\qquad\textbf{(C)}\ \text{There cannot be two scores of }0\text{.}\\<br />
\qquad\textbf{(D)}\ \text{The sum of the scores must be at least }100\text{.}\\<br />
\qquad\textbf{(E)}\ \text{The highest score must be at least }12\text{.}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
What is the value of <math>a</math> for which <math>\frac{1}{\text{log}_2a} + \frac{1}{\text{log}_3a} + \frac{1}{\text{log}_4a} = 1</math>?<br />
<br />
[[2015 AMC 12A Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
What is the minimum number of digits to the right of the decimal point needed to express the fraction <math>\frac{123456789}{2^{26}\cdot 5^4}</math> as a decimal?<br />
<br />
<math> \textbf{(A)}\ 4\qquad\textbf{(B)}\ 22\qquad\textbf{(C)}\ 26\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 104</math><br />
<br />
[[2015 AMC 12A Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
Tetrahedron <math>ABCD</math> has <math>AB=5,AC=3,BC=4,BD=4,AD=3,</math> and <math>CD=\frac{12}{5}\sqrt{2}</math>. What is the volume of the tetrahedron?<br />
<br />
<math> \textbf{(A)}\ 3\sqrt{2}\qquad\textbf{(B)}\ 2\sqrt{5}\qquad\textbf{(C)}\ \frac{24}{5}\qquad\textbf{(D)}\ 3\sqrt{3}\qquad\textbf{(E)}\ \frac{24}{5}\sqrt{2}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?<br />
<br />
<math> \textbf{(A)}\ \frac{47}{256} \qquad\textbf{(B)}\ \frac{3}{16} \qquad\textbf{(C)}\ \frac{49}{256} \qquad\textbf{(D)}\ \frac{25}{128} \qquad\textbf{(E)}\ \frac{51}{256}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
The zeros of the function <math>f(x) = x^2-ax+2a</math> are integers. What is the sum of the possible values of <math>a</math>?<br />
<br />
<math> \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 16 \qquad\textbf{(D)}\ 17 \qquad\textbf{(E)}\ 18</math><br />
<br />
[[2015 AMC 12A Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
For some positive integers <math>p</math>, there is a quadrilateral <math>ABCD</math> with positive integer side lengths, perimeter <math>p</math>, right angles at <math>B</math> and <math>C</math>, <math>AB=2</math>, and <math>CD=AD</math>. How many different values of <math>p<2015</math> are possible?<br />
<br />
<math> \textbf{(A)}\ 30 \qquad\textbf{(B)}\ 31 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}\ 62 \qquad\textbf{(E)}\ 63</math><br />
<br />
[[2015 AMC 12A Problems/Problem 19|Solution]]<br />
<br />
==Problem 20==<br />
<br />
Isosceles triangles <math>T</math> and <math>T'</math> are not congruent but have the same area and the same perimeter. The sides of <math>T</math> have lengths of <math>5,5,</math> and <math>8</math>, while those of <math>T'</math> have lengths of <math>a,a,</math> and <math>b</math>. Which of the following numbers is closest to <math>b</math>?<br />
<br />
<math> \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}}\ 6 \qquad\textbf{(E)}\ 8</math><br />
<br />
[[2015 AMC 12A Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
A circle of radius <math>r</math> passes through both foci of, and exactly four points on, the ellipse with equation <math>x^2+16y^2=16</math>. The set of all possible values of <math>r</math> is an interval <math>[a,b)</math>. What is <math>a+b</math>?<br />
<br />
<math> \textbf{(A)}\ 5\sqrt{2}+4 \qquad\textbf{(B)}\ \sqrt{17}+7 \qquad\textbf{(C)}\ 6\sqrt{2}+3 \qquad\textbf{(D)}}\ \sqrt{15}+8 \qquad\textbf{(E)}\ 12</math><br />
<br />
[[2015 AMC 12A Problems/Problem 21|Solution]]<br />
<br />
==Problem 22==<br />
<br />
For each positive integer <math>n</math>, let <math>S(n)</math> be the number of sequences of length <math>n</math> consisting solely of the letters <math>A</math> and <math>B</math>, with no more than three <math>A</math>s in a row and no more than three <math>B</math>s in a row. What is the remainder when <math>S(2015)</math> is divided by 12?<br />
<br />
<math> \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}}\ 8 \qquad\textbf{(E)}\ 10</math><br />
<br />
[[2015 AMC 12A Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
Let <math>S</math> be a square of side length 1. Two points are chosen independently at random on the sides of <math>S</math>. The probability that the straight-line distance between the points is at least <math>\frac12</math> is <math>\frac{a-b\pi}{c}</math>, where <math>a,b,</math> and <math>c</math> are positive integers and <math>\text{gcd}(a,b,c) = 1</math>. What is <math>a+b+c</math>?<br />
<br />
<math> \textbf{(A)}\ 59 \qquad\textbf{(B)}\ 60 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}}\ 62 \qquad\textbf{(E)}\ 63</math><br />
<br />
[[2015 AMC 12A Problems/Problem 23|Solution]]<br />
<br />
==Problem 24==<br />
<br />
Rational numbers <math>a</math> and <math>b</math> are chosen at random among all rational numbers in the interval <math>[0,2)</math> that can be written as fractions <math>\frac{n}{d}</math> where <math>n</math> and <math>d</math> are integers with <math>1 \le d \le 5</math>. What is the probability that<br />
<cmath>(\text{cos}(a\pi)+i\text{sin}(b\pi))^4</cmath><br />
is a real number?<br />
<br />
<math> \textbf{(A)}\ \frac{3}{5} \qquad\textbf{(B)}\ \frac{4}{25} \qquad\textbf{(C)}\ \frac{41}{200} \qquad\textbf{(D)}}\ \frac{6}{25} \qquad\textbf{(E)}\ \frac{13}{50}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 24|Solution]]<br />
<br />
==Problem 25==<br />
<br />
A collection of circles in the upper half-plane, all tangent to the <math>x</math>-axis, is constructed in layers as follows. Layer <math>L_0</math> consists of two circles of radii <math>70^2</math> and <math>73^2</math> that are externally tangent. For <math>k\ge1</math>, the circles in <math>\bigcup_{j=0}^{k-1}L_j</math> are ordered according to their points of tangency with the <math>x</math>-axis. For every pair of consecutive circles in this order, a new circle is constructed externally tangent to each of the two circles in the pair. Layer <math>L_k</math> consists of the <math>2^{k-1}</math> circles constructed in this way. Let <math>S=\bigcup_{j=0}^{6}L_j</math>, and for every circle <math>C</math> denote by <math>r(C)</math> its radius. What is<br />
<cmath>\sum_{C\in S} \frac{1}{\sqrt{r(C)}}?</cmath><br />
<br />
<asy><br />
import olympiad;<br />
size(350);<br />
defaultpen(linewidth(0.7));<br />
// define a bunch of arrays and starting points<br />
pair[] coord = new pair[65];<br />
int[] trav = {32,16,8,4,2,1};<br />
coord[0] = (0,73^2); coord[64] = (2*73*70,70^2);<br />
// draw the big circles and the bottom line<br />
path arc1 = arc(coord[0],coord[0].y,260,360);<br />
path arc2 = arc(coord[64],coord[64].y,175,280);<br />
fill((coord[0].x-910,coord[0].y)--arc1--cycle,gray(0.75));<br />
fill((coord[64].x+870,coord[64].y+425)--arc2--cycle,gray(0.75));<br />
draw(arc1^^arc2);<br />
draw((-930,0)--(70^2+73^2+850,0));<br />
// We now apply the findCenter function 63 times to get<br />
// the location of the centers of all 63 constructed circles.<br />
// The complicated array setup ensures that all the circles<br />
// will be taken in the right order<br />
for(int i = 0;i<=5;i=i+1)<br />
{<br />
int skip = trav[i];<br />
for(int k=skip;k<=64 - skip; k = k + 2*skip)<br />
{<br />
pair cent1 = coord[k-skip], cent2 = coord[k+skip];<br />
real r1 = cent1.y, r2 = cent2.y, rn=r1*r2/((sqrt(r1)+sqrt(r2))^2);<br />
real shiftx = cent1.x + sqrt(4*r1*rn);<br />
coord[k] = (shiftx,rn);<br />
}<br />
// Draw the remaining 63 circles<br />
}<br />
for(int i=1;i<=63;i=i+1)<br />
{<br />
filldraw(circle(coord[i],coord[i].y),gray(0.75));<br />
}</asy><br />
<br />
<br />
$ \textbf{(A)}\ \frac{286}{35} \qquad\textbf{(B)}\ \frac{583}{70} \qquad\textbf{(C)}\ \frac{715}{73} \qquad\textbf{(D)}}\ \frac{143}{14} \qquad\textbf{(E)}\ \frac{1573}{146}$<br />
<br />
[[2015 AMC 12A Problems/Problem 25|Solution]]<br />
<br />
== See also ==<br />
* [[AMC Problems and Solutions]]<br />
{{MAA Notice}}</div>Devenwarehttps://artofproblemsolving.com/wiki/index.php?title=2015_AMC_12A_Problems&diff=680202015 AMC 12A Problems2015-03-01T23:28:38Z<p>Devenware: /* Problem 18 */</p>
<hr />
<div>==Problem 1==<br />
<br />
What is the value of <math>(2^0-1+5^2-0)^{-1}\times5?</math><br />
<br />
<math> \textbf{(A)}\ -125\qquad\textbf{(B)}\ -120\qquad\textbf{(C)}\ \frac{1}{5}\qquad\textbf{(D)}\ \frac{5}{24}\qquad\textbf{(E)}\ 25 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
<br />
Two of the three sides of a triangle are 20 and 15. Which of the following numbers is not a possible perimeter of the triangle?<br />
<br />
<math> \textbf{(A)}\ 52\qquad\textbf{(B)}\ 57\qquad\textbf{(C)}\ 62\qquad\textbf{(D)}\ 67\qquad\textbf{(E)}\ 72 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
<br />
Mr. Patrick teaches math to 15 students. He was grading tests and found that when he graded everyone's test except Payton's, the average grade for the class was 80. after he graded Payton's test, the class average became 81. What was Payton's score on the test?<br />
<br />
<math> \textbf{(A)}\ 81\qquad\textbf{(B)}\ 85\qquad\textbf{(C)}\ 91\qquad\textbf{(D)}\ 94\qquad\textbf{(E)}\ 95 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
The sum of two positive numbers is 5 times their difference. What is the ratio of the larger number to the smaller?<br />
<br />
<math> \textbf{(A)}\ \frac54 \qquad\textbf{(B)}\ \frac32 \qquad\textbf{(C)}\ \frac95 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ \frac52 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
Amelia needs to estimate the quantity <math>\frac{a}{b} - c</math>, where <math>a, b,</math> and <math>c</math> are large positive integers. She rounds each of the integers so that the calculation will be easier to do mentally. In which of these situations will her answer necessarily be greater than the exact value of <math>\frac{a}{b} - c</math>?<br />
<br />
<math> \textbf{(A)}\ \text{She rounds all three numbers up.}\\<br />
\qquad\textbf{(B)}\ \text{She rounds } a \text{ and } b \text{ up, and she rounds } c \text{ down.}\\<br />
\qquad\textbf{(C)}\ \text{She rounds } a \text{ and } c \text{ up, and she rounds } b \text{ down.} \\<br />
\qquad\textbf{(D)}\ \text{She rounds } a \text{ up, and she rounds } b \text{ and } c \text{ down.}\\<br />
\qquad\textbf{(E)}\ \text{She rounds } c \text{ up, and she rounds } a \text{ and } b \text{ down.} </math><br />
<br />
[[2015 AMC 12A Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be <math>2 : 1</math>?<br />
<br />
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 8</math><br />
<br />
[[2015 AMC 12A Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
Two right circular cylinders have the same volume. The radius of the second cylinder is <math>10\%</math> more than the radius of the first. What is the relationship between the heights of the two cylinders?<br />
<br />
<math>\textbf{(A)}\ \text{The second height is } 10\% \text{ less than the first.} \\ \textbf{(B)}\ \text{The first height is } 10\% \text{ more than the second.}\\ \textbf{(C)}\ \text{The second height is } 21\% \text{ less than the first.} \\ \textbf{(D)}\ \text{The first height is } 21\% \text{ more than the second.}\\ \textbf{(E)}\ \text{The second height is } 80\% \text{ of the first.}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
The ratio of the length to the width of a rectangle is <math>4</math> : <math>3</math>. If the rectangle has diagonal of length <math>d</math>, then the area may be expressed as <math>kd^2</math> for some constant <math>k</math>. What is <math>k</math>?<br />
<br />
<math> \textbf{(A)}\ \frac27 \qquad\textbf{(B)}\ \frac37 \qquad\textbf{(C)}\ \frac{12}{25} \qquad\textbf{(D)}\ \frac{16}{26} \qquad\textbf{(E)}\ \frac34</math><br />
<br />
[[2015 AMC 12A Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
A box contains 2 red marbles, 2 green marbles, and 2 yellow marbles. Carol takes 2 marbles from the box at random; then Claudia takes 2 of the remaining marbles at random; and then Cheryl takes the last 2 marbles. What is the probability that Cheryl gets 2 marbles of the same color?<br />
<br />
<math> \textbf{(A)}\ \frac{1}{10} \qquad\textbf{(B)}\ \frac16 \qquad\textbf{(C)}\ \frac15 \qquad\textbf{(D)}\ \frac13 \qquad\textbf{(E)}\ \frac12</math><br />
<br />
[[2015 AMC 12A Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
Integers <math>x</math> and <math>y</math> with <math>x>y>0</math> satisfy <math>x+y+xy=80</math>. What is <math>x</math>?<br />
<br />
<math> \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 15 \qquad\textbf{(D)}\ 18 \qquad\textbf{(E)}\ 26</math><br />
<br />
[[2015 AMC 12A Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
On a sheet of paper, Isabella draws a circle of radius <math>2</math>, a circle of radius <math>3</math>, and all possible lines simultaneously tangent to both circles. Isabella notices that she has drawn exactly <math>k \ge 0</math> lines. How many different values of <math>k</math> are possible?<br />
<br />
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6</math><br />
<br />
[[2015 AMC 12A Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
The parabolas <math>y=ax^2 - 2</math> and <math>y=4 - bx^2</math> intersect the coordinate axes in exactly four points, and these four points are the vertices of a kite of area <math>12</math>. What is <math>a+b</math>?<br />
<br />
<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 1.5\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 2.5\qquad\textbf{(E)}\ 3</math><br />
<br />
[[2015 AMC 12A Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
<br />
A league with 12 teams holds a round-robin tournament, with each team playing every other team exactly once. Games either end with one team victorious or else end in a draw. A team scores 2 points for every game it wins and 1 point for every game it draws. Which of the following is NOT a true statement about the list of 12 scores?<br />
<br />
<math> \textbf{(A)}\ \text{There must be an even number of odd scores.}\\<br />
\qquad\textbf{(B)}\ \text{There must be an even number of even scores.}\\<br />
\qquad\textbf{(C)}\ \text{There cannot be two scores of }0\text{.}\\<br />
\qquad\textbf{(D)}\ \text{The sum of the scores must be at least }100\text{.}\\<br />
\qquad\textbf{(E)}\ \text{The highest score must be at least }12\text{.}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
What is the value of <math>a</math> for which <math>\frac{1}{\text{log}_2a} + \frac{1}{\text{log}_3a} + \frac{1}{\text{log}_4a} = 1</math>?<br />
<br />
[[2015 AMC 12A Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
What is the minimum number of digits to the right of the decimal point needed to express the fraction <math>\frac{123456789}{2^{26}\cdot 5^4}</math> as a decimal?<br />
<br />
<math> \textbf{(A)}\ 4\qquad\textbf{(B)}\ 22\qquad\textbf{(C)}\ 26\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 104</math><br />
<br />
[[2015 AMC 12A Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
Tetrahedron <math>ABCD</math> has <math>AB=5,AC=3,BC=4,BD=4,AD=3,</math> and <math>CD=\frac{12}{5}\sqrt{2}</math>. What is the volume of the tetrahedron?<br />
<br />
<math> \textbf{(A)}\ 3\sqrt{2}\qquad\textbf{(B)}\ 2\sqrt{5}\qquad\textbf{(C)}\ \frac{24}{5}\qquad\textbf{(D)}\ 3\sqrt{3}\qquad\textbf{(E)}\ \frac{24}{5}\sqrt{2}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?<br />
<br />
<math> \textbf{(A)}\ \frac{47}{256} \qquad\textbf{(B)}\ \frac{3}{16} \qquad\textbf{(C)}\ \frac{49}{256} \qquad\textbf{(D)}\ \frac{25}{128} \qquad\textbf{(E)}\ \frac{51}{256}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
The zeros of the function <math>f(x) = x^2-ax+2a</math> are integers. What is the sum of the possible values of <math>a</math>?<br />
<br />
<math> \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 16 \qquad\textbf{(D)}\ 17 \qquad\textbf{(E)}\ 18</math><br />
<br />
[[2015 AMC 12A Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
For some positive integers <math>p</math>, there is a quadrilateral <math>ABCD</math> with positive integer side lengths, perimeter <math>p</math>, right angles at <math>B</math> and <math>C</math>, <math>AB=2</math>, and <math>CD=AD</math>. How many different values of <math>p<2015</math> are possible?<br />
<br />
<math> \textbf{(A)}\ 30 \qquad\textbf{(B)}\ 31 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}}\ 62 \qquad\textbf{(E)}\ 63</math><br />
<br />
[[2015 AMC 12A Problems/Problem 19|Solution]]<br />
<br />
==Problem 20==<br />
<br />
Isosceles triangles <math>T</math> and <math>T'</math> are not congruent but have the same area and the same perimeter. The sides of <math>T</math> have lengths of <math>5,5,</math> and <math>8</math>, while those of <math>T'</math> have lengths of <math>a,a,</math> and <math>b</math>. Which of the following numbers is closest to <math>b</math>?<br />
<br />
<math> \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}}\ 6 \qquad\textbf{(E)}\ 8</math><br />
<br />
[[2015 AMC 12A Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
A circle of radius <math>r</math> passes through both foci of, and exactly four points on, the ellipse with equation <math>x^2+16y^2=16</math>. The set of all possible values of <math>r</math> is an interval <math>[a,b)</math>. What is <math>a+b</math>?<br />
<br />
<math> \textbf{(A)}\ 5\sqrt{2}+4 \qquad\textbf{(B)}\ \sqrt{17}+7 \qquad\textbf{(C)}\ 6\sqrt{2}+3 \qquad\textbf{(D)}}\ \sqrt{15}+8 \qquad\textbf{(E)}\ 12</math><br />
<br />
[[2015 AMC 12A Problems/Problem 21|Solution]]<br />
<br />
==Problem 22==<br />
<br />
For each positive integer <math>n</math>, let <math>S(n)</math> be the number of sequences of length <math>n</math> consisting solely of the letters <math>A</math> and <math>B</math>, with no more than three <math>A</math>s in a row and no more than three <math>B</math>s in a row. What is the remainder when <math>S(2015)</math> is divided by 12?<br />
<br />
<math> \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}}\ 8 \qquad\textbf{(E)}\ 10</math><br />
<br />
[[2015 AMC 12A Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
Let <math>S</math> be a square of side length 1. Two points are chosen independently at random on the sides of <math>S</math>. The probability that the straight-line distance between the points is at least <math>\frac12</math> is <math>\frac{a-b\pi}{c}</math>, where <math>a,b,</math> and <math>c</math> are positive integers and <math>\text{gcd}(a,b,c) = 1</math>. What is <math>a+b+c</math>?<br />
<br />
<math> \textbf{(A)}\ 59 \qquad\textbf{(B)}\ 60 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}}\ 62 \qquad\textbf{(E)}\ 63</math><br />
<br />
[[2015 AMC 12A Problems/Problem 23|Solution]]<br />
<br />
==Problem 24==<br />
<br />
Rational numbers <math>a</math> and <math>b</math> are chosen at random among all rational numbers in the interval <math>[0,2)</math> that can be written as fractions <math>\frac{n}{d}</math> where <math>n</math> and <math>d</math> are integers with <math>1 \le d \le 5</math>. What is the probability that<br />
<cmath>(\text{cos}(a\pi)+i\text{sin}(b\pi))^4</cmath><br />
is a real number?<br />
<br />
<math> \textbf{(A)}\ \frac{3}{5} \qquad\textbf{(B)}\ \frac{4}{25} \qquad\textbf{(C)}\ \frac{41}{200} \qquad\textbf{(D)}}\ \frac{6}{25} \qquad\textbf{(E)}\ \frac{13}{50}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 24|Solution]]<br />
<br />
==Problem 25==<br />
<br />
A collection of circles in the upper half-plane, all tangent to the <math>x</math>-axis, is constructed in layers as follows. Layer <math>L_0</math> consists of two circles of radii <math>70^2</math> and <math>73^2</math> that are externally tangent. For <math>k\ge1</math>, the circles in <math>\bigcup_{j=0}^{k-1}L_j</math> are ordered according to their points of tangency with the <math>x</math>-axis. For every pair of consecutive circles in this order, a new circle is constructed externally tangent to each of the two circles in the pair. Layer <math>L_k</math> consists of the <math>2^{k-1}</math> circles constructed in this way. Let <math>S=\bigcup_{j=0}^{6}L_j</math>, and for every circle <math>C</math> denote by <math>r(C)</math> its radius. What is<br />
<cmath>\sum_{C\in S} \frac{1}{\sqrt{r(C)}}?</cmath><br />
<br />
<asy><br />
import olympiad;<br />
size(350);<br />
defaultpen(linewidth(0.7));<br />
// define a bunch of arrays and starting points<br />
pair[] coord = new pair[65];<br />
int[] trav = {32,16,8,4,2,1};<br />
coord[0] = (0,73^2); coord[64] = (2*73*70,70^2);<br />
// draw the big circles and the bottom line<br />
path arc1 = arc(coord[0],coord[0].y,260,360);<br />
path arc2 = arc(coord[64],coord[64].y,175,280);<br />
fill((coord[0].x-910,coord[0].y)--arc1--cycle,gray(0.75));<br />
fill((coord[64].x+870,coord[64].y+425)--arc2--cycle,gray(0.75));<br />
draw(arc1^^arc2);<br />
draw((-930,0)--(70^2+73^2+850,0));<br />
// We now apply the findCenter function 63 times to get<br />
// the location of the centers of all 63 constructed circles.<br />
// The complicated array setup ensures that all the circles<br />
// will be taken in the right order<br />
for(int i = 0;i<=5;i=i+1)<br />
{<br />
int skip = trav[i];<br />
for(int k=skip;k<=64 - skip; k = k + 2*skip)<br />
{<br />
pair cent1 = coord[k-skip], cent2 = coord[k+skip];<br />
real r1 = cent1.y, r2 = cent2.y, rn=r1*r2/((sqrt(r1)+sqrt(r2))^2);<br />
real shiftx = cent1.x + sqrt(4*r1*rn);<br />
coord[k] = (shiftx,rn);<br />
}<br />
// Draw the remaining 63 circles<br />
}<br />
for(int i=1;i<=63;i=i+1)<br />
{<br />
filldraw(circle(coord[i],coord[i].y),gray(0.75));<br />
}</asy><br />
<br />
<br />
$ \textbf{(A)}\ \frac{286}{35} \qquad\textbf{(B)}\ \frac{583}{70} \qquad\textbf{(C)}\ \frac{715}{73} \qquad\textbf{(D)}}\ \frac{143}{14} \qquad\textbf{(E)}\ \frac{1573}{146}$<br />
<br />
[[2015 AMC 12A Problems/Problem 25|Solution]]<br />
<br />
== See also ==<br />
* [[AMC Problems and Solutions]]<br />
{{MAA Notice}}</div>Devenwarehttps://artofproblemsolving.com/wiki/index.php?title=2015_AMC_12A_Problems&diff=680192015 AMC 12A Problems2015-03-01T23:28:21Z<p>Devenware: /* Problem 17 */</p>
<hr />
<div>==Problem 1==<br />
<br />
What is the value of <math>(2^0-1+5^2-0)^{-1}\times5?</math><br />
<br />
<math> \textbf{(A)}\ -125\qquad\textbf{(B)}\ -120\qquad\textbf{(C)}\ \frac{1}{5}\qquad\textbf{(D)}\ \frac{5}{24}\qquad\textbf{(E)}\ 25 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
<br />
Two of the three sides of a triangle are 20 and 15. Which of the following numbers is not a possible perimeter of the triangle?<br />
<br />
<math> \textbf{(A)}\ 52\qquad\textbf{(B)}\ 57\qquad\textbf{(C)}\ 62\qquad\textbf{(D)}\ 67\qquad\textbf{(E)}\ 72 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
<br />
Mr. Patrick teaches math to 15 students. He was grading tests and found that when he graded everyone's test except Payton's, the average grade for the class was 80. after he graded Payton's test, the class average became 81. What was Payton's score on the test?<br />
<br />
<math> \textbf{(A)}\ 81\qquad\textbf{(B)}\ 85\qquad\textbf{(C)}\ 91\qquad\textbf{(D)}\ 94\qquad\textbf{(E)}\ 95 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
The sum of two positive numbers is 5 times their difference. What is the ratio of the larger number to the smaller?<br />
<br />
<math> \textbf{(A)}\ \frac54 \qquad\textbf{(B)}\ \frac32 \qquad\textbf{(C)}\ \frac95 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ \frac52 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
Amelia needs to estimate the quantity <math>\frac{a}{b} - c</math>, where <math>a, b,</math> and <math>c</math> are large positive integers. She rounds each of the integers so that the calculation will be easier to do mentally. In which of these situations will her answer necessarily be greater than the exact value of <math>\frac{a}{b} - c</math>?<br />
<br />
<math> \textbf{(A)}\ \text{She rounds all three numbers up.}\\<br />
\qquad\textbf{(B)}\ \text{She rounds } a \text{ and } b \text{ up, and she rounds } c \text{ down.}\\<br />
\qquad\textbf{(C)}\ \text{She rounds } a \text{ and } c \text{ up, and she rounds } b \text{ down.} \\<br />
\qquad\textbf{(D)}\ \text{She rounds } a \text{ up, and she rounds } b \text{ and } c \text{ down.}\\<br />
\qquad\textbf{(E)}\ \text{She rounds } c \text{ up, and she rounds } a \text{ and } b \text{ down.} </math><br />
<br />
[[2015 AMC 12A Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be <math>2 : 1</math>?<br />
<br />
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 8</math><br />
<br />
[[2015 AMC 12A Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
Two right circular cylinders have the same volume. The radius of the second cylinder is <math>10\%</math> more than the radius of the first. What is the relationship between the heights of the two cylinders?<br />
<br />
<math>\textbf{(A)}\ \text{The second height is } 10\% \text{ less than the first.} \\ \textbf{(B)}\ \text{The first height is } 10\% \text{ more than the second.}\\ \textbf{(C)}\ \text{The second height is } 21\% \text{ less than the first.} \\ \textbf{(D)}\ \text{The first height is } 21\% \text{ more than the second.}\\ \textbf{(E)}\ \text{The second height is } 80\% \text{ of the first.}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
The ratio of the length to the width of a rectangle is <math>4</math> : <math>3</math>. If the rectangle has diagonal of length <math>d</math>, then the area may be expressed as <math>kd^2</math> for some constant <math>k</math>. What is <math>k</math>?<br />
<br />
<math> \textbf{(A)}\ \frac27 \qquad\textbf{(B)}\ \frac37 \qquad\textbf{(C)}\ \frac{12}{25} \qquad\textbf{(D)}\ \frac{16}{26} \qquad\textbf{(E)}\ \frac34</math><br />
<br />
[[2015 AMC 12A Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
A box contains 2 red marbles, 2 green marbles, and 2 yellow marbles. Carol takes 2 marbles from the box at random; then Claudia takes 2 of the remaining marbles at random; and then Cheryl takes the last 2 marbles. What is the probability that Cheryl gets 2 marbles of the same color?<br />
<br />
<math> \textbf{(A)}\ \frac{1}{10} \qquad\textbf{(B)}\ \frac16 \qquad\textbf{(C)}\ \frac15 \qquad\textbf{(D)}\ \frac13 \qquad\textbf{(E)}\ \frac12</math><br />
<br />
[[2015 AMC 12A Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
Integers <math>x</math> and <math>y</math> with <math>x>y>0</math> satisfy <math>x+y+xy=80</math>. What is <math>x</math>?<br />
<br />
<math> \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 15 \qquad\textbf{(D)}\ 18 \qquad\textbf{(E)}\ 26</math><br />
<br />
[[2015 AMC 12A Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
On a sheet of paper, Isabella draws a circle of radius <math>2</math>, a circle of radius <math>3</math>, and all possible lines simultaneously tangent to both circles. Isabella notices that she has drawn exactly <math>k \ge 0</math> lines. How many different values of <math>k</math> are possible?<br />
<br />
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6</math><br />
<br />
[[2015 AMC 12A Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
The parabolas <math>y=ax^2 - 2</math> and <math>y=4 - bx^2</math> intersect the coordinate axes in exactly four points, and these four points are the vertices of a kite of area <math>12</math>. What is <math>a+b</math>?<br />
<br />
<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 1.5\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 2.5\qquad\textbf{(E)}\ 3</math><br />
<br />
[[2015 AMC 12A Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
<br />
A league with 12 teams holds a round-robin tournament, with each team playing every other team exactly once. Games either end with one team victorious or else end in a draw. A team scores 2 points for every game it wins and 1 point for every game it draws. Which of the following is NOT a true statement about the list of 12 scores?<br />
<br />
<math> \textbf{(A)}\ \text{There must be an even number of odd scores.}\\<br />
\qquad\textbf{(B)}\ \text{There must be an even number of even scores.}\\<br />
\qquad\textbf{(C)}\ \text{There cannot be two scores of }0\text{.}\\<br />
\qquad\textbf{(D)}\ \text{The sum of the scores must be at least }100\text{.}\\<br />
\qquad\textbf{(E)}\ \text{The highest score must be at least }12\text{.}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
What is the value of <math>a</math> for which <math>\frac{1}{\text{log}_2a} + \frac{1}{\text{log}_3a} + \frac{1}{\text{log}_4a} = 1</math>?<br />
<br />
[[2015 AMC 12A Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
What is the minimum number of digits to the right of the decimal point needed to express the fraction <math>\frac{123456789}{2^{26}\cdot 5^4}</math> as a decimal?<br />
<br />
<math> \textbf{(A)}\ 4\qquad\textbf{(B)}\ 22\qquad\textbf{(C)}\ 26\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 104</math><br />
<br />
[[2015 AMC 12A Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
Tetrahedron <math>ABCD</math> has <math>AB=5,AC=3,BC=4,BD=4,AD=3,</math> and <math>CD=\frac{12}{5}\sqrt{2}</math>. What is the volume of the tetrahedron?<br />
<br />
<math> \textbf{(A)}\ 3\sqrt{2}\qquad\textbf{(B)}\ 2\sqrt{5}\qquad\textbf{(C)}\ \frac{24}{5}\qquad\textbf{(D)}\ 3\sqrt{3}\qquad\textbf{(E)}\ \frac{24}{5}\sqrt{2}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?<br />
<br />
<math> \textbf{(A)}\ \frac{47}{256} \qquad\textbf{(B)}\ \frac{3}{16} \qquad\textbf{(C)}\ \frac{49}{256} \qquad\textbf{(D)}\ \frac{25}{128} \qquad\textbf{(E)}\ \frac{51}{256}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
The zeros of the function <math>f(x) = x^2-ax+2a</math> are integers. What is the sum of the possible values of <math>a</math>?<br />
<br />
<math> \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 16 \qquad\textbf{(D)}}\ 17 \qquad\textbf{(E)}\ 18</math><br />
<br />
[[2015 AMC 12A Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
For some positive integers <math>p</math>, there is a quadrilateral <math>ABCD</math> with positive integer side lengths, perimeter <math>p</math>, right angles at <math>B</math> and <math>C</math>, <math>AB=2</math>, and <math>CD=AD</math>. How many different values of <math>p<2015</math> are possible?<br />
<br />
<math> \textbf{(A)}\ 30 \qquad\textbf{(B)}\ 31 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}}\ 62 \qquad\textbf{(E)}\ 63</math><br />
<br />
[[2015 AMC 12A Problems/Problem 19|Solution]]<br />
<br />
==Problem 20==<br />
<br />
Isosceles triangles <math>T</math> and <math>T'</math> are not congruent but have the same area and the same perimeter. The sides of <math>T</math> have lengths of <math>5,5,</math> and <math>8</math>, while those of <math>T'</math> have lengths of <math>a,a,</math> and <math>b</math>. Which of the following numbers is closest to <math>b</math>?<br />
<br />
<math> \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}}\ 6 \qquad\textbf{(E)}\ 8</math><br />
<br />
[[2015 AMC 12A Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
A circle of radius <math>r</math> passes through both foci of, and exactly four points on, the ellipse with equation <math>x^2+16y^2=16</math>. The set of all possible values of <math>r</math> is an interval <math>[a,b)</math>. What is <math>a+b</math>?<br />
<br />
<math> \textbf{(A)}\ 5\sqrt{2}+4 \qquad\textbf{(B)}\ \sqrt{17}+7 \qquad\textbf{(C)}\ 6\sqrt{2}+3 \qquad\textbf{(D)}}\ \sqrt{15}+8 \qquad\textbf{(E)}\ 12</math><br />
<br />
[[2015 AMC 12A Problems/Problem 21|Solution]]<br />
<br />
==Problem 22==<br />
<br />
For each positive integer <math>n</math>, let <math>S(n)</math> be the number of sequences of length <math>n</math> consisting solely of the letters <math>A</math> and <math>B</math>, with no more than three <math>A</math>s in a row and no more than three <math>B</math>s in a row. What is the remainder when <math>S(2015)</math> is divided by 12?<br />
<br />
<math> \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}}\ 8 \qquad\textbf{(E)}\ 10</math><br />
<br />
[[2015 AMC 12A Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
Let <math>S</math> be a square of side length 1. Two points are chosen independently at random on the sides of <math>S</math>. The probability that the straight-line distance between the points is at least <math>\frac12</math> is <math>\frac{a-b\pi}{c}</math>, where <math>a,b,</math> and <math>c</math> are positive integers and <math>\text{gcd}(a,b,c) = 1</math>. What is <math>a+b+c</math>?<br />
<br />
<math> \textbf{(A)}\ 59 \qquad\textbf{(B)}\ 60 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}}\ 62 \qquad\textbf{(E)}\ 63</math><br />
<br />
[[2015 AMC 12A Problems/Problem 23|Solution]]<br />
<br />
==Problem 24==<br />
<br />
Rational numbers <math>a</math> and <math>b</math> are chosen at random among all rational numbers in the interval <math>[0,2)</math> that can be written as fractions <math>\frac{n}{d}</math> where <math>n</math> and <math>d</math> are integers with <math>1 \le d \le 5</math>. What is the probability that<br />
<cmath>(\text{cos}(a\pi)+i\text{sin}(b\pi))^4</cmath><br />
is a real number?<br />
<br />
<math> \textbf{(A)}\ \frac{3}{5} \qquad\textbf{(B)}\ \frac{4}{25} \qquad\textbf{(C)}\ \frac{41}{200} \qquad\textbf{(D)}}\ \frac{6}{25} \qquad\textbf{(E)}\ \frac{13}{50}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 24|Solution]]<br />
<br />
==Problem 25==<br />
<br />
A collection of circles in the upper half-plane, all tangent to the <math>x</math>-axis, is constructed in layers as follows. Layer <math>L_0</math> consists of two circles of radii <math>70^2</math> and <math>73^2</math> that are externally tangent. For <math>k\ge1</math>, the circles in <math>\bigcup_{j=0}^{k-1}L_j</math> are ordered according to their points of tangency with the <math>x</math>-axis. For every pair of consecutive circles in this order, a new circle is constructed externally tangent to each of the two circles in the pair. Layer <math>L_k</math> consists of the <math>2^{k-1}</math> circles constructed in this way. Let <math>S=\bigcup_{j=0}^{6}L_j</math>, and for every circle <math>C</math> denote by <math>r(C)</math> its radius. What is<br />
<cmath>\sum_{C\in S} \frac{1}{\sqrt{r(C)}}?</cmath><br />
<br />
<asy><br />
import olympiad;<br />
size(350);<br />
defaultpen(linewidth(0.7));<br />
// define a bunch of arrays and starting points<br />
pair[] coord = new pair[65];<br />
int[] trav = {32,16,8,4,2,1};<br />
coord[0] = (0,73^2); coord[64] = (2*73*70,70^2);<br />
// draw the big circles and the bottom line<br />
path arc1 = arc(coord[0],coord[0].y,260,360);<br />
path arc2 = arc(coord[64],coord[64].y,175,280);<br />
fill((coord[0].x-910,coord[0].y)--arc1--cycle,gray(0.75));<br />
fill((coord[64].x+870,coord[64].y+425)--arc2--cycle,gray(0.75));<br />
draw(arc1^^arc2);<br />
draw((-930,0)--(70^2+73^2+850,0));<br />
// We now apply the findCenter function 63 times to get<br />
// the location of the centers of all 63 constructed circles.<br />
// The complicated array setup ensures that all the circles<br />
// will be taken in the right order<br />
for(int i = 0;i<=5;i=i+1)<br />
{<br />
int skip = trav[i];<br />
for(int k=skip;k<=64 - skip; k = k + 2*skip)<br />
{<br />
pair cent1 = coord[k-skip], cent2 = coord[k+skip];<br />
real r1 = cent1.y, r2 = cent2.y, rn=r1*r2/((sqrt(r1)+sqrt(r2))^2);<br />
real shiftx = cent1.x + sqrt(4*r1*rn);<br />
coord[k] = (shiftx,rn);<br />
}<br />
// Draw the remaining 63 circles<br />
}<br />
for(int i=1;i<=63;i=i+1)<br />
{<br />
filldraw(circle(coord[i],coord[i].y),gray(0.75));<br />
}</asy><br />
<br />
<br />
$ \textbf{(A)}\ \frac{286}{35} \qquad\textbf{(B)}\ \frac{583}{70} \qquad\textbf{(C)}\ \frac{715}{73} \qquad\textbf{(D)}}\ \frac{143}{14} \qquad\textbf{(E)}\ \frac{1573}{146}$<br />
<br />
[[2015 AMC 12A Problems/Problem 25|Solution]]<br />
<br />
== See also ==<br />
* [[AMC Problems and Solutions]]<br />
{{MAA Notice}}</div>Devenwarehttps://artofproblemsolving.com/wiki/index.php?title=2015_AMC_12A_Problems&diff=680182015 AMC 12A Problems2015-03-01T23:28:04Z<p>Devenware: /* Problem 16 */</p>
<hr />
<div>==Problem 1==<br />
<br />
What is the value of <math>(2^0-1+5^2-0)^{-1}\times5?</math><br />
<br />
<math> \textbf{(A)}\ -125\qquad\textbf{(B)}\ -120\qquad\textbf{(C)}\ \frac{1}{5}\qquad\textbf{(D)}\ \frac{5}{24}\qquad\textbf{(E)}\ 25 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
<br />
Two of the three sides of a triangle are 20 and 15. Which of the following numbers is not a possible perimeter of the triangle?<br />
<br />
<math> \textbf{(A)}\ 52\qquad\textbf{(B)}\ 57\qquad\textbf{(C)}\ 62\qquad\textbf{(D)}\ 67\qquad\textbf{(E)}\ 72 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
<br />
Mr. Patrick teaches math to 15 students. He was grading tests and found that when he graded everyone's test except Payton's, the average grade for the class was 80. after he graded Payton's test, the class average became 81. What was Payton's score on the test?<br />
<br />
<math> \textbf{(A)}\ 81\qquad\textbf{(B)}\ 85\qquad\textbf{(C)}\ 91\qquad\textbf{(D)}\ 94\qquad\textbf{(E)}\ 95 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
The sum of two positive numbers is 5 times their difference. What is the ratio of the larger number to the smaller?<br />
<br />
<math> \textbf{(A)}\ \frac54 \qquad\textbf{(B)}\ \frac32 \qquad\textbf{(C)}\ \frac95 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ \frac52 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
Amelia needs to estimate the quantity <math>\frac{a}{b} - c</math>, where <math>a, b,</math> and <math>c</math> are large positive integers. She rounds each of the integers so that the calculation will be easier to do mentally. In which of these situations will her answer necessarily be greater than the exact value of <math>\frac{a}{b} - c</math>?<br />
<br />
<math> \textbf{(A)}\ \text{She rounds all three numbers up.}\\<br />
\qquad\textbf{(B)}\ \text{She rounds } a \text{ and } b \text{ up, and she rounds } c \text{ down.}\\<br />
\qquad\textbf{(C)}\ \text{She rounds } a \text{ and } c \text{ up, and she rounds } b \text{ down.} \\<br />
\qquad\textbf{(D)}\ \text{She rounds } a \text{ up, and she rounds } b \text{ and } c \text{ down.}\\<br />
\qquad\textbf{(E)}\ \text{She rounds } c \text{ up, and she rounds } a \text{ and } b \text{ down.} </math><br />
<br />
[[2015 AMC 12A Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be <math>2 : 1</math>?<br />
<br />
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 8</math><br />
<br />
[[2015 AMC 12A Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
Two right circular cylinders have the same volume. The radius of the second cylinder is <math>10\%</math> more than the radius of the first. What is the relationship between the heights of the two cylinders?<br />
<br />
<math>\textbf{(A)}\ \text{The second height is } 10\% \text{ less than the first.} \\ \textbf{(B)}\ \text{The first height is } 10\% \text{ more than the second.}\\ \textbf{(C)}\ \text{The second height is } 21\% \text{ less than the first.} \\ \textbf{(D)}\ \text{The first height is } 21\% \text{ more than the second.}\\ \textbf{(E)}\ \text{The second height is } 80\% \text{ of the first.}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
The ratio of the length to the width of a rectangle is <math>4</math> : <math>3</math>. If the rectangle has diagonal of length <math>d</math>, then the area may be expressed as <math>kd^2</math> for some constant <math>k</math>. What is <math>k</math>?<br />
<br />
<math> \textbf{(A)}\ \frac27 \qquad\textbf{(B)}\ \frac37 \qquad\textbf{(C)}\ \frac{12}{25} \qquad\textbf{(D)}\ \frac{16}{26} \qquad\textbf{(E)}\ \frac34</math><br />
<br />
[[2015 AMC 12A Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
A box contains 2 red marbles, 2 green marbles, and 2 yellow marbles. Carol takes 2 marbles from the box at random; then Claudia takes 2 of the remaining marbles at random; and then Cheryl takes the last 2 marbles. What is the probability that Cheryl gets 2 marbles of the same color?<br />
<br />
<math> \textbf{(A)}\ \frac{1}{10} \qquad\textbf{(B)}\ \frac16 \qquad\textbf{(C)}\ \frac15 \qquad\textbf{(D)}\ \frac13 \qquad\textbf{(E)}\ \frac12</math><br />
<br />
[[2015 AMC 12A Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
Integers <math>x</math> and <math>y</math> with <math>x>y>0</math> satisfy <math>x+y+xy=80</math>. What is <math>x</math>?<br />
<br />
<math> \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 15 \qquad\textbf{(D)}\ 18 \qquad\textbf{(E)}\ 26</math><br />
<br />
[[2015 AMC 12A Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
On a sheet of paper, Isabella draws a circle of radius <math>2</math>, a circle of radius <math>3</math>, and all possible lines simultaneously tangent to both circles. Isabella notices that she has drawn exactly <math>k \ge 0</math> lines. How many different values of <math>k</math> are possible?<br />
<br />
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6</math><br />
<br />
[[2015 AMC 12A Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
The parabolas <math>y=ax^2 - 2</math> and <math>y=4 - bx^2</math> intersect the coordinate axes in exactly four points, and these four points are the vertices of a kite of area <math>12</math>. What is <math>a+b</math>?<br />
<br />
<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 1.5\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 2.5\qquad\textbf{(E)}\ 3</math><br />
<br />
[[2015 AMC 12A Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
<br />
A league with 12 teams holds a round-robin tournament, with each team playing every other team exactly once. Games either end with one team victorious or else end in a draw. A team scores 2 points for every game it wins and 1 point for every game it draws. Which of the following is NOT a true statement about the list of 12 scores?<br />
<br />
<math> \textbf{(A)}\ \text{There must be an even number of odd scores.}\\<br />
\qquad\textbf{(B)}\ \text{There must be an even number of even scores.}\\<br />
\qquad\textbf{(C)}\ \text{There cannot be two scores of }0\text{.}\\<br />
\qquad\textbf{(D)}\ \text{The sum of the scores must be at least }100\text{.}\\<br />
\qquad\textbf{(E)}\ \text{The highest score must be at least }12\text{.}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
What is the value of <math>a</math> for which <math>\frac{1}{\text{log}_2a} + \frac{1}{\text{log}_3a} + \frac{1}{\text{log}_4a} = 1</math>?<br />
<br />
[[2015 AMC 12A Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
What is the minimum number of digits to the right of the decimal point needed to express the fraction <math>\frac{123456789}{2^{26}\cdot 5^4}</math> as a decimal?<br />
<br />
<math> \textbf{(A)}\ 4\qquad\textbf{(B)}\ 22\qquad\textbf{(C)}\ 26\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 104</math><br />
<br />
[[2015 AMC 12A Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
Tetrahedron <math>ABCD</math> has <math>AB=5,AC=3,BC=4,BD=4,AD=3,</math> and <math>CD=\frac{12}{5}\sqrt{2}</math>. What is the volume of the tetrahedron?<br />
<br />
<math> \textbf{(A)}\ 3\sqrt{2}\qquad\textbf{(B)}\ 2\sqrt{5}\qquad\textbf{(C)}\ \frac{24}{5}\qquad\textbf{(D)}\ 3\sqrt{3}\qquad\textbf{(E)}\ \frac{24}{5}\sqrt{2}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?<br />
<br />
<math> \textbf{(A)}\ \frac{47}{256} \qquad\textbf{(B)}\ \frac{3}{16} \qquad\textbf{(C)}\ \frac{49}{256} \qquad\textbf{(D)}}\ \frac{25}{128} \qquad\textbf{(E)}\ \frac{51}{256}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
The zeros of the function <math>f(x) = x^2-ax+2a</math> are integers. What is the sum of the possible values of <math>a</math>?<br />
<br />
<math> \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 16 \qquad\textbf{(D)}}\ 17 \qquad\textbf{(E)}\ 18</math><br />
<br />
[[2015 AMC 12A Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
For some positive integers <math>p</math>, there is a quadrilateral <math>ABCD</math> with positive integer side lengths, perimeter <math>p</math>, right angles at <math>B</math> and <math>C</math>, <math>AB=2</math>, and <math>CD=AD</math>. How many different values of <math>p<2015</math> are possible?<br />
<br />
<math> \textbf{(A)}\ 30 \qquad\textbf{(B)}\ 31 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}}\ 62 \qquad\textbf{(E)}\ 63</math><br />
<br />
[[2015 AMC 12A Problems/Problem 19|Solution]]<br />
<br />
==Problem 20==<br />
<br />
Isosceles triangles <math>T</math> and <math>T'</math> are not congruent but have the same area and the same perimeter. The sides of <math>T</math> have lengths of <math>5,5,</math> and <math>8</math>, while those of <math>T'</math> have lengths of <math>a,a,</math> and <math>b</math>. Which of the following numbers is closest to <math>b</math>?<br />
<br />
<math> \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}}\ 6 \qquad\textbf{(E)}\ 8</math><br />
<br />
[[2015 AMC 12A Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
A circle of radius <math>r</math> passes through both foci of, and exactly four points on, the ellipse with equation <math>x^2+16y^2=16</math>. The set of all possible values of <math>r</math> is an interval <math>[a,b)</math>. What is <math>a+b</math>?<br />
<br />
<math> \textbf{(A)}\ 5\sqrt{2}+4 \qquad\textbf{(B)}\ \sqrt{17}+7 \qquad\textbf{(C)}\ 6\sqrt{2}+3 \qquad\textbf{(D)}}\ \sqrt{15}+8 \qquad\textbf{(E)}\ 12</math><br />
<br />
[[2015 AMC 12A Problems/Problem 21|Solution]]<br />
<br />
==Problem 22==<br />
<br />
For each positive integer <math>n</math>, let <math>S(n)</math> be the number of sequences of length <math>n</math> consisting solely of the letters <math>A</math> and <math>B</math>, with no more than three <math>A</math>s in a row and no more than three <math>B</math>s in a row. What is the remainder when <math>S(2015)</math> is divided by 12?<br />
<br />
<math> \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}}\ 8 \qquad\textbf{(E)}\ 10</math><br />
<br />
[[2015 AMC 12A Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
Let <math>S</math> be a square of side length 1. Two points are chosen independently at random on the sides of <math>S</math>. The probability that the straight-line distance between the points is at least <math>\frac12</math> is <math>\frac{a-b\pi}{c}</math>, where <math>a,b,</math> and <math>c</math> are positive integers and <math>\text{gcd}(a,b,c) = 1</math>. What is <math>a+b+c</math>?<br />
<br />
<math> \textbf{(A)}\ 59 \qquad\textbf{(B)}\ 60 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}}\ 62 \qquad\textbf{(E)}\ 63</math><br />
<br />
[[2015 AMC 12A Problems/Problem 23|Solution]]<br />
<br />
==Problem 24==<br />
<br />
Rational numbers <math>a</math> and <math>b</math> are chosen at random among all rational numbers in the interval <math>[0,2)</math> that can be written as fractions <math>\frac{n}{d}</math> where <math>n</math> and <math>d</math> are integers with <math>1 \le d \le 5</math>. What is the probability that<br />
<cmath>(\text{cos}(a\pi)+i\text{sin}(b\pi))^4</cmath><br />
is a real number?<br />
<br />
<math> \textbf{(A)}\ \frac{3}{5} \qquad\textbf{(B)}\ \frac{4}{25} \qquad\textbf{(C)}\ \frac{41}{200} \qquad\textbf{(D)}}\ \frac{6}{25} \qquad\textbf{(E)}\ \frac{13}{50}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 24|Solution]]<br />
<br />
==Problem 25==<br />
<br />
A collection of circles in the upper half-plane, all tangent to the <math>x</math>-axis, is constructed in layers as follows. Layer <math>L_0</math> consists of two circles of radii <math>70^2</math> and <math>73^2</math> that are externally tangent. For <math>k\ge1</math>, the circles in <math>\bigcup_{j=0}^{k-1}L_j</math> are ordered according to their points of tangency with the <math>x</math>-axis. For every pair of consecutive circles in this order, a new circle is constructed externally tangent to each of the two circles in the pair. Layer <math>L_k</math> consists of the <math>2^{k-1}</math> circles constructed in this way. Let <math>S=\bigcup_{j=0}^{6}L_j</math>, and for every circle <math>C</math> denote by <math>r(C)</math> its radius. What is<br />
<cmath>\sum_{C\in S} \frac{1}{\sqrt{r(C)}}?</cmath><br />
<br />
<asy><br />
import olympiad;<br />
size(350);<br />
defaultpen(linewidth(0.7));<br />
// define a bunch of arrays and starting points<br />
pair[] coord = new pair[65];<br />
int[] trav = {32,16,8,4,2,1};<br />
coord[0] = (0,73^2); coord[64] = (2*73*70,70^2);<br />
// draw the big circles and the bottom line<br />
path arc1 = arc(coord[0],coord[0].y,260,360);<br />
path arc2 = arc(coord[64],coord[64].y,175,280);<br />
fill((coord[0].x-910,coord[0].y)--arc1--cycle,gray(0.75));<br />
fill((coord[64].x+870,coord[64].y+425)--arc2--cycle,gray(0.75));<br />
draw(arc1^^arc2);<br />
draw((-930,0)--(70^2+73^2+850,0));<br />
// We now apply the findCenter function 63 times to get<br />
// the location of the centers of all 63 constructed circles.<br />
// The complicated array setup ensures that all the circles<br />
// will be taken in the right order<br />
for(int i = 0;i<=5;i=i+1)<br />
{<br />
int skip = trav[i];<br />
for(int k=skip;k<=64 - skip; k = k + 2*skip)<br />
{<br />
pair cent1 = coord[k-skip], cent2 = coord[k+skip];<br />
real r1 = cent1.y, r2 = cent2.y, rn=r1*r2/((sqrt(r1)+sqrt(r2))^2);<br />
real shiftx = cent1.x + sqrt(4*r1*rn);<br />
coord[k] = (shiftx,rn);<br />
}<br />
// Draw the remaining 63 circles<br />
}<br />
for(int i=1;i<=63;i=i+1)<br />
{<br />
filldraw(circle(coord[i],coord[i].y),gray(0.75));<br />
}</asy><br />
<br />
<br />
$ \textbf{(A)}\ \frac{286}{35} \qquad\textbf{(B)}\ \frac{583}{70} \qquad\textbf{(C)}\ \frac{715}{73} \qquad\textbf{(D)}}\ \frac{143}{14} \qquad\textbf{(E)}\ \frac{1573}{146}$<br />
<br />
[[2015 AMC 12A Problems/Problem 25|Solution]]<br />
<br />
== See also ==<br />
* [[AMC Problems and Solutions]]<br />
{{MAA Notice}}</div>Devenwarehttps://artofproblemsolving.com/wiki/index.php?title=2015_AMC_12A_Problems&diff=680172015 AMC 12A Problems2015-03-01T23:27:18Z<p>Devenware: /* Problem 15 */</p>
<hr />
<div>==Problem 1==<br />
<br />
What is the value of <math>(2^0-1+5^2-0)^{-1}\times5?</math><br />
<br />
<math> \textbf{(A)}\ -125\qquad\textbf{(B)}\ -120\qquad\textbf{(C)}\ \frac{1}{5}\qquad\textbf{(D)}\ \frac{5}{24}\qquad\textbf{(E)}\ 25 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
<br />
Two of the three sides of a triangle are 20 and 15. Which of the following numbers is not a possible perimeter of the triangle?<br />
<br />
<math> \textbf{(A)}\ 52\qquad\textbf{(B)}\ 57\qquad\textbf{(C)}\ 62\qquad\textbf{(D)}\ 67\qquad\textbf{(E)}\ 72 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
<br />
Mr. Patrick teaches math to 15 students. He was grading tests and found that when he graded everyone's test except Payton's, the average grade for the class was 80. after he graded Payton's test, the class average became 81. What was Payton's score on the test?<br />
<br />
<math> \textbf{(A)}\ 81\qquad\textbf{(B)}\ 85\qquad\textbf{(C)}\ 91\qquad\textbf{(D)}\ 94\qquad\textbf{(E)}\ 95 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
The sum of two positive numbers is 5 times their difference. What is the ratio of the larger number to the smaller?<br />
<br />
<math> \textbf{(A)}\ \frac54 \qquad\textbf{(B)}\ \frac32 \qquad\textbf{(C)}\ \frac95 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ \frac52 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
Amelia needs to estimate the quantity <math>\frac{a}{b} - c</math>, where <math>a, b,</math> and <math>c</math> are large positive integers. She rounds each of the integers so that the calculation will be easier to do mentally. In which of these situations will her answer necessarily be greater than the exact value of <math>\frac{a}{b} - c</math>?<br />
<br />
<math> \textbf{(A)}\ \text{She rounds all three numbers up.}\\<br />
\qquad\textbf{(B)}\ \text{She rounds } a \text{ and } b \text{ up, and she rounds } c \text{ down.}\\<br />
\qquad\textbf{(C)}\ \text{She rounds } a \text{ and } c \text{ up, and she rounds } b \text{ down.} \\<br />
\qquad\textbf{(D)}\ \text{She rounds } a \text{ up, and she rounds } b \text{ and } c \text{ down.}\\<br />
\qquad\textbf{(E)}\ \text{She rounds } c \text{ up, and she rounds } a \text{ and } b \text{ down.} </math><br />
<br />
[[2015 AMC 12A Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be <math>2 : 1</math>?<br />
<br />
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 8</math><br />
<br />
[[2015 AMC 12A Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
Two right circular cylinders have the same volume. The radius of the second cylinder is <math>10\%</math> more than the radius of the first. What is the relationship between the heights of the two cylinders?<br />
<br />
<math>\textbf{(A)}\ \text{The second height is } 10\% \text{ less than the first.} \\ \textbf{(B)}\ \text{The first height is } 10\% \text{ more than the second.}\\ \textbf{(C)}\ \text{The second height is } 21\% \text{ less than the first.} \\ \textbf{(D)}\ \text{The first height is } 21\% \text{ more than the second.}\\ \textbf{(E)}\ \text{The second height is } 80\% \text{ of the first.}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
The ratio of the length to the width of a rectangle is <math>4</math> : <math>3</math>. If the rectangle has diagonal of length <math>d</math>, then the area may be expressed as <math>kd^2</math> for some constant <math>k</math>. What is <math>k</math>?<br />
<br />
<math> \textbf{(A)}\ \frac27 \qquad\textbf{(B)}\ \frac37 \qquad\textbf{(C)}\ \frac{12}{25} \qquad\textbf{(D)}\ \frac{16}{26} \qquad\textbf{(E)}\ \frac34</math><br />
<br />
[[2015 AMC 12A Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
A box contains 2 red marbles, 2 green marbles, and 2 yellow marbles. Carol takes 2 marbles from the box at random; then Claudia takes 2 of the remaining marbles at random; and then Cheryl takes the last 2 marbles. What is the probability that Cheryl gets 2 marbles of the same color?<br />
<br />
<math> \textbf{(A)}\ \frac{1}{10} \qquad\textbf{(B)}\ \frac16 \qquad\textbf{(C)}\ \frac15 \qquad\textbf{(D)}\ \frac13 \qquad\textbf{(E)}\ \frac12</math><br />
<br />
[[2015 AMC 12A Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
Integers <math>x</math> and <math>y</math> with <math>x>y>0</math> satisfy <math>x+y+xy=80</math>. What is <math>x</math>?<br />
<br />
<math> \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 15 \qquad\textbf{(D)}\ 18 \qquad\textbf{(E)}\ 26</math><br />
<br />
[[2015 AMC 12A Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
On a sheet of paper, Isabella draws a circle of radius <math>2</math>, a circle of radius <math>3</math>, and all possible lines simultaneously tangent to both circles. Isabella notices that she has drawn exactly <math>k \ge 0</math> lines. How many different values of <math>k</math> are possible?<br />
<br />
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6</math><br />
<br />
[[2015 AMC 12A Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
The parabolas <math>y=ax^2 - 2</math> and <math>y=4 - bx^2</math> intersect the coordinate axes in exactly four points, and these four points are the vertices of a kite of area <math>12</math>. What is <math>a+b</math>?<br />
<br />
<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 1.5\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 2.5\qquad\textbf{(E)}\ 3</math><br />
<br />
[[2015 AMC 12A Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
<br />
A league with 12 teams holds a round-robin tournament, with each team playing every other team exactly once. Games either end with one team victorious or else end in a draw. A team scores 2 points for every game it wins and 1 point for every game it draws. Which of the following is NOT a true statement about the list of 12 scores?<br />
<br />
<math> \textbf{(A)}\ \text{There must be an even number of odd scores.}\\<br />
\qquad\textbf{(B)}\ \text{There must be an even number of even scores.}\\<br />
\qquad\textbf{(C)}\ \text{There cannot be two scores of }0\text{.}\\<br />
\qquad\textbf{(D)}\ \text{The sum of the scores must be at least }100\text{.}\\<br />
\qquad\textbf{(E)}\ \text{The highest score must be at least }12\text{.}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
What is the value of <math>a</math> for which <math>\frac{1}{\text{log}_2a} + \frac{1}{\text{log}_3a} + \frac{1}{\text{log}_4a} = 1</math>?<br />
<br />
[[2015 AMC 12A Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
What is the minimum number of digits to the right of the decimal point needed to express the fraction <math>\frac{123456789}{2^{26}\cdot 5^4}</math> as a decimal?<br />
<br />
<math> \textbf{(A)}\ 4\qquad\textbf{(B)}\ 22\qquad\textbf{(C)}\ 26\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 104</math><br />
<br />
[[2015 AMC 12A Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
Tetrahedron <math>ABCD</math> has <math>AB=5,AC=3,BC=4,BD=4,AD=3,</math> and <math>CD=\frac{12}{5}\sqrt{2}</math>. What is the volume of the tetrahedron?<br />
<br />
<math> \textbf{(A)}\ 3\sqrt{2}\qquad\textbf{(B)}\ 2\sqrt{5}\qquad\textbf{(C)}\ \frac{24}{5}\qquad\textbf{(D)}}\ 3\sqrt{3}\qquad\textbf{(E)}\ \frac{24}{5}\sqrt{2}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?<br />
<br />
<math> \textbf{(A)}\ \frac{47}{256} \qquad\textbf{(B)}\ \frac{3}{16} \qquad\textbf{(C)}\ \frac{49}{256} \qquad\textbf{(D)}}\ \frac{25}{128} \qquad\textbf{(E)}\ \frac{51}{256}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
The zeros of the function <math>f(x) = x^2-ax+2a</math> are integers. What is the sum of the possible values of <math>a</math>?<br />
<br />
<math> \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 16 \qquad\textbf{(D)}}\ 17 \qquad\textbf{(E)}\ 18</math><br />
<br />
[[2015 AMC 12A Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
For some positive integers <math>p</math>, there is a quadrilateral <math>ABCD</math> with positive integer side lengths, perimeter <math>p</math>, right angles at <math>B</math> and <math>C</math>, <math>AB=2</math>, and <math>CD=AD</math>. How many different values of <math>p<2015</math> are possible?<br />
<br />
<math> \textbf{(A)}\ 30 \qquad\textbf{(B)}\ 31 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}}\ 62 \qquad\textbf{(E)}\ 63</math><br />
<br />
[[2015 AMC 12A Problems/Problem 19|Solution]]<br />
<br />
==Problem 20==<br />
<br />
Isosceles triangles <math>T</math> and <math>T'</math> are not congruent but have the same area and the same perimeter. The sides of <math>T</math> have lengths of <math>5,5,</math> and <math>8</math>, while those of <math>T'</math> have lengths of <math>a,a,</math> and <math>b</math>. Which of the following numbers is closest to <math>b</math>?<br />
<br />
<math> \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}}\ 6 \qquad\textbf{(E)}\ 8</math><br />
<br />
[[2015 AMC 12A Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
A circle of radius <math>r</math> passes through both foci of, and exactly four points on, the ellipse with equation <math>x^2+16y^2=16</math>. The set of all possible values of <math>r</math> is an interval <math>[a,b)</math>. What is <math>a+b</math>?<br />
<br />
<math> \textbf{(A)}\ 5\sqrt{2}+4 \qquad\textbf{(B)}\ \sqrt{17}+7 \qquad\textbf{(C)}\ 6\sqrt{2}+3 \qquad\textbf{(D)}}\ \sqrt{15}+8 \qquad\textbf{(E)}\ 12</math><br />
<br />
[[2015 AMC 12A Problems/Problem 21|Solution]]<br />
<br />
==Problem 22==<br />
<br />
For each positive integer <math>n</math>, let <math>S(n)</math> be the number of sequences of length <math>n</math> consisting solely of the letters <math>A</math> and <math>B</math>, with no more than three <math>A</math>s in a row and no more than three <math>B</math>s in a row. What is the remainder when <math>S(2015)</math> is divided by 12?<br />
<br />
<math> \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}}\ 8 \qquad\textbf{(E)}\ 10</math><br />
<br />
[[2015 AMC 12A Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
Let <math>S</math> be a square of side length 1. Two points are chosen independently at random on the sides of <math>S</math>. The probability that the straight-line distance between the points is at least <math>\frac12</math> is <math>\frac{a-b\pi}{c}</math>, where <math>a,b,</math> and <math>c</math> are positive integers and <math>\text{gcd}(a,b,c) = 1</math>. What is <math>a+b+c</math>?<br />
<br />
<math> \textbf{(A)}\ 59 \qquad\textbf{(B)}\ 60 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}}\ 62 \qquad\textbf{(E)}\ 63</math><br />
<br />
[[2015 AMC 12A Problems/Problem 23|Solution]]<br />
<br />
==Problem 24==<br />
<br />
Rational numbers <math>a</math> and <math>b</math> are chosen at random among all rational numbers in the interval <math>[0,2)</math> that can be written as fractions <math>\frac{n}{d}</math> where <math>n</math> and <math>d</math> are integers with <math>1 \le d \le 5</math>. What is the probability that<br />
<cmath>(\text{cos}(a\pi)+i\text{sin}(b\pi))^4</cmath><br />
is a real number?<br />
<br />
<math> \textbf{(A)}\ \frac{3}{5} \qquad\textbf{(B)}\ \frac{4}{25} \qquad\textbf{(C)}\ \frac{41}{200} \qquad\textbf{(D)}}\ \frac{6}{25} \qquad\textbf{(E)}\ \frac{13}{50}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 24|Solution]]<br />
<br />
==Problem 25==<br />
<br />
A collection of circles in the upper half-plane, all tangent to the <math>x</math>-axis, is constructed in layers as follows. Layer <math>L_0</math> consists of two circles of radii <math>70^2</math> and <math>73^2</math> that are externally tangent. For <math>k\ge1</math>, the circles in <math>\bigcup_{j=0}^{k-1}L_j</math> are ordered according to their points of tangency with the <math>x</math>-axis. For every pair of consecutive circles in this order, a new circle is constructed externally tangent to each of the two circles in the pair. Layer <math>L_k</math> consists of the <math>2^{k-1}</math> circles constructed in this way. Let <math>S=\bigcup_{j=0}^{6}L_j</math>, and for every circle <math>C</math> denote by <math>r(C)</math> its radius. What is<br />
<cmath>\sum_{C\in S} \frac{1}{\sqrt{r(C)}}?</cmath><br />
<br />
<asy><br />
import olympiad;<br />
size(350);<br />
defaultpen(linewidth(0.7));<br />
// define a bunch of arrays and starting points<br />
pair[] coord = new pair[65];<br />
int[] trav = {32,16,8,4,2,1};<br />
coord[0] = (0,73^2); coord[64] = (2*73*70,70^2);<br />
// draw the big circles and the bottom line<br />
path arc1 = arc(coord[0],coord[0].y,260,360);<br />
path arc2 = arc(coord[64],coord[64].y,175,280);<br />
fill((coord[0].x-910,coord[0].y)--arc1--cycle,gray(0.75));<br />
fill((coord[64].x+870,coord[64].y+425)--arc2--cycle,gray(0.75));<br />
draw(arc1^^arc2);<br />
draw((-930,0)--(70^2+73^2+850,0));<br />
// We now apply the findCenter function 63 times to get<br />
// the location of the centers of all 63 constructed circles.<br />
// The complicated array setup ensures that all the circles<br />
// will be taken in the right order<br />
for(int i = 0;i<=5;i=i+1)<br />
{<br />
int skip = trav[i];<br />
for(int k=skip;k<=64 - skip; k = k + 2*skip)<br />
{<br />
pair cent1 = coord[k-skip], cent2 = coord[k+skip];<br />
real r1 = cent1.y, r2 = cent2.y, rn=r1*r2/((sqrt(r1)+sqrt(r2))^2);<br />
real shiftx = cent1.x + sqrt(4*r1*rn);<br />
coord[k] = (shiftx,rn);<br />
}<br />
// Draw the remaining 63 circles<br />
}<br />
for(int i=1;i<=63;i=i+1)<br />
{<br />
filldraw(circle(coord[i],coord[i].y),gray(0.75));<br />
}</asy><br />
<br />
<br />
$ \textbf{(A)}\ \frac{286}{35} \qquad\textbf{(B)}\ \frac{583}{70} \qquad\textbf{(C)}\ \frac{715}{73} \qquad\textbf{(D)}}\ \frac{143}{14} \qquad\textbf{(E)}\ \frac{1573}{146}$<br />
<br />
[[2015 AMC 12A Problems/Problem 25|Solution]]<br />
<br />
== See also ==<br />
* [[AMC Problems and Solutions]]<br />
{{MAA Notice}}</div>Devenwarehttps://artofproblemsolving.com/wiki/index.php?title=2015_AMC_12A_Problems&diff=680162015 AMC 12A Problems2015-03-01T23:26:52Z<p>Devenware: /* Problem 13 */</p>
<hr />
<div>==Problem 1==<br />
<br />
What is the value of <math>(2^0-1+5^2-0)^{-1}\times5?</math><br />
<br />
<math> \textbf{(A)}\ -125\qquad\textbf{(B)}\ -120\qquad\textbf{(C)}\ \frac{1}{5}\qquad\textbf{(D)}\ \frac{5}{24}\qquad\textbf{(E)}\ 25 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
<br />
Two of the three sides of a triangle are 20 and 15. Which of the following numbers is not a possible perimeter of the triangle?<br />
<br />
<math> \textbf{(A)}\ 52\qquad\textbf{(B)}\ 57\qquad\textbf{(C)}\ 62\qquad\textbf{(D)}\ 67\qquad\textbf{(E)}\ 72 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
<br />
Mr. Patrick teaches math to 15 students. He was grading tests and found that when he graded everyone's test except Payton's, the average grade for the class was 80. after he graded Payton's test, the class average became 81. What was Payton's score on the test?<br />
<br />
<math> \textbf{(A)}\ 81\qquad\textbf{(B)}\ 85\qquad\textbf{(C)}\ 91\qquad\textbf{(D)}\ 94\qquad\textbf{(E)}\ 95 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
The sum of two positive numbers is 5 times their difference. What is the ratio of the larger number to the smaller?<br />
<br />
<math> \textbf{(A)}\ \frac54 \qquad\textbf{(B)}\ \frac32 \qquad\textbf{(C)}\ \frac95 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ \frac52 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
Amelia needs to estimate the quantity <math>\frac{a}{b} - c</math>, where <math>a, b,</math> and <math>c</math> are large positive integers. She rounds each of the integers so that the calculation will be easier to do mentally. In which of these situations will her answer necessarily be greater than the exact value of <math>\frac{a}{b} - c</math>?<br />
<br />
<math> \textbf{(A)}\ \text{She rounds all three numbers up.}\\<br />
\qquad\textbf{(B)}\ \text{She rounds } a \text{ and } b \text{ up, and she rounds } c \text{ down.}\\<br />
\qquad\textbf{(C)}\ \text{She rounds } a \text{ and } c \text{ up, and she rounds } b \text{ down.} \\<br />
\qquad\textbf{(D)}\ \text{She rounds } a \text{ up, and she rounds } b \text{ and } c \text{ down.}\\<br />
\qquad\textbf{(E)}\ \text{She rounds } c \text{ up, and she rounds } a \text{ and } b \text{ down.} </math><br />
<br />
[[2015 AMC 12A Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be <math>2 : 1</math>?<br />
<br />
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 8</math><br />
<br />
[[2015 AMC 12A Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
Two right circular cylinders have the same volume. The radius of the second cylinder is <math>10\%</math> more than the radius of the first. What is the relationship between the heights of the two cylinders?<br />
<br />
<math>\textbf{(A)}\ \text{The second height is } 10\% \text{ less than the first.} \\ \textbf{(B)}\ \text{The first height is } 10\% \text{ more than the second.}\\ \textbf{(C)}\ \text{The second height is } 21\% \text{ less than the first.} \\ \textbf{(D)}\ \text{The first height is } 21\% \text{ more than the second.}\\ \textbf{(E)}\ \text{The second height is } 80\% \text{ of the first.}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
The ratio of the length to the width of a rectangle is <math>4</math> : <math>3</math>. If the rectangle has diagonal of length <math>d</math>, then the area may be expressed as <math>kd^2</math> for some constant <math>k</math>. What is <math>k</math>?<br />
<br />
<math> \textbf{(A)}\ \frac27 \qquad\textbf{(B)}\ \frac37 \qquad\textbf{(C)}\ \frac{12}{25} \qquad\textbf{(D)}\ \frac{16}{26} \qquad\textbf{(E)}\ \frac34</math><br />
<br />
[[2015 AMC 12A Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
A box contains 2 red marbles, 2 green marbles, and 2 yellow marbles. Carol takes 2 marbles from the box at random; then Claudia takes 2 of the remaining marbles at random; and then Cheryl takes the last 2 marbles. What is the probability that Cheryl gets 2 marbles of the same color?<br />
<br />
<math> \textbf{(A)}\ \frac{1}{10} \qquad\textbf{(B)}\ \frac16 \qquad\textbf{(C)}\ \frac15 \qquad\textbf{(D)}\ \frac13 \qquad\textbf{(E)}\ \frac12</math><br />
<br />
[[2015 AMC 12A Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
Integers <math>x</math> and <math>y</math> with <math>x>y>0</math> satisfy <math>x+y+xy=80</math>. What is <math>x</math>?<br />
<br />
<math> \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 15 \qquad\textbf{(D)}\ 18 \qquad\textbf{(E)}\ 26</math><br />
<br />
[[2015 AMC 12A Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
On a sheet of paper, Isabella draws a circle of radius <math>2</math>, a circle of radius <math>3</math>, and all possible lines simultaneously tangent to both circles. Isabella notices that she has drawn exactly <math>k \ge 0</math> lines. How many different values of <math>k</math> are possible?<br />
<br />
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6</math><br />
<br />
[[2015 AMC 12A Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
The parabolas <math>y=ax^2 - 2</math> and <math>y=4 - bx^2</math> intersect the coordinate axes in exactly four points, and these four points are the vertices of a kite of area <math>12</math>. What is <math>a+b</math>?<br />
<br />
<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 1.5\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 2.5\qquad\textbf{(E)}\ 3</math><br />
<br />
[[2015 AMC 12A Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
<br />
A league with 12 teams holds a round-robin tournament, with each team playing every other team exactly once. Games either end with one team victorious or else end in a draw. A team scores 2 points for every game it wins and 1 point for every game it draws. Which of the following is NOT a true statement about the list of 12 scores?<br />
<br />
<math> \textbf{(A)}\ \text{There must be an even number of odd scores.}\\<br />
\qquad\textbf{(B)}\ \text{There must be an even number of even scores.}\\<br />
\qquad\textbf{(C)}\ \text{There cannot be two scores of }0\text{.}\\<br />
\qquad\textbf{(D)}\ \text{The sum of the scores must be at least }100\text{.}\\<br />
\qquad\textbf{(E)}\ \text{The highest score must be at least }12\text{.}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
What is the value of <math>a</math> for which <math>\frac{1}{\text{log}_2a} + \frac{1}{\text{log}_3a} + \frac{1}{\text{log}_4a} = 1</math>?<br />
<br />
[[2015 AMC 12A Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
What is the minimum number of digits to the right of the decimal point needed to express the fraction <math>\frac{123456789}{2^{26}\cdot 5^4}</math> as a decimal?<br />
<br />
<math> \textbf{(A)}\ 4\qquad\textbf{(B)}\ 22\qquad\textbf{(C)}\ 26\qquad\textbf{(D)}}\ 30\qquad\textbf{(E)}\ 104</math><br />
<br />
[[2015 AMC 12A Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
Tetrahedron <math>ABCD</math> has <math>AB=5,AC=3,BC=4,BD=4,AD=3,</math> and <math>CD=\frac{12}{5}\sqrt{2}</math>. What is the volume of the tetrahedron?<br />
<br />
<math> \textbf{(A)}\ 3\sqrt{2}\qquad\textbf{(B)}\ 2\sqrt{5}\qquad\textbf{(C)}\ \frac{24}{5}\qquad\textbf{(D)}}\ 3\sqrt{3}\qquad\textbf{(E)}\ \frac{24}{5}\sqrt{2}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?<br />
<br />
<math> \textbf{(A)}\ \frac{47}{256} \qquad\textbf{(B)}\ \frac{3}{16} \qquad\textbf{(C)}\ \frac{49}{256} \qquad\textbf{(D)}}\ \frac{25}{128} \qquad\textbf{(E)}\ \frac{51}{256}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
The zeros of the function <math>f(x) = x^2-ax+2a</math> are integers. What is the sum of the possible values of <math>a</math>?<br />
<br />
<math> \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 16 \qquad\textbf{(D)}}\ 17 \qquad\textbf{(E)}\ 18</math><br />
<br />
[[2015 AMC 12A Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
For some positive integers <math>p</math>, there is a quadrilateral <math>ABCD</math> with positive integer side lengths, perimeter <math>p</math>, right angles at <math>B</math> and <math>C</math>, <math>AB=2</math>, and <math>CD=AD</math>. How many different values of <math>p<2015</math> are possible?<br />
<br />
<math> \textbf{(A)}\ 30 \qquad\textbf{(B)}\ 31 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}}\ 62 \qquad\textbf{(E)}\ 63</math><br />
<br />
[[2015 AMC 12A Problems/Problem 19|Solution]]<br />
<br />
==Problem 20==<br />
<br />
Isosceles triangles <math>T</math> and <math>T'</math> are not congruent but have the same area and the same perimeter. The sides of <math>T</math> have lengths of <math>5,5,</math> and <math>8</math>, while those of <math>T'</math> have lengths of <math>a,a,</math> and <math>b</math>. Which of the following numbers is closest to <math>b</math>?<br />
<br />
<math> \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}}\ 6 \qquad\textbf{(E)}\ 8</math><br />
<br />
[[2015 AMC 12A Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
A circle of radius <math>r</math> passes through both foci of, and exactly four points on, the ellipse with equation <math>x^2+16y^2=16</math>. The set of all possible values of <math>r</math> is an interval <math>[a,b)</math>. What is <math>a+b</math>?<br />
<br />
<math> \textbf{(A)}\ 5\sqrt{2}+4 \qquad\textbf{(B)}\ \sqrt{17}+7 \qquad\textbf{(C)}\ 6\sqrt{2}+3 \qquad\textbf{(D)}}\ \sqrt{15}+8 \qquad\textbf{(E)}\ 12</math><br />
<br />
[[2015 AMC 12A Problems/Problem 21|Solution]]<br />
<br />
==Problem 22==<br />
<br />
For each positive integer <math>n</math>, let <math>S(n)</math> be the number of sequences of length <math>n</math> consisting solely of the letters <math>A</math> and <math>B</math>, with no more than three <math>A</math>s in a row and no more than three <math>B</math>s in a row. What is the remainder when <math>S(2015)</math> is divided by 12?<br />
<br />
<math> \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}}\ 8 \qquad\textbf{(E)}\ 10</math><br />
<br />
[[2015 AMC 12A Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
Let <math>S</math> be a square of side length 1. Two points are chosen independently at random on the sides of <math>S</math>. The probability that the straight-line distance between the points is at least <math>\frac12</math> is <math>\frac{a-b\pi}{c}</math>, where <math>a,b,</math> and <math>c</math> are positive integers and <math>\text{gcd}(a,b,c) = 1</math>. What is <math>a+b+c</math>?<br />
<br />
<math> \textbf{(A)}\ 59 \qquad\textbf{(B)}\ 60 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}}\ 62 \qquad\textbf{(E)}\ 63</math><br />
<br />
[[2015 AMC 12A Problems/Problem 23|Solution]]<br />
<br />
==Problem 24==<br />
<br />
Rational numbers <math>a</math> and <math>b</math> are chosen at random among all rational numbers in the interval <math>[0,2)</math> that can be written as fractions <math>\frac{n}{d}</math> where <math>n</math> and <math>d</math> are integers with <math>1 \le d \le 5</math>. What is the probability that<br />
<cmath>(\text{cos}(a\pi)+i\text{sin}(b\pi))^4</cmath><br />
is a real number?<br />
<br />
<math> \textbf{(A)}\ \frac{3}{5} \qquad\textbf{(B)}\ \frac{4}{25} \qquad\textbf{(C)}\ \frac{41}{200} \qquad\textbf{(D)}}\ \frac{6}{25} \qquad\textbf{(E)}\ \frac{13}{50}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 24|Solution]]<br />
<br />
==Problem 25==<br />
<br />
A collection of circles in the upper half-plane, all tangent to the <math>x</math>-axis, is constructed in layers as follows. Layer <math>L_0</math> consists of two circles of radii <math>70^2</math> and <math>73^2</math> that are externally tangent. For <math>k\ge1</math>, the circles in <math>\bigcup_{j=0}^{k-1}L_j</math> are ordered according to their points of tangency with the <math>x</math>-axis. For every pair of consecutive circles in this order, a new circle is constructed externally tangent to each of the two circles in the pair. Layer <math>L_k</math> consists of the <math>2^{k-1}</math> circles constructed in this way. Let <math>S=\bigcup_{j=0}^{6}L_j</math>, and for every circle <math>C</math> denote by <math>r(C)</math> its radius. What is<br />
<cmath>\sum_{C\in S} \frac{1}{\sqrt{r(C)}}?</cmath><br />
<br />
<asy><br />
import olympiad;<br />
size(350);<br />
defaultpen(linewidth(0.7));<br />
// define a bunch of arrays and starting points<br />
pair[] coord = new pair[65];<br />
int[] trav = {32,16,8,4,2,1};<br />
coord[0] = (0,73^2); coord[64] = (2*73*70,70^2);<br />
// draw the big circles and the bottom line<br />
path arc1 = arc(coord[0],coord[0].y,260,360);<br />
path arc2 = arc(coord[64],coord[64].y,175,280);<br />
fill((coord[0].x-910,coord[0].y)--arc1--cycle,gray(0.75));<br />
fill((coord[64].x+870,coord[64].y+425)--arc2--cycle,gray(0.75));<br />
draw(arc1^^arc2);<br />
draw((-930,0)--(70^2+73^2+850,0));<br />
// We now apply the findCenter function 63 times to get<br />
// the location of the centers of all 63 constructed circles.<br />
// The complicated array setup ensures that all the circles<br />
// will be taken in the right order<br />
for(int i = 0;i<=5;i=i+1)<br />
{<br />
int skip = trav[i];<br />
for(int k=skip;k<=64 - skip; k = k + 2*skip)<br />
{<br />
pair cent1 = coord[k-skip], cent2 = coord[k+skip];<br />
real r1 = cent1.y, r2 = cent2.y, rn=r1*r2/((sqrt(r1)+sqrt(r2))^2);<br />
real shiftx = cent1.x + sqrt(4*r1*rn);<br />
coord[k] = (shiftx,rn);<br />
}<br />
// Draw the remaining 63 circles<br />
}<br />
for(int i=1;i<=63;i=i+1)<br />
{<br />
filldraw(circle(coord[i],coord[i].y),gray(0.75));<br />
}</asy><br />
<br />
<br />
$ \textbf{(A)}\ \frac{286}{35} \qquad\textbf{(B)}\ \frac{583}{70} \qquad\textbf{(C)}\ \frac{715}{73} \qquad\textbf{(D)}}\ \frac{143}{14} \qquad\textbf{(E)}\ \frac{1573}{146}$<br />
<br />
[[2015 AMC 12A Problems/Problem 25|Solution]]<br />
<br />
== See also ==<br />
* [[AMC Problems and Solutions]]<br />
{{MAA Notice}}</div>Devenwarehttps://artofproblemsolving.com/wiki/index.php?title=2015_AMC_12A_Problems&diff=680152015 AMC 12A Problems2015-03-01T23:26:35Z<p>Devenware: /* Problem 12 */</p>
<hr />
<div>==Problem 1==<br />
<br />
What is the value of <math>(2^0-1+5^2-0)^{-1}\times5?</math><br />
<br />
<math> \textbf{(A)}\ -125\qquad\textbf{(B)}\ -120\qquad\textbf{(C)}\ \frac{1}{5}\qquad\textbf{(D)}\ \frac{5}{24}\qquad\textbf{(E)}\ 25 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
<br />
Two of the three sides of a triangle are 20 and 15. Which of the following numbers is not a possible perimeter of the triangle?<br />
<br />
<math> \textbf{(A)}\ 52\qquad\textbf{(B)}\ 57\qquad\textbf{(C)}\ 62\qquad\textbf{(D)}\ 67\qquad\textbf{(E)}\ 72 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
<br />
Mr. Patrick teaches math to 15 students. He was grading tests and found that when he graded everyone's test except Payton's, the average grade for the class was 80. after he graded Payton's test, the class average became 81. What was Payton's score on the test?<br />
<br />
<math> \textbf{(A)}\ 81\qquad\textbf{(B)}\ 85\qquad\textbf{(C)}\ 91\qquad\textbf{(D)}\ 94\qquad\textbf{(E)}\ 95 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
The sum of two positive numbers is 5 times their difference. What is the ratio of the larger number to the smaller?<br />
<br />
<math> \textbf{(A)}\ \frac54 \qquad\textbf{(B)}\ \frac32 \qquad\textbf{(C)}\ \frac95 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ \frac52 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
Amelia needs to estimate the quantity <math>\frac{a}{b} - c</math>, where <math>a, b,</math> and <math>c</math> are large positive integers. She rounds each of the integers so that the calculation will be easier to do mentally. In which of these situations will her answer necessarily be greater than the exact value of <math>\frac{a}{b} - c</math>?<br />
<br />
<math> \textbf{(A)}\ \text{She rounds all three numbers up.}\\<br />
\qquad\textbf{(B)}\ \text{She rounds } a \text{ and } b \text{ up, and she rounds } c \text{ down.}\\<br />
\qquad\textbf{(C)}\ \text{She rounds } a \text{ and } c \text{ up, and she rounds } b \text{ down.} \\<br />
\qquad\textbf{(D)}\ \text{She rounds } a \text{ up, and she rounds } b \text{ and } c \text{ down.}\\<br />
\qquad\textbf{(E)}\ \text{She rounds } c \text{ up, and she rounds } a \text{ and } b \text{ down.} </math><br />
<br />
[[2015 AMC 12A Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be <math>2 : 1</math>?<br />
<br />
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 8</math><br />
<br />
[[2015 AMC 12A Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
Two right circular cylinders have the same volume. The radius of the second cylinder is <math>10\%</math> more than the radius of the first. What is the relationship between the heights of the two cylinders?<br />
<br />
<math>\textbf{(A)}\ \text{The second height is } 10\% \text{ less than the first.} \\ \textbf{(B)}\ \text{The first height is } 10\% \text{ more than the second.}\\ \textbf{(C)}\ \text{The second height is } 21\% \text{ less than the first.} \\ \textbf{(D)}\ \text{The first height is } 21\% \text{ more than the second.}\\ \textbf{(E)}\ \text{The second height is } 80\% \text{ of the first.}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
The ratio of the length to the width of a rectangle is <math>4</math> : <math>3</math>. If the rectangle has diagonal of length <math>d</math>, then the area may be expressed as <math>kd^2</math> for some constant <math>k</math>. What is <math>k</math>?<br />
<br />
<math> \textbf{(A)}\ \frac27 \qquad\textbf{(B)}\ \frac37 \qquad\textbf{(C)}\ \frac{12}{25} \qquad\textbf{(D)}\ \frac{16}{26} \qquad\textbf{(E)}\ \frac34</math><br />
<br />
[[2015 AMC 12A Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
A box contains 2 red marbles, 2 green marbles, and 2 yellow marbles. Carol takes 2 marbles from the box at random; then Claudia takes 2 of the remaining marbles at random; and then Cheryl takes the last 2 marbles. What is the probability that Cheryl gets 2 marbles of the same color?<br />
<br />
<math> \textbf{(A)}\ \frac{1}{10} \qquad\textbf{(B)}\ \frac16 \qquad\textbf{(C)}\ \frac15 \qquad\textbf{(D)}\ \frac13 \qquad\textbf{(E)}\ \frac12</math><br />
<br />
[[2015 AMC 12A Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
Integers <math>x</math> and <math>y</math> with <math>x>y>0</math> satisfy <math>x+y+xy=80</math>. What is <math>x</math>?<br />
<br />
<math> \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 15 \qquad\textbf{(D)}\ 18 \qquad\textbf{(E)}\ 26</math><br />
<br />
[[2015 AMC 12A Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
On a sheet of paper, Isabella draws a circle of radius <math>2</math>, a circle of radius <math>3</math>, and all possible lines simultaneously tangent to both circles. Isabella notices that she has drawn exactly <math>k \ge 0</math> lines. How many different values of <math>k</math> are possible?<br />
<br />
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6</math><br />
<br />
[[2015 AMC 12A Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
The parabolas <math>y=ax^2 - 2</math> and <math>y=4 - bx^2</math> intersect the coordinate axes in exactly four points, and these four points are the vertices of a kite of area <math>12</math>. What is <math>a+b</math>?<br />
<br />
<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 1.5\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 2.5\qquad\textbf{(E)}\ 3</math><br />
<br />
[[2015 AMC 12A Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
<br />
A league with 12 teams holds a round-robin tournament, with each team playing every other team exactly once. Games either end with one team victorious or else end in a draw. A team scores 2 points for every game it wins and 1 point for every game it draws. Which of the following is NOT a true statement about the list of 12 scores?<br />
<br />
<math> \textbf{(A)}\ \text{There must be an even number of odd scores.}\\<br />
\qquad\textbf{(B)}\ \text{There must be an even number of even scores.}\\<br />
\qquad\textbf{(C)}\ \text{There cannot be two scores of }0\text{.}\\<br />
\qquad\textbf{(D)}}\ \text{The sum of the scores must be at least }100\text{.}\\<br />
\qquad\textbf{(E)}\ \text{The highest score must be at least }12\text{.}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
What is the value of <math>a</math> for which <math>\frac{1}{\text{log}_2a} + \frac{1}{\text{log}_3a} + \frac{1}{\text{log}_4a} = 1</math>?<br />
<br />
[[2015 AMC 12A Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
What is the minimum number of digits to the right of the decimal point needed to express the fraction <math>\frac{123456789}{2^{26}\cdot 5^4}</math> as a decimal?<br />
<br />
<math> \textbf{(A)}\ 4\qquad\textbf{(B)}\ 22\qquad\textbf{(C)}\ 26\qquad\textbf{(D)}}\ 30\qquad\textbf{(E)}\ 104</math><br />
<br />
[[2015 AMC 12A Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
Tetrahedron <math>ABCD</math> has <math>AB=5,AC=3,BC=4,BD=4,AD=3,</math> and <math>CD=\frac{12}{5}\sqrt{2}</math>. What is the volume of the tetrahedron?<br />
<br />
<math> \textbf{(A)}\ 3\sqrt{2}\qquad\textbf{(B)}\ 2\sqrt{5}\qquad\textbf{(C)}\ \frac{24}{5}\qquad\textbf{(D)}}\ 3\sqrt{3}\qquad\textbf{(E)}\ \frac{24}{5}\sqrt{2}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?<br />
<br />
<math> \textbf{(A)}\ \frac{47}{256} \qquad\textbf{(B)}\ \frac{3}{16} \qquad\textbf{(C)}\ \frac{49}{256} \qquad\textbf{(D)}}\ \frac{25}{128} \qquad\textbf{(E)}\ \frac{51}{256}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
The zeros of the function <math>f(x) = x^2-ax+2a</math> are integers. What is the sum of the possible values of <math>a</math>?<br />
<br />
<math> \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 16 \qquad\textbf{(D)}}\ 17 \qquad\textbf{(E)}\ 18</math><br />
<br />
[[2015 AMC 12A Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
For some positive integers <math>p</math>, there is a quadrilateral <math>ABCD</math> with positive integer side lengths, perimeter <math>p</math>, right angles at <math>B</math> and <math>C</math>, <math>AB=2</math>, and <math>CD=AD</math>. How many different values of <math>p<2015</math> are possible?<br />
<br />
<math> \textbf{(A)}\ 30 \qquad\textbf{(B)}\ 31 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}}\ 62 \qquad\textbf{(E)}\ 63</math><br />
<br />
[[2015 AMC 12A Problems/Problem 19|Solution]]<br />
<br />
==Problem 20==<br />
<br />
Isosceles triangles <math>T</math> and <math>T'</math> are not congruent but have the same area and the same perimeter. The sides of <math>T</math> have lengths of <math>5,5,</math> and <math>8</math>, while those of <math>T'</math> have lengths of <math>a,a,</math> and <math>b</math>. Which of the following numbers is closest to <math>b</math>?<br />
<br />
<math> \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}}\ 6 \qquad\textbf{(E)}\ 8</math><br />
<br />
[[2015 AMC 12A Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
A circle of radius <math>r</math> passes through both foci of, and exactly four points on, the ellipse with equation <math>x^2+16y^2=16</math>. The set of all possible values of <math>r</math> is an interval <math>[a,b)</math>. What is <math>a+b</math>?<br />
<br />
<math> \textbf{(A)}\ 5\sqrt{2}+4 \qquad\textbf{(B)}\ \sqrt{17}+7 \qquad\textbf{(C)}\ 6\sqrt{2}+3 \qquad\textbf{(D)}}\ \sqrt{15}+8 \qquad\textbf{(E)}\ 12</math><br />
<br />
[[2015 AMC 12A Problems/Problem 21|Solution]]<br />
<br />
==Problem 22==<br />
<br />
For each positive integer <math>n</math>, let <math>S(n)</math> be the number of sequences of length <math>n</math> consisting solely of the letters <math>A</math> and <math>B</math>, with no more than three <math>A</math>s in a row and no more than three <math>B</math>s in a row. What is the remainder when <math>S(2015)</math> is divided by 12?<br />
<br />
<math> \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}}\ 8 \qquad\textbf{(E)}\ 10</math><br />
<br />
[[2015 AMC 12A Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
Let <math>S</math> be a square of side length 1. Two points are chosen independently at random on the sides of <math>S</math>. The probability that the straight-line distance between the points is at least <math>\frac12</math> is <math>\frac{a-b\pi}{c}</math>, where <math>a,b,</math> and <math>c</math> are positive integers and <math>\text{gcd}(a,b,c) = 1</math>. What is <math>a+b+c</math>?<br />
<br />
<math> \textbf{(A)}\ 59 \qquad\textbf{(B)}\ 60 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}}\ 62 \qquad\textbf{(E)}\ 63</math><br />
<br />
[[2015 AMC 12A Problems/Problem 23|Solution]]<br />
<br />
==Problem 24==<br />
<br />
Rational numbers <math>a</math> and <math>b</math> are chosen at random among all rational numbers in the interval <math>[0,2)</math> that can be written as fractions <math>\frac{n}{d}</math> where <math>n</math> and <math>d</math> are integers with <math>1 \le d \le 5</math>. What is the probability that<br />
<cmath>(\text{cos}(a\pi)+i\text{sin}(b\pi))^4</cmath><br />
is a real number?<br />
<br />
<math> \textbf{(A)}\ \frac{3}{5} \qquad\textbf{(B)}\ \frac{4}{25} \qquad\textbf{(C)}\ \frac{41}{200} \qquad\textbf{(D)}}\ \frac{6}{25} \qquad\textbf{(E)}\ \frac{13}{50}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 24|Solution]]<br />
<br />
==Problem 25==<br />
<br />
A collection of circles in the upper half-plane, all tangent to the <math>x</math>-axis, is constructed in layers as follows. Layer <math>L_0</math> consists of two circles of radii <math>70^2</math> and <math>73^2</math> that are externally tangent. For <math>k\ge1</math>, the circles in <math>\bigcup_{j=0}^{k-1}L_j</math> are ordered according to their points of tangency with the <math>x</math>-axis. For every pair of consecutive circles in this order, a new circle is constructed externally tangent to each of the two circles in the pair. Layer <math>L_k</math> consists of the <math>2^{k-1}</math> circles constructed in this way. Let <math>S=\bigcup_{j=0}^{6}L_j</math>, and for every circle <math>C</math> denote by <math>r(C)</math> its radius. What is<br />
<cmath>\sum_{C\in S} \frac{1}{\sqrt{r(C)}}?</cmath><br />
<br />
<asy><br />
import olympiad;<br />
size(350);<br />
defaultpen(linewidth(0.7));<br />
// define a bunch of arrays and starting points<br />
pair[] coord = new pair[65];<br />
int[] trav = {32,16,8,4,2,1};<br />
coord[0] = (0,73^2); coord[64] = (2*73*70,70^2);<br />
// draw the big circles and the bottom line<br />
path arc1 = arc(coord[0],coord[0].y,260,360);<br />
path arc2 = arc(coord[64],coord[64].y,175,280);<br />
fill((coord[0].x-910,coord[0].y)--arc1--cycle,gray(0.75));<br />
fill((coord[64].x+870,coord[64].y+425)--arc2--cycle,gray(0.75));<br />
draw(arc1^^arc2);<br />
draw((-930,0)--(70^2+73^2+850,0));<br />
// We now apply the findCenter function 63 times to get<br />
// the location of the centers of all 63 constructed circles.<br />
// The complicated array setup ensures that all the circles<br />
// will be taken in the right order<br />
for(int i = 0;i<=5;i=i+1)<br />
{<br />
int skip = trav[i];<br />
for(int k=skip;k<=64 - skip; k = k + 2*skip)<br />
{<br />
pair cent1 = coord[k-skip], cent2 = coord[k+skip];<br />
real r1 = cent1.y, r2 = cent2.y, rn=r1*r2/((sqrt(r1)+sqrt(r2))^2);<br />
real shiftx = cent1.x + sqrt(4*r1*rn);<br />
coord[k] = (shiftx,rn);<br />
}<br />
// Draw the remaining 63 circles<br />
}<br />
for(int i=1;i<=63;i=i+1)<br />
{<br />
filldraw(circle(coord[i],coord[i].y),gray(0.75));<br />
}</asy><br />
<br />
<br />
$ \textbf{(A)}\ \frac{286}{35} \qquad\textbf{(B)}\ \frac{583}{70} \qquad\textbf{(C)}\ \frac{715}{73} \qquad\textbf{(D)}}\ \frac{143}{14} \qquad\textbf{(E)}\ \frac{1573}{146}$<br />
<br />
[[2015 AMC 12A Problems/Problem 25|Solution]]<br />
<br />
== See also ==<br />
* [[AMC Problems and Solutions]]<br />
{{MAA Notice}}</div>Devenwarehttps://artofproblemsolving.com/wiki/index.php?title=2015_AMC_12A_Problems&diff=680142015 AMC 12A Problems2015-03-01T23:26:07Z<p>Devenware: /* Problem 11 */</p>
<hr />
<div>==Problem 1==<br />
<br />
What is the value of <math>(2^0-1+5^2-0)^{-1}\times5?</math><br />
<br />
<math> \textbf{(A)}\ -125\qquad\textbf{(B)}\ -120\qquad\textbf{(C)}\ \frac{1}{5}\qquad\textbf{(D)}\ \frac{5}{24}\qquad\textbf{(E)}\ 25 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
<br />
Two of the three sides of a triangle are 20 and 15. Which of the following numbers is not a possible perimeter of the triangle?<br />
<br />
<math> \textbf{(A)}\ 52\qquad\textbf{(B)}\ 57\qquad\textbf{(C)}\ 62\qquad\textbf{(D)}\ 67\qquad\textbf{(E)}\ 72 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
<br />
Mr. Patrick teaches math to 15 students. He was grading tests and found that when he graded everyone's test except Payton's, the average grade for the class was 80. after he graded Payton's test, the class average became 81. What was Payton's score on the test?<br />
<br />
<math> \textbf{(A)}\ 81\qquad\textbf{(B)}\ 85\qquad\textbf{(C)}\ 91\qquad\textbf{(D)}\ 94\qquad\textbf{(E)}\ 95 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
The sum of two positive numbers is 5 times their difference. What is the ratio of the larger number to the smaller?<br />
<br />
<math> \textbf{(A)}\ \frac54 \qquad\textbf{(B)}\ \frac32 \qquad\textbf{(C)}\ \frac95 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ \frac52 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
Amelia needs to estimate the quantity <math>\frac{a}{b} - c</math>, where <math>a, b,</math> and <math>c</math> are large positive integers. She rounds each of the integers so that the calculation will be easier to do mentally. In which of these situations will her answer necessarily be greater than the exact value of <math>\frac{a}{b} - c</math>?<br />
<br />
<math> \textbf{(A)}\ \text{She rounds all three numbers up.}\\<br />
\qquad\textbf{(B)}\ \text{She rounds } a \text{ and } b \text{ up, and she rounds } c \text{ down.}\\<br />
\qquad\textbf{(C)}\ \text{She rounds } a \text{ and } c \text{ up, and she rounds } b \text{ down.} \\<br />
\qquad\textbf{(D)}\ \text{She rounds } a \text{ up, and she rounds } b \text{ and } c \text{ down.}\\<br />
\qquad\textbf{(E)}\ \text{She rounds } c \text{ up, and she rounds } a \text{ and } b \text{ down.} </math><br />
<br />
[[2015 AMC 12A Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be <math>2 : 1</math>?<br />
<br />
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 8</math><br />
<br />
[[2015 AMC 12A Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
Two right circular cylinders have the same volume. The radius of the second cylinder is <math>10\%</math> more than the radius of the first. What is the relationship between the heights of the two cylinders?<br />
<br />
<math>\textbf{(A)}\ \text{The second height is } 10\% \text{ less than the first.} \\ \textbf{(B)}\ \text{The first height is } 10\% \text{ more than the second.}\\ \textbf{(C)}\ \text{The second height is } 21\% \text{ less than the first.} \\ \textbf{(D)}\ \text{The first height is } 21\% \text{ more than the second.}\\ \textbf{(E)}\ \text{The second height is } 80\% \text{ of the first.}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
The ratio of the length to the width of a rectangle is <math>4</math> : <math>3</math>. If the rectangle has diagonal of length <math>d</math>, then the area may be expressed as <math>kd^2</math> for some constant <math>k</math>. What is <math>k</math>?<br />
<br />
<math> \textbf{(A)}\ \frac27 \qquad\textbf{(B)}\ \frac37 \qquad\textbf{(C)}\ \frac{12}{25} \qquad\textbf{(D)}\ \frac{16}{26} \qquad\textbf{(E)}\ \frac34</math><br />
<br />
[[2015 AMC 12A Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
A box contains 2 red marbles, 2 green marbles, and 2 yellow marbles. Carol takes 2 marbles from the box at random; then Claudia takes 2 of the remaining marbles at random; and then Cheryl takes the last 2 marbles. What is the probability that Cheryl gets 2 marbles of the same color?<br />
<br />
<math> \textbf{(A)}\ \frac{1}{10} \qquad\textbf{(B)}\ \frac16 \qquad\textbf{(C)}\ \frac15 \qquad\textbf{(D)}\ \frac13 \qquad\textbf{(E)}\ \frac12</math><br />
<br />
[[2015 AMC 12A Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
Integers <math>x</math> and <math>y</math> with <math>x>y>0</math> satisfy <math>x+y+xy=80</math>. What is <math>x</math>?<br />
<br />
<math> \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 15 \qquad\textbf{(D)}\ 18 \qquad\textbf{(E)}\ 26</math><br />
<br />
[[2015 AMC 12A Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
On a sheet of paper, Isabella draws a circle of radius <math>2</math>, a circle of radius <math>3</math>, and all possible lines simultaneously tangent to both circles. Isabella notices that she has drawn exactly <math>k \ge 0</math> lines. How many different values of <math>k</math> are possible?<br />
<br />
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6</math><br />
<br />
[[2015 AMC 12A Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
The parabolas <math>y=ax^2 - 2</math> and <math>y=4 - bx^2</math> intersect the coordinate axes in exactly four points, and these four points are the vertices of a kite of area <math>12</math>. What is <math>a+b</math>?<br />
<br />
<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 1.5\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}}\ 2.5\qquad\textbf{(E)}\ 3</math><br />
<br />
[[2015 AMC 12A Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
<br />
A league with 12 teams holds a round-robin tournament, with each team playing every other team exactly once. Games either end with one team victorious or else end in a draw. A team scores 2 points for every game it wins and 1 point for every game it draws. Which of the following is NOT a true statement about the list of 12 scores?<br />
<br />
<math> \textbf{(A)}\ \text{There must be an even number of odd scores.}\\<br />
\qquad\textbf{(B)}\ \text{There must be an even number of even scores.}\\<br />
\qquad\textbf{(C)}\ \text{There cannot be two scores of }0\text{.}\\<br />
\qquad\textbf{(D)}}\ \text{The sum of the scores must be at least }100\text{.}\\<br />
\qquad\textbf{(E)}\ \text{The highest score must be at least }12\text{.}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
What is the value of <math>a</math> for which <math>\frac{1}{\text{log}_2a} + \frac{1}{\text{log}_3a} + \frac{1}{\text{log}_4a} = 1</math>?<br />
<br />
[[2015 AMC 12A Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
What is the minimum number of digits to the right of the decimal point needed to express the fraction <math>\frac{123456789}{2^{26}\cdot 5^4}</math> as a decimal?<br />
<br />
<math> \textbf{(A)}\ 4\qquad\textbf{(B)}\ 22\qquad\textbf{(C)}\ 26\qquad\textbf{(D)}}\ 30\qquad\textbf{(E)}\ 104</math><br />
<br />
[[2015 AMC 12A Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
Tetrahedron <math>ABCD</math> has <math>AB=5,AC=3,BC=4,BD=4,AD=3,</math> and <math>CD=\frac{12}{5}\sqrt{2}</math>. What is the volume of the tetrahedron?<br />
<br />
<math> \textbf{(A)}\ 3\sqrt{2}\qquad\textbf{(B)}\ 2\sqrt{5}\qquad\textbf{(C)}\ \frac{24}{5}\qquad\textbf{(D)}}\ 3\sqrt{3}\qquad\textbf{(E)}\ \frac{24}{5}\sqrt{2}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?<br />
<br />
<math> \textbf{(A)}\ \frac{47}{256} \qquad\textbf{(B)}\ \frac{3}{16} \qquad\textbf{(C)}\ \frac{49}{256} \qquad\textbf{(D)}}\ \frac{25}{128} \qquad\textbf{(E)}\ \frac{51}{256}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
The zeros of the function <math>f(x) = x^2-ax+2a</math> are integers. What is the sum of the possible values of <math>a</math>?<br />
<br />
<math> \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 16 \qquad\textbf{(D)}}\ 17 \qquad\textbf{(E)}\ 18</math><br />
<br />
[[2015 AMC 12A Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
For some positive integers <math>p</math>, there is a quadrilateral <math>ABCD</math> with positive integer side lengths, perimeter <math>p</math>, right angles at <math>B</math> and <math>C</math>, <math>AB=2</math>, and <math>CD=AD</math>. How many different values of <math>p<2015</math> are possible?<br />
<br />
<math> \textbf{(A)}\ 30 \qquad\textbf{(B)}\ 31 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}}\ 62 \qquad\textbf{(E)}\ 63</math><br />
<br />
[[2015 AMC 12A Problems/Problem 19|Solution]]<br />
<br />
==Problem 20==<br />
<br />
Isosceles triangles <math>T</math> and <math>T'</math> are not congruent but have the same area and the same perimeter. The sides of <math>T</math> have lengths of <math>5,5,</math> and <math>8</math>, while those of <math>T'</math> have lengths of <math>a,a,</math> and <math>b</math>. Which of the following numbers is closest to <math>b</math>?<br />
<br />
<math> \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}}\ 6 \qquad\textbf{(E)}\ 8</math><br />
<br />
[[2015 AMC 12A Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
A circle of radius <math>r</math> passes through both foci of, and exactly four points on, the ellipse with equation <math>x^2+16y^2=16</math>. The set of all possible values of <math>r</math> is an interval <math>[a,b)</math>. What is <math>a+b</math>?<br />
<br />
<math> \textbf{(A)}\ 5\sqrt{2}+4 \qquad\textbf{(B)}\ \sqrt{17}+7 \qquad\textbf{(C)}\ 6\sqrt{2}+3 \qquad\textbf{(D)}}\ \sqrt{15}+8 \qquad\textbf{(E)}\ 12</math><br />
<br />
[[2015 AMC 12A Problems/Problem 21|Solution]]<br />
<br />
==Problem 22==<br />
<br />
For each positive integer <math>n</math>, let <math>S(n)</math> be the number of sequences of length <math>n</math> consisting solely of the letters <math>A</math> and <math>B</math>, with no more than three <math>A</math>s in a row and no more than three <math>B</math>s in a row. What is the remainder when <math>S(2015)</math> is divided by 12?<br />
<br />
<math> \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}}\ 8 \qquad\textbf{(E)}\ 10</math><br />
<br />
[[2015 AMC 12A Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
Let <math>S</math> be a square of side length 1. Two points are chosen independently at random on the sides of <math>S</math>. The probability that the straight-line distance between the points is at least <math>\frac12</math> is <math>\frac{a-b\pi}{c}</math>, where <math>a,b,</math> and <math>c</math> are positive integers and <math>\text{gcd}(a,b,c) = 1</math>. What is <math>a+b+c</math>?<br />
<br />
<math> \textbf{(A)}\ 59 \qquad\textbf{(B)}\ 60 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}}\ 62 \qquad\textbf{(E)}\ 63</math><br />
<br />
[[2015 AMC 12A Problems/Problem 23|Solution]]<br />
<br />
==Problem 24==<br />
<br />
Rational numbers <math>a</math> and <math>b</math> are chosen at random among all rational numbers in the interval <math>[0,2)</math> that can be written as fractions <math>\frac{n}{d}</math> where <math>n</math> and <math>d</math> are integers with <math>1 \le d \le 5</math>. What is the probability that<br />
<cmath>(\text{cos}(a\pi)+i\text{sin}(b\pi))^4</cmath><br />
is a real number?<br />
<br />
<math> \textbf{(A)}\ \frac{3}{5} \qquad\textbf{(B)}\ \frac{4}{25} \qquad\textbf{(C)}\ \frac{41}{200} \qquad\textbf{(D)}}\ \frac{6}{25} \qquad\textbf{(E)}\ \frac{13}{50}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 24|Solution]]<br />
<br />
==Problem 25==<br />
<br />
A collection of circles in the upper half-plane, all tangent to the <math>x</math>-axis, is constructed in layers as follows. Layer <math>L_0</math> consists of two circles of radii <math>70^2</math> and <math>73^2</math> that are externally tangent. For <math>k\ge1</math>, the circles in <math>\bigcup_{j=0}^{k-1}L_j</math> are ordered according to their points of tangency with the <math>x</math>-axis. For every pair of consecutive circles in this order, a new circle is constructed externally tangent to each of the two circles in the pair. Layer <math>L_k</math> consists of the <math>2^{k-1}</math> circles constructed in this way. Let <math>S=\bigcup_{j=0}^{6}L_j</math>, and for every circle <math>C</math> denote by <math>r(C)</math> its radius. What is<br />
<cmath>\sum_{C\in S} \frac{1}{\sqrt{r(C)}}?</cmath><br />
<br />
<asy><br />
import olympiad;<br />
size(350);<br />
defaultpen(linewidth(0.7));<br />
// define a bunch of arrays and starting points<br />
pair[] coord = new pair[65];<br />
int[] trav = {32,16,8,4,2,1};<br />
coord[0] = (0,73^2); coord[64] = (2*73*70,70^2);<br />
// draw the big circles and the bottom line<br />
path arc1 = arc(coord[0],coord[0].y,260,360);<br />
path arc2 = arc(coord[64],coord[64].y,175,280);<br />
fill((coord[0].x-910,coord[0].y)--arc1--cycle,gray(0.75));<br />
fill((coord[64].x+870,coord[64].y+425)--arc2--cycle,gray(0.75));<br />
draw(arc1^^arc2);<br />
draw((-930,0)--(70^2+73^2+850,0));<br />
// We now apply the findCenter function 63 times to get<br />
// the location of the centers of all 63 constructed circles.<br />
// The complicated array setup ensures that all the circles<br />
// will be taken in the right order<br />
for(int i = 0;i<=5;i=i+1)<br />
{<br />
int skip = trav[i];<br />
for(int k=skip;k<=64 - skip; k = k + 2*skip)<br />
{<br />
pair cent1 = coord[k-skip], cent2 = coord[k+skip];<br />
real r1 = cent1.y, r2 = cent2.y, rn=r1*r2/((sqrt(r1)+sqrt(r2))^2);<br />
real shiftx = cent1.x + sqrt(4*r1*rn);<br />
coord[k] = (shiftx,rn);<br />
}<br />
// Draw the remaining 63 circles<br />
}<br />
for(int i=1;i<=63;i=i+1)<br />
{<br />
filldraw(circle(coord[i],coord[i].y),gray(0.75));<br />
}</asy><br />
<br />
<br />
$ \textbf{(A)}\ \frac{286}{35} \qquad\textbf{(B)}\ \frac{583}{70} \qquad\textbf{(C)}\ \frac{715}{73} \qquad\textbf{(D)}}\ \frac{143}{14} \qquad\textbf{(E)}\ \frac{1573}{146}$<br />
<br />
[[2015 AMC 12A Problems/Problem 25|Solution]]<br />
<br />
== See also ==<br />
* [[AMC Problems and Solutions]]<br />
{{MAA Notice}}</div>Devenwarehttps://artofproblemsolving.com/wiki/index.php?title=2015_AMC_12A_Problems&diff=680132015 AMC 12A Problems2015-03-01T23:25:37Z<p>Devenware: /* Problem 10 */</p>
<hr />
<div>==Problem 1==<br />
<br />
What is the value of <math>(2^0-1+5^2-0)^{-1}\times5?</math><br />
<br />
<math> \textbf{(A)}\ -125\qquad\textbf{(B)}\ -120\qquad\textbf{(C)}\ \frac{1}{5}\qquad\textbf{(D)}\ \frac{5}{24}\qquad\textbf{(E)}\ 25 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
<br />
Two of the three sides of a triangle are 20 and 15. Which of the following numbers is not a possible perimeter of the triangle?<br />
<br />
<math> \textbf{(A)}\ 52\qquad\textbf{(B)}\ 57\qquad\textbf{(C)}\ 62\qquad\textbf{(D)}\ 67\qquad\textbf{(E)}\ 72 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
<br />
Mr. Patrick teaches math to 15 students. He was grading tests and found that when he graded everyone's test except Payton's, the average grade for the class was 80. after he graded Payton's test, the class average became 81. What was Payton's score on the test?<br />
<br />
<math> \textbf{(A)}\ 81\qquad\textbf{(B)}\ 85\qquad\textbf{(C)}\ 91\qquad\textbf{(D)}\ 94\qquad\textbf{(E)}\ 95 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
The sum of two positive numbers is 5 times their difference. What is the ratio of the larger number to the smaller?<br />
<br />
<math> \textbf{(A)}\ \frac54 \qquad\textbf{(B)}\ \frac32 \qquad\textbf{(C)}\ \frac95 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ \frac52 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
Amelia needs to estimate the quantity <math>\frac{a}{b} - c</math>, where <math>a, b,</math> and <math>c</math> are large positive integers. She rounds each of the integers so that the calculation will be easier to do mentally. In which of these situations will her answer necessarily be greater than the exact value of <math>\frac{a}{b} - c</math>?<br />
<br />
<math> \textbf{(A)}\ \text{She rounds all three numbers up.}\\<br />
\qquad\textbf{(B)}\ \text{She rounds } a \text{ and } b \text{ up, and she rounds } c \text{ down.}\\<br />
\qquad\textbf{(C)}\ \text{She rounds } a \text{ and } c \text{ up, and she rounds } b \text{ down.} \\<br />
\qquad\textbf{(D)}\ \text{She rounds } a \text{ up, and she rounds } b \text{ and } c \text{ down.}\\<br />
\qquad\textbf{(E)}\ \text{She rounds } c \text{ up, and she rounds } a \text{ and } b \text{ down.} </math><br />
<br />
[[2015 AMC 12A Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be <math>2 : 1</math>?<br />
<br />
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 8</math><br />
<br />
[[2015 AMC 12A Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
Two right circular cylinders have the same volume. The radius of the second cylinder is <math>10\%</math> more than the radius of the first. What is the relationship between the heights of the two cylinders?<br />
<br />
<math>\textbf{(A)}\ \text{The second height is } 10\% \text{ less than the first.} \\ \textbf{(B)}\ \text{The first height is } 10\% \text{ more than the second.}\\ \textbf{(C)}\ \text{The second height is } 21\% \text{ less than the first.} \\ \textbf{(D)}\ \text{The first height is } 21\% \text{ more than the second.}\\ \textbf{(E)}\ \text{The second height is } 80\% \text{ of the first.}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
The ratio of the length to the width of a rectangle is <math>4</math> : <math>3</math>. If the rectangle has diagonal of length <math>d</math>, then the area may be expressed as <math>kd^2</math> for some constant <math>k</math>. What is <math>k</math>?<br />
<br />
<math> \textbf{(A)}\ \frac27 \qquad\textbf{(B)}\ \frac37 \qquad\textbf{(C)}\ \frac{12}{25} \qquad\textbf{(D)}\ \frac{16}{26} \qquad\textbf{(E)}\ \frac34</math><br />
<br />
[[2015 AMC 12A Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
A box contains 2 red marbles, 2 green marbles, and 2 yellow marbles. Carol takes 2 marbles from the box at random; then Claudia takes 2 of the remaining marbles at random; and then Cheryl takes the last 2 marbles. What is the probability that Cheryl gets 2 marbles of the same color?<br />
<br />
<math> \textbf{(A)}\ \frac{1}{10} \qquad\textbf{(B)}\ \frac16 \qquad\textbf{(C)}\ \frac15 \qquad\textbf{(D)}\ \frac13 \qquad\textbf{(E)}\ \frac12</math><br />
<br />
[[2015 AMC 12A Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
Integers <math>x</math> and <math>y</math> with <math>x>y>0</math> satisfy <math>x+y+xy=80</math>. What is <math>x</math>?<br />
<br />
<math> \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 15 \qquad\textbf{(D)}\ 18 \qquad\textbf{(E)}\ 26</math><br />
<br />
[[2015 AMC 12A Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
On a sheet of paper, Isabella draws a circle of radius <math>2</math>, a circle of radius <math>3</math>, and all possible lines simultaneously tangent to both circles. Isabella notices that she has drawn exactly <math>k \ge 0</math> lines. How many different values of <math>k</math> are possible?<br />
<br />
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}}\ 5\qquad\textbf{(E)}\ 6</math><br />
<br />
[[2015 AMC 12A Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
The parabolas <math>y=ax^2 - 2</math> and <math>y=4 - bx^2</math> intersect the coordinate axes in exactly four points, and these four points are the vertices of a kite of area <math>12</math>. What is <math>a+b</math>?<br />
<br />
<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 1.5\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}}\ 2.5\qquad\textbf{(E)}\ 3</math><br />
<br />
[[2015 AMC 12A Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
<br />
A league with 12 teams holds a round-robin tournament, with each team playing every other team exactly once. Games either end with one team victorious or else end in a draw. A team scores 2 points for every game it wins and 1 point for every game it draws. Which of the following is NOT a true statement about the list of 12 scores?<br />
<br />
<math> \textbf{(A)}\ \text{There must be an even number of odd scores.}\\<br />
\qquad\textbf{(B)}\ \text{There must be an even number of even scores.}\\<br />
\qquad\textbf{(C)}\ \text{There cannot be two scores of }0\text{.}\\<br />
\qquad\textbf{(D)}}\ \text{The sum of the scores must be at least }100\text{.}\\<br />
\qquad\textbf{(E)}\ \text{The highest score must be at least }12\text{.}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
What is the value of <math>a</math> for which <math>\frac{1}{\text{log}_2a} + \frac{1}{\text{log}_3a} + \frac{1}{\text{log}_4a} = 1</math>?<br />
<br />
[[2015 AMC 12A Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
What is the minimum number of digits to the right of the decimal point needed to express the fraction <math>\frac{123456789}{2^{26}\cdot 5^4}</math> as a decimal?<br />
<br />
<math> \textbf{(A)}\ 4\qquad\textbf{(B)}\ 22\qquad\textbf{(C)}\ 26\qquad\textbf{(D)}}\ 30\qquad\textbf{(E)}\ 104</math><br />
<br />
[[2015 AMC 12A Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
Tetrahedron <math>ABCD</math> has <math>AB=5,AC=3,BC=4,BD=4,AD=3,</math> and <math>CD=\frac{12}{5}\sqrt{2}</math>. What is the volume of the tetrahedron?<br />
<br />
<math> \textbf{(A)}\ 3\sqrt{2}\qquad\textbf{(B)}\ 2\sqrt{5}\qquad\textbf{(C)}\ \frac{24}{5}\qquad\textbf{(D)}}\ 3\sqrt{3}\qquad\textbf{(E)}\ \frac{24}{5}\sqrt{2}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?<br />
<br />
<math> \textbf{(A)}\ \frac{47}{256} \qquad\textbf{(B)}\ \frac{3}{16} \qquad\textbf{(C)}\ \frac{49}{256} \qquad\textbf{(D)}}\ \frac{25}{128} \qquad\textbf{(E)}\ \frac{51}{256}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
The zeros of the function <math>f(x) = x^2-ax+2a</math> are integers. What is the sum of the possible values of <math>a</math>?<br />
<br />
<math> \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 16 \qquad\textbf{(D)}}\ 17 \qquad\textbf{(E)}\ 18</math><br />
<br />
[[2015 AMC 12A Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
For some positive integers <math>p</math>, there is a quadrilateral <math>ABCD</math> with positive integer side lengths, perimeter <math>p</math>, right angles at <math>B</math> and <math>C</math>, <math>AB=2</math>, and <math>CD=AD</math>. How many different values of <math>p<2015</math> are possible?<br />
<br />
<math> \textbf{(A)}\ 30 \qquad\textbf{(B)}\ 31 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}}\ 62 \qquad\textbf{(E)}\ 63</math><br />
<br />
[[2015 AMC 12A Problems/Problem 19|Solution]]<br />
<br />
==Problem 20==<br />
<br />
Isosceles triangles <math>T</math> and <math>T'</math> are not congruent but have the same area and the same perimeter. The sides of <math>T</math> have lengths of <math>5,5,</math> and <math>8</math>, while those of <math>T'</math> have lengths of <math>a,a,</math> and <math>b</math>. Which of the following numbers is closest to <math>b</math>?<br />
<br />
<math> \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}}\ 6 \qquad\textbf{(E)}\ 8</math><br />
<br />
[[2015 AMC 12A Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
A circle of radius <math>r</math> passes through both foci of, and exactly four points on, the ellipse with equation <math>x^2+16y^2=16</math>. The set of all possible values of <math>r</math> is an interval <math>[a,b)</math>. What is <math>a+b</math>?<br />
<br />
<math> \textbf{(A)}\ 5\sqrt{2}+4 \qquad\textbf{(B)}\ \sqrt{17}+7 \qquad\textbf{(C)}\ 6\sqrt{2}+3 \qquad\textbf{(D)}}\ \sqrt{15}+8 \qquad\textbf{(E)}\ 12</math><br />
<br />
[[2015 AMC 12A Problems/Problem 21|Solution]]<br />
<br />
==Problem 22==<br />
<br />
For each positive integer <math>n</math>, let <math>S(n)</math> be the number of sequences of length <math>n</math> consisting solely of the letters <math>A</math> and <math>B</math>, with no more than three <math>A</math>s in a row and no more than three <math>B</math>s in a row. What is the remainder when <math>S(2015)</math> is divided by 12?<br />
<br />
<math> \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}}\ 8 \qquad\textbf{(E)}\ 10</math><br />
<br />
[[2015 AMC 12A Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
Let <math>S</math> be a square of side length 1. Two points are chosen independently at random on the sides of <math>S</math>. The probability that the straight-line distance between the points is at least <math>\frac12</math> is <math>\frac{a-b\pi}{c}</math>, where <math>a,b,</math> and <math>c</math> are positive integers and <math>\text{gcd}(a,b,c) = 1</math>. What is <math>a+b+c</math>?<br />
<br />
<math> \textbf{(A)}\ 59 \qquad\textbf{(B)}\ 60 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}}\ 62 \qquad\textbf{(E)}\ 63</math><br />
<br />
[[2015 AMC 12A Problems/Problem 23|Solution]]<br />
<br />
==Problem 24==<br />
<br />
Rational numbers <math>a</math> and <math>b</math> are chosen at random among all rational numbers in the interval <math>[0,2)</math> that can be written as fractions <math>\frac{n}{d}</math> where <math>n</math> and <math>d</math> are integers with <math>1 \le d \le 5</math>. What is the probability that<br />
<cmath>(\text{cos}(a\pi)+i\text{sin}(b\pi))^4</cmath><br />
is a real number?<br />
<br />
<math> \textbf{(A)}\ \frac{3}{5} \qquad\textbf{(B)}\ \frac{4}{25} \qquad\textbf{(C)}\ \frac{41}{200} \qquad\textbf{(D)}}\ \frac{6}{25} \qquad\textbf{(E)}\ \frac{13}{50}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 24|Solution]]<br />
<br />
==Problem 25==<br />
<br />
A collection of circles in the upper half-plane, all tangent to the <math>x</math>-axis, is constructed in layers as follows. Layer <math>L_0</math> consists of two circles of radii <math>70^2</math> and <math>73^2</math> that are externally tangent. For <math>k\ge1</math>, the circles in <math>\bigcup_{j=0}^{k-1}L_j</math> are ordered according to their points of tangency with the <math>x</math>-axis. For every pair of consecutive circles in this order, a new circle is constructed externally tangent to each of the two circles in the pair. Layer <math>L_k</math> consists of the <math>2^{k-1}</math> circles constructed in this way. Let <math>S=\bigcup_{j=0}^{6}L_j</math>, and for every circle <math>C</math> denote by <math>r(C)</math> its radius. What is<br />
<cmath>\sum_{C\in S} \frac{1}{\sqrt{r(C)}}?</cmath><br />
<br />
<asy><br />
import olympiad;<br />
size(350);<br />
defaultpen(linewidth(0.7));<br />
// define a bunch of arrays and starting points<br />
pair[] coord = new pair[65];<br />
int[] trav = {32,16,8,4,2,1};<br />
coord[0] = (0,73^2); coord[64] = (2*73*70,70^2);<br />
// draw the big circles and the bottom line<br />
path arc1 = arc(coord[0],coord[0].y,260,360);<br />
path arc2 = arc(coord[64],coord[64].y,175,280);<br />
fill((coord[0].x-910,coord[0].y)--arc1--cycle,gray(0.75));<br />
fill((coord[64].x+870,coord[64].y+425)--arc2--cycle,gray(0.75));<br />
draw(arc1^^arc2);<br />
draw((-930,0)--(70^2+73^2+850,0));<br />
// We now apply the findCenter function 63 times to get<br />
// the location of the centers of all 63 constructed circles.<br />
// The complicated array setup ensures that all the circles<br />
// will be taken in the right order<br />
for(int i = 0;i<=5;i=i+1)<br />
{<br />
int skip = trav[i];<br />
for(int k=skip;k<=64 - skip; k = k + 2*skip)<br />
{<br />
pair cent1 = coord[k-skip], cent2 = coord[k+skip];<br />
real r1 = cent1.y, r2 = cent2.y, rn=r1*r2/((sqrt(r1)+sqrt(r2))^2);<br />
real shiftx = cent1.x + sqrt(4*r1*rn);<br />
coord[k] = (shiftx,rn);<br />
}<br />
// Draw the remaining 63 circles<br />
}<br />
for(int i=1;i<=63;i=i+1)<br />
{<br />
filldraw(circle(coord[i],coord[i].y),gray(0.75));<br />
}</asy><br />
<br />
<br />
$ \textbf{(A)}\ \frac{286}{35} \qquad\textbf{(B)}\ \frac{583}{70} \qquad\textbf{(C)}\ \frac{715}{73} \qquad\textbf{(D)}}\ \frac{143}{14} \qquad\textbf{(E)}\ \frac{1573}{146}$<br />
<br />
[[2015 AMC 12A Problems/Problem 25|Solution]]<br />
<br />
== See also ==<br />
* [[AMC Problems and Solutions]]<br />
{{MAA Notice}}</div>Devenwarehttps://artofproblemsolving.com/wiki/index.php?title=2015_AMC_12A_Problems&diff=680122015 AMC 12A Problems2015-03-01T23:25:16Z<p>Devenware: /* Problem 9 */</p>
<hr />
<div>==Problem 1==<br />
<br />
What is the value of <math>(2^0-1+5^2-0)^{-1}\times5?</math><br />
<br />
<math> \textbf{(A)}\ -125\qquad\textbf{(B)}\ -120\qquad\textbf{(C)}\ \frac{1}{5}\qquad\textbf{(D)}\ \frac{5}{24}\qquad\textbf{(E)}\ 25 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
<br />
Two of the three sides of a triangle are 20 and 15. Which of the following numbers is not a possible perimeter of the triangle?<br />
<br />
<math> \textbf{(A)}\ 52\qquad\textbf{(B)}\ 57\qquad\textbf{(C)}\ 62\qquad\textbf{(D)}\ 67\qquad\textbf{(E)}\ 72 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
<br />
Mr. Patrick teaches math to 15 students. He was grading tests and found that when he graded everyone's test except Payton's, the average grade for the class was 80. after he graded Payton's test, the class average became 81. What was Payton's score on the test?<br />
<br />
<math> \textbf{(A)}\ 81\qquad\textbf{(B)}\ 85\qquad\textbf{(C)}\ 91\qquad\textbf{(D)}\ 94\qquad\textbf{(E)}\ 95 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
The sum of two positive numbers is 5 times their difference. What is the ratio of the larger number to the smaller?<br />
<br />
<math> \textbf{(A)}\ \frac54 \qquad\textbf{(B)}\ \frac32 \qquad\textbf{(C)}\ \frac95 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ \frac52 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
Amelia needs to estimate the quantity <math>\frac{a}{b} - c</math>, where <math>a, b,</math> and <math>c</math> are large positive integers. She rounds each of the integers so that the calculation will be easier to do mentally. In which of these situations will her answer necessarily be greater than the exact value of <math>\frac{a}{b} - c</math>?<br />
<br />
<math> \textbf{(A)}\ \text{She rounds all three numbers up.}\\<br />
\qquad\textbf{(B)}\ \text{She rounds } a \text{ and } b \text{ up, and she rounds } c \text{ down.}\\<br />
\qquad\textbf{(C)}\ \text{She rounds } a \text{ and } c \text{ up, and she rounds } b \text{ down.} \\<br />
\qquad\textbf{(D)}\ \text{She rounds } a \text{ up, and she rounds } b \text{ and } c \text{ down.}\\<br />
\qquad\textbf{(E)}\ \text{She rounds } c \text{ up, and she rounds } a \text{ and } b \text{ down.} </math><br />
<br />
[[2015 AMC 12A Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be <math>2 : 1</math>?<br />
<br />
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 8</math><br />
<br />
[[2015 AMC 12A Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
Two right circular cylinders have the same volume. The radius of the second cylinder is <math>10\%</math> more than the radius of the first. What is the relationship between the heights of the two cylinders?<br />
<br />
<math>\textbf{(A)}\ \text{The second height is } 10\% \text{ less than the first.} \\ \textbf{(B)}\ \text{The first height is } 10\% \text{ more than the second.}\\ \textbf{(C)}\ \text{The second height is } 21\% \text{ less than the first.} \\ \textbf{(D)}\ \text{The first height is } 21\% \text{ more than the second.}\\ \textbf{(E)}\ \text{The second height is } 80\% \text{ of the first.}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
The ratio of the length to the width of a rectangle is <math>4</math> : <math>3</math>. If the rectangle has diagonal of length <math>d</math>, then the area may be expressed as <math>kd^2</math> for some constant <math>k</math>. What is <math>k</math>?<br />
<br />
<math> \textbf{(A)}\ \frac27 \qquad\textbf{(B)}\ \frac37 \qquad\textbf{(C)}\ \frac{12}{25} \qquad\textbf{(D)}\ \frac{16}{26} \qquad\textbf{(E)}\ \frac34</math><br />
<br />
[[2015 AMC 12A Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
A box contains 2 red marbles, 2 green marbles, and 2 yellow marbles. Carol takes 2 marbles from the box at random; then Claudia takes 2 of the remaining marbles at random; and then Cheryl takes the last 2 marbles. What is the probability that Cheryl gets 2 marbles of the same color?<br />
<br />
<math> \textbf{(A)}\ \frac{1}{10} \qquad\textbf{(B)}\ \frac16 \qquad\textbf{(C)}\ \frac15 \qquad\textbf{(D)}\ \frac13 \qquad\textbf{(E)}\ \frac12</math><br />
<br />
[[2015 AMC 12A Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
Integers <math>x</math> and <math>y</math> with <math>x>y>0</math> satisfy <math>x+y+xy=80</math>. What is <math>x</math>?<br />
<br />
<math> \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 15 \qquad\textbf{(D)}}\ 18 \qquad\textbf{(E)}\ 26</math><br />
<br />
[[2015 AMC 12A Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
On a sheet of paper, Isabella draws a circle of radius <math>2</math>, a circle of radius <math>3</math>, and all possible lines simultaneously tangent to both circles. Isabella notices that she has drawn exactly <math>k \ge 0</math> lines. How many different values of <math>k</math> are possible?<br />
<br />
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}}\ 5\qquad\textbf{(E)}\ 6</math><br />
<br />
[[2015 AMC 12A Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
The parabolas <math>y=ax^2 - 2</math> and <math>y=4 - bx^2</math> intersect the coordinate axes in exactly four points, and these four points are the vertices of a kite of area <math>12</math>. What is <math>a+b</math>?<br />
<br />
<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 1.5\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}}\ 2.5\qquad\textbf{(E)}\ 3</math><br />
<br />
[[2015 AMC 12A Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
<br />
A league with 12 teams holds a round-robin tournament, with each team playing every other team exactly once. Games either end with one team victorious or else end in a draw. A team scores 2 points for every game it wins and 1 point for every game it draws. Which of the following is NOT a true statement about the list of 12 scores?<br />
<br />
<math> \textbf{(A)}\ \text{There must be an even number of odd scores.}\\<br />
\qquad\textbf{(B)}\ \text{There must be an even number of even scores.}\\<br />
\qquad\textbf{(C)}\ \text{There cannot be two scores of }0\text{.}\\<br />
\qquad\textbf{(D)}}\ \text{The sum of the scores must be at least }100\text{.}\\<br />
\qquad\textbf{(E)}\ \text{The highest score must be at least }12\text{.}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
What is the value of <math>a</math> for which <math>\frac{1}{\text{log}_2a} + \frac{1}{\text{log}_3a} + \frac{1}{\text{log}_4a} = 1</math>?<br />
<br />
[[2015 AMC 12A Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
What is the minimum number of digits to the right of the decimal point needed to express the fraction <math>\frac{123456789}{2^{26}\cdot 5^4}</math> as a decimal?<br />
<br />
<math> \textbf{(A)}\ 4\qquad\textbf{(B)}\ 22\qquad\textbf{(C)}\ 26\qquad\textbf{(D)}}\ 30\qquad\textbf{(E)}\ 104</math><br />
<br />
[[2015 AMC 12A Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
Tetrahedron <math>ABCD</math> has <math>AB=5,AC=3,BC=4,BD=4,AD=3,</math> and <math>CD=\frac{12}{5}\sqrt{2}</math>. What is the volume of the tetrahedron?<br />
<br />
<math> \textbf{(A)}\ 3\sqrt{2}\qquad\textbf{(B)}\ 2\sqrt{5}\qquad\textbf{(C)}\ \frac{24}{5}\qquad\textbf{(D)}}\ 3\sqrt{3}\qquad\textbf{(E)}\ \frac{24}{5}\sqrt{2}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?<br />
<br />
<math> \textbf{(A)}\ \frac{47}{256} \qquad\textbf{(B)}\ \frac{3}{16} \qquad\textbf{(C)}\ \frac{49}{256} \qquad\textbf{(D)}}\ \frac{25}{128} \qquad\textbf{(E)}\ \frac{51}{256}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
The zeros of the function <math>f(x) = x^2-ax+2a</math> are integers. What is the sum of the possible values of <math>a</math>?<br />
<br />
<math> \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 16 \qquad\textbf{(D)}}\ 17 \qquad\textbf{(E)}\ 18</math><br />
<br />
[[2015 AMC 12A Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
For some positive integers <math>p</math>, there is a quadrilateral <math>ABCD</math> with positive integer side lengths, perimeter <math>p</math>, right angles at <math>B</math> and <math>C</math>, <math>AB=2</math>, and <math>CD=AD</math>. How many different values of <math>p<2015</math> are possible?<br />
<br />
<math> \textbf{(A)}\ 30 \qquad\textbf{(B)}\ 31 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}}\ 62 \qquad\textbf{(E)}\ 63</math><br />
<br />
[[2015 AMC 12A Problems/Problem 19|Solution]]<br />
<br />
==Problem 20==<br />
<br />
Isosceles triangles <math>T</math> and <math>T'</math> are not congruent but have the same area and the same perimeter. The sides of <math>T</math> have lengths of <math>5,5,</math> and <math>8</math>, while those of <math>T'</math> have lengths of <math>a,a,</math> and <math>b</math>. Which of the following numbers is closest to <math>b</math>?<br />
<br />
<math> \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}}\ 6 \qquad\textbf{(E)}\ 8</math><br />
<br />
[[2015 AMC 12A Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
A circle of radius <math>r</math> passes through both foci of, and exactly four points on, the ellipse with equation <math>x^2+16y^2=16</math>. The set of all possible values of <math>r</math> is an interval <math>[a,b)</math>. What is <math>a+b</math>?<br />
<br />
<math> \textbf{(A)}\ 5\sqrt{2}+4 \qquad\textbf{(B)}\ \sqrt{17}+7 \qquad\textbf{(C)}\ 6\sqrt{2}+3 \qquad\textbf{(D)}}\ \sqrt{15}+8 \qquad\textbf{(E)}\ 12</math><br />
<br />
[[2015 AMC 12A Problems/Problem 21|Solution]]<br />
<br />
==Problem 22==<br />
<br />
For each positive integer <math>n</math>, let <math>S(n)</math> be the number of sequences of length <math>n</math> consisting solely of the letters <math>A</math> and <math>B</math>, with no more than three <math>A</math>s in a row and no more than three <math>B</math>s in a row. What is the remainder when <math>S(2015)</math> is divided by 12?<br />
<br />
<math> \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}}\ 8 \qquad\textbf{(E)}\ 10</math><br />
<br />
[[2015 AMC 12A Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
Let <math>S</math> be a square of side length 1. Two points are chosen independently at random on the sides of <math>S</math>. The probability that the straight-line distance between the points is at least <math>\frac12</math> is <math>\frac{a-b\pi}{c}</math>, where <math>a,b,</math> and <math>c</math> are positive integers and <math>\text{gcd}(a,b,c) = 1</math>. What is <math>a+b+c</math>?<br />
<br />
<math> \textbf{(A)}\ 59 \qquad\textbf{(B)}\ 60 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}}\ 62 \qquad\textbf{(E)}\ 63</math><br />
<br />
[[2015 AMC 12A Problems/Problem 23|Solution]]<br />
<br />
==Problem 24==<br />
<br />
Rational numbers <math>a</math> and <math>b</math> are chosen at random among all rational numbers in the interval <math>[0,2)</math> that can be written as fractions <math>\frac{n}{d}</math> where <math>n</math> and <math>d</math> are integers with <math>1 \le d \le 5</math>. What is the probability that<br />
<cmath>(\text{cos}(a\pi)+i\text{sin}(b\pi))^4</cmath><br />
is a real number?<br />
<br />
<math> \textbf{(A)}\ \frac{3}{5} \qquad\textbf{(B)}\ \frac{4}{25} \qquad\textbf{(C)}\ \frac{41}{200} \qquad\textbf{(D)}}\ \frac{6}{25} \qquad\textbf{(E)}\ \frac{13}{50}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 24|Solution]]<br />
<br />
==Problem 25==<br />
<br />
A collection of circles in the upper half-plane, all tangent to the <math>x</math>-axis, is constructed in layers as follows. Layer <math>L_0</math> consists of two circles of radii <math>70^2</math> and <math>73^2</math> that are externally tangent. For <math>k\ge1</math>, the circles in <math>\bigcup_{j=0}^{k-1}L_j</math> are ordered according to their points of tangency with the <math>x</math>-axis. For every pair of consecutive circles in this order, a new circle is constructed externally tangent to each of the two circles in the pair. Layer <math>L_k</math> consists of the <math>2^{k-1}</math> circles constructed in this way. Let <math>S=\bigcup_{j=0}^{6}L_j</math>, and for every circle <math>C</math> denote by <math>r(C)</math> its radius. What is<br />
<cmath>\sum_{C\in S} \frac{1}{\sqrt{r(C)}}?</cmath><br />
<br />
<asy><br />
import olympiad;<br />
size(350);<br />
defaultpen(linewidth(0.7));<br />
// define a bunch of arrays and starting points<br />
pair[] coord = new pair[65];<br />
int[] trav = {32,16,8,4,2,1};<br />
coord[0] = (0,73^2); coord[64] = (2*73*70,70^2);<br />
// draw the big circles and the bottom line<br />
path arc1 = arc(coord[0],coord[0].y,260,360);<br />
path arc2 = arc(coord[64],coord[64].y,175,280);<br />
fill((coord[0].x-910,coord[0].y)--arc1--cycle,gray(0.75));<br />
fill((coord[64].x+870,coord[64].y+425)--arc2--cycle,gray(0.75));<br />
draw(arc1^^arc2);<br />
draw((-930,0)--(70^2+73^2+850,0));<br />
// We now apply the findCenter function 63 times to get<br />
// the location of the centers of all 63 constructed circles.<br />
// The complicated array setup ensures that all the circles<br />
// will be taken in the right order<br />
for(int i = 0;i<=5;i=i+1)<br />
{<br />
int skip = trav[i];<br />
for(int k=skip;k<=64 - skip; k = k + 2*skip)<br />
{<br />
pair cent1 = coord[k-skip], cent2 = coord[k+skip];<br />
real r1 = cent1.y, r2 = cent2.y, rn=r1*r2/((sqrt(r1)+sqrt(r2))^2);<br />
real shiftx = cent1.x + sqrt(4*r1*rn);<br />
coord[k] = (shiftx,rn);<br />
}<br />
// Draw the remaining 63 circles<br />
}<br />
for(int i=1;i<=63;i=i+1)<br />
{<br />
filldraw(circle(coord[i],coord[i].y),gray(0.75));<br />
}</asy><br />
<br />
<br />
$ \textbf{(A)}\ \frac{286}{35} \qquad\textbf{(B)}\ \frac{583}{70} \qquad\textbf{(C)}\ \frac{715}{73} \qquad\textbf{(D)}}\ \frac{143}{14} \qquad\textbf{(E)}\ \frac{1573}{146}$<br />
<br />
[[2015 AMC 12A Problems/Problem 25|Solution]]<br />
<br />
== See also ==<br />
* [[AMC Problems and Solutions]]<br />
{{MAA Notice}}</div>Devenwarehttps://artofproblemsolving.com/wiki/index.php?title=2015_AMC_12A_Problems&diff=680112015 AMC 12A Problems2015-03-01T23:24:56Z<p>Devenware: /* Problem 8 */</p>
<hr />
<div>==Problem 1==<br />
<br />
What is the value of <math>(2^0-1+5^2-0)^{-1}\times5?</math><br />
<br />
<math> \textbf{(A)}\ -125\qquad\textbf{(B)}\ -120\qquad\textbf{(C)}\ \frac{1}{5}\qquad\textbf{(D)}\ \frac{5}{24}\qquad\textbf{(E)}\ 25 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
<br />
Two of the three sides of a triangle are 20 and 15. Which of the following numbers is not a possible perimeter of the triangle?<br />
<br />
<math> \textbf{(A)}\ 52\qquad\textbf{(B)}\ 57\qquad\textbf{(C)}\ 62\qquad\textbf{(D)}\ 67\qquad\textbf{(E)}\ 72 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
<br />
Mr. Patrick teaches math to 15 students. He was grading tests and found that when he graded everyone's test except Payton's, the average grade for the class was 80. after he graded Payton's test, the class average became 81. What was Payton's score on the test?<br />
<br />
<math> \textbf{(A)}\ 81\qquad\textbf{(B)}\ 85\qquad\textbf{(C)}\ 91\qquad\textbf{(D)}\ 94\qquad\textbf{(E)}\ 95 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
The sum of two positive numbers is 5 times their difference. What is the ratio of the larger number to the smaller?<br />
<br />
<math> \textbf{(A)}\ \frac54 \qquad\textbf{(B)}\ \frac32 \qquad\textbf{(C)}\ \frac95 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ \frac52 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
Amelia needs to estimate the quantity <math>\frac{a}{b} - c</math>, where <math>a, b,</math> and <math>c</math> are large positive integers. She rounds each of the integers so that the calculation will be easier to do mentally. In which of these situations will her answer necessarily be greater than the exact value of <math>\frac{a}{b} - c</math>?<br />
<br />
<math> \textbf{(A)}\ \text{She rounds all three numbers up.}\\<br />
\qquad\textbf{(B)}\ \text{She rounds } a \text{ and } b \text{ up, and she rounds } c \text{ down.}\\<br />
\qquad\textbf{(C)}\ \text{She rounds } a \text{ and } c \text{ up, and she rounds } b \text{ down.} \\<br />
\qquad\textbf{(D)}\ \text{She rounds } a \text{ up, and she rounds } b \text{ and } c \text{ down.}\\<br />
\qquad\textbf{(E)}\ \text{She rounds } c \text{ up, and she rounds } a \text{ and } b \text{ down.} </math><br />
<br />
[[2015 AMC 12A Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be <math>2 : 1</math>?<br />
<br />
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 8</math><br />
<br />
[[2015 AMC 12A Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
Two right circular cylinders have the same volume. The radius of the second cylinder is <math>10\%</math> more than the radius of the first. What is the relationship between the heights of the two cylinders?<br />
<br />
<math>\textbf{(A)}\ \text{The second height is } 10\% \text{ less than the first.} \\ \textbf{(B)}\ \text{The first height is } 10\% \text{ more than the second.}\\ \textbf{(C)}\ \text{The second height is } 21\% \text{ less than the first.} \\ \textbf{(D)}\ \text{The first height is } 21\% \text{ more than the second.}\\ \textbf{(E)}\ \text{The second height is } 80\% \text{ of the first.}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
The ratio of the length to the width of a rectangle is <math>4</math> : <math>3</math>. If the rectangle has diagonal of length <math>d</math>, then the area may be expressed as <math>kd^2</math> for some constant <math>k</math>. What is <math>k</math>?<br />
<br />
<math> \textbf{(A)}\ \frac27 \qquad\textbf{(B)}\ \frac37 \qquad\textbf{(C)}\ \frac{12}{25} \qquad\textbf{(D)}\ \frac{16}{26} \qquad\textbf{(E)}\ \frac34</math><br />
<br />
[[2015 AMC 12A Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
A box contains 2 red marbles, 2 green marbles, and 2 yellow marbles. Carol takes 2 marbles from the box at random; then Claudia takes 2 of the remaining marbles at random; and then Cheryl takes the last 2 marbles. What is the probability that Cheryl gets 2 marbles of the same color?<br />
<br />
<math> \textbf{(A)}\ \frac{1}{10} \qquad\textbf{(B)}\ \frac16 \qquad\textbf{(C)}\ \frac15 \qquad\textbf{(D)}}\ \frac13 \qquad\textbf{(E)}\ \frac12</math><br />
<br />
[[2015 AMC 12A Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
Integers <math>x</math> and <math>y</math> with <math>x>y>0</math> satisfy <math>x+y+xy=80</math>. What is <math>x</math>?<br />
<br />
<math> \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 15 \qquad\textbf{(D)}}\ 18 \qquad\textbf{(E)}\ 26</math><br />
<br />
[[2015 AMC 12A Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
On a sheet of paper, Isabella draws a circle of radius <math>2</math>, a circle of radius <math>3</math>, and all possible lines simultaneously tangent to both circles. Isabella notices that she has drawn exactly <math>k \ge 0</math> lines. How many different values of <math>k</math> are possible?<br />
<br />
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}}\ 5\qquad\textbf{(E)}\ 6</math><br />
<br />
[[2015 AMC 12A Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
The parabolas <math>y=ax^2 - 2</math> and <math>y=4 - bx^2</math> intersect the coordinate axes in exactly four points, and these four points are the vertices of a kite of area <math>12</math>. What is <math>a+b</math>?<br />
<br />
<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 1.5\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}}\ 2.5\qquad\textbf{(E)}\ 3</math><br />
<br />
[[2015 AMC 12A Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
<br />
A league with 12 teams holds a round-robin tournament, with each team playing every other team exactly once. Games either end with one team victorious or else end in a draw. A team scores 2 points for every game it wins and 1 point for every game it draws. Which of the following is NOT a true statement about the list of 12 scores?<br />
<br />
<math> \textbf{(A)}\ \text{There must be an even number of odd scores.}\\<br />
\qquad\textbf{(B)}\ \text{There must be an even number of even scores.}\\<br />
\qquad\textbf{(C)}\ \text{There cannot be two scores of }0\text{.}\\<br />
\qquad\textbf{(D)}}\ \text{The sum of the scores must be at least }100\text{.}\\<br />
\qquad\textbf{(E)}\ \text{The highest score must be at least }12\text{.}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
What is the value of <math>a</math> for which <math>\frac{1}{\text{log}_2a} + \frac{1}{\text{log}_3a} + \frac{1}{\text{log}_4a} = 1</math>?<br />
<br />
[[2015 AMC 12A Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
What is the minimum number of digits to the right of the decimal point needed to express the fraction <math>\frac{123456789}{2^{26}\cdot 5^4}</math> as a decimal?<br />
<br />
<math> \textbf{(A)}\ 4\qquad\textbf{(B)}\ 22\qquad\textbf{(C)}\ 26\qquad\textbf{(D)}}\ 30\qquad\textbf{(E)}\ 104</math><br />
<br />
[[2015 AMC 12A Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
Tetrahedron <math>ABCD</math> has <math>AB=5,AC=3,BC=4,BD=4,AD=3,</math> and <math>CD=\frac{12}{5}\sqrt{2}</math>. What is the volume of the tetrahedron?<br />
<br />
<math> \textbf{(A)}\ 3\sqrt{2}\qquad\textbf{(B)}\ 2\sqrt{5}\qquad\textbf{(C)}\ \frac{24}{5}\qquad\textbf{(D)}}\ 3\sqrt{3}\qquad\textbf{(E)}\ \frac{24}{5}\sqrt{2}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?<br />
<br />
<math> \textbf{(A)}\ \frac{47}{256} \qquad\textbf{(B)}\ \frac{3}{16} \qquad\textbf{(C)}\ \frac{49}{256} \qquad\textbf{(D)}}\ \frac{25}{128} \qquad\textbf{(E)}\ \frac{51}{256}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
The zeros of the function <math>f(x) = x^2-ax+2a</math> are integers. What is the sum of the possible values of <math>a</math>?<br />
<br />
<math> \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 16 \qquad\textbf{(D)}}\ 17 \qquad\textbf{(E)}\ 18</math><br />
<br />
[[2015 AMC 12A Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
For some positive integers <math>p</math>, there is a quadrilateral <math>ABCD</math> with positive integer side lengths, perimeter <math>p</math>, right angles at <math>B</math> and <math>C</math>, <math>AB=2</math>, and <math>CD=AD</math>. How many different values of <math>p<2015</math> are possible?<br />
<br />
<math> \textbf{(A)}\ 30 \qquad\textbf{(B)}\ 31 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}}\ 62 \qquad\textbf{(E)}\ 63</math><br />
<br />
[[2015 AMC 12A Problems/Problem 19|Solution]]<br />
<br />
==Problem 20==<br />
<br />
Isosceles triangles <math>T</math> and <math>T'</math> are not congruent but have the same area and the same perimeter. The sides of <math>T</math> have lengths of <math>5,5,</math> and <math>8</math>, while those of <math>T'</math> have lengths of <math>a,a,</math> and <math>b</math>. Which of the following numbers is closest to <math>b</math>?<br />
<br />
<math> \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}}\ 6 \qquad\textbf{(E)}\ 8</math><br />
<br />
[[2015 AMC 12A Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
A circle of radius <math>r</math> passes through both foci of, and exactly four points on, the ellipse with equation <math>x^2+16y^2=16</math>. The set of all possible values of <math>r</math> is an interval <math>[a,b)</math>. What is <math>a+b</math>?<br />
<br />
<math> \textbf{(A)}\ 5\sqrt{2}+4 \qquad\textbf{(B)}\ \sqrt{17}+7 \qquad\textbf{(C)}\ 6\sqrt{2}+3 \qquad\textbf{(D)}}\ \sqrt{15}+8 \qquad\textbf{(E)}\ 12</math><br />
<br />
[[2015 AMC 12A Problems/Problem 21|Solution]]<br />
<br />
==Problem 22==<br />
<br />
For each positive integer <math>n</math>, let <math>S(n)</math> be the number of sequences of length <math>n</math> consisting solely of the letters <math>A</math> and <math>B</math>, with no more than three <math>A</math>s in a row and no more than three <math>B</math>s in a row. What is the remainder when <math>S(2015)</math> is divided by 12?<br />
<br />
<math> \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}}\ 8 \qquad\textbf{(E)}\ 10</math><br />
<br />
[[2015 AMC 12A Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
Let <math>S</math> be a square of side length 1. Two points are chosen independently at random on the sides of <math>S</math>. The probability that the straight-line distance between the points is at least <math>\frac12</math> is <math>\frac{a-b\pi}{c}</math>, where <math>a,b,</math> and <math>c</math> are positive integers and <math>\text{gcd}(a,b,c) = 1</math>. What is <math>a+b+c</math>?<br />
<br />
<math> \textbf{(A)}\ 59 \qquad\textbf{(B)}\ 60 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}}\ 62 \qquad\textbf{(E)}\ 63</math><br />
<br />
[[2015 AMC 12A Problems/Problem 23|Solution]]<br />
<br />
==Problem 24==<br />
<br />
Rational numbers <math>a</math> and <math>b</math> are chosen at random among all rational numbers in the interval <math>[0,2)</math> that can be written as fractions <math>\frac{n}{d}</math> where <math>n</math> and <math>d</math> are integers with <math>1 \le d \le 5</math>. What is the probability that<br />
<cmath>(\text{cos}(a\pi)+i\text{sin}(b\pi))^4</cmath><br />
is a real number?<br />
<br />
<math> \textbf{(A)}\ \frac{3}{5} \qquad\textbf{(B)}\ \frac{4}{25} \qquad\textbf{(C)}\ \frac{41}{200} \qquad\textbf{(D)}}\ \frac{6}{25} \qquad\textbf{(E)}\ \frac{13}{50}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 24|Solution]]<br />
<br />
==Problem 25==<br />
<br />
A collection of circles in the upper half-plane, all tangent to the <math>x</math>-axis, is constructed in layers as follows. Layer <math>L_0</math> consists of two circles of radii <math>70^2</math> and <math>73^2</math> that are externally tangent. For <math>k\ge1</math>, the circles in <math>\bigcup_{j=0}^{k-1}L_j</math> are ordered according to their points of tangency with the <math>x</math>-axis. For every pair of consecutive circles in this order, a new circle is constructed externally tangent to each of the two circles in the pair. Layer <math>L_k</math> consists of the <math>2^{k-1}</math> circles constructed in this way. Let <math>S=\bigcup_{j=0}^{6}L_j</math>, and for every circle <math>C</math> denote by <math>r(C)</math> its radius. What is<br />
<cmath>\sum_{C\in S} \frac{1}{\sqrt{r(C)}}?</cmath><br />
<br />
<asy><br />
import olympiad;<br />
size(350);<br />
defaultpen(linewidth(0.7));<br />
// define a bunch of arrays and starting points<br />
pair[] coord = new pair[65];<br />
int[] trav = {32,16,8,4,2,1};<br />
coord[0] = (0,73^2); coord[64] = (2*73*70,70^2);<br />
// draw the big circles and the bottom line<br />
path arc1 = arc(coord[0],coord[0].y,260,360);<br />
path arc2 = arc(coord[64],coord[64].y,175,280);<br />
fill((coord[0].x-910,coord[0].y)--arc1--cycle,gray(0.75));<br />
fill((coord[64].x+870,coord[64].y+425)--arc2--cycle,gray(0.75));<br />
draw(arc1^^arc2);<br />
draw((-930,0)--(70^2+73^2+850,0));<br />
// We now apply the findCenter function 63 times to get<br />
// the location of the centers of all 63 constructed circles.<br />
// The complicated array setup ensures that all the circles<br />
// will be taken in the right order<br />
for(int i = 0;i<=5;i=i+1)<br />
{<br />
int skip = trav[i];<br />
for(int k=skip;k<=64 - skip; k = k + 2*skip)<br />
{<br />
pair cent1 = coord[k-skip], cent2 = coord[k+skip];<br />
real r1 = cent1.y, r2 = cent2.y, rn=r1*r2/((sqrt(r1)+sqrt(r2))^2);<br />
real shiftx = cent1.x + sqrt(4*r1*rn);<br />
coord[k] = (shiftx,rn);<br />
}<br />
// Draw the remaining 63 circles<br />
}<br />
for(int i=1;i<=63;i=i+1)<br />
{<br />
filldraw(circle(coord[i],coord[i].y),gray(0.75));<br />
}</asy><br />
<br />
<br />
$ \textbf{(A)}\ \frac{286}{35} \qquad\textbf{(B)}\ \frac{583}{70} \qquad\textbf{(C)}\ \frac{715}{73} \qquad\textbf{(D)}}\ \frac{143}{14} \qquad\textbf{(E)}\ \frac{1573}{146}$<br />
<br />
[[2015 AMC 12A Problems/Problem 25|Solution]]<br />
<br />
== See also ==<br />
* [[AMC Problems and Solutions]]<br />
{{MAA Notice}}</div>Devenwarehttps://artofproblemsolving.com/wiki/index.php?title=2015_AMC_12A_Problems&diff=680102015 AMC 12A Problems2015-03-01T23:24:36Z<p>Devenware: /* Problem 7 */</p>
<hr />
<div>==Problem 1==<br />
<br />
What is the value of <math>(2^0-1+5^2-0)^{-1}\times5?</math><br />
<br />
<math> \textbf{(A)}\ -125\qquad\textbf{(B)}\ -120\qquad\textbf{(C)}\ \frac{1}{5}\qquad\textbf{(D)}\ \frac{5}{24}\qquad\textbf{(E)}\ 25 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
<br />
Two of the three sides of a triangle are 20 and 15. Which of the following numbers is not a possible perimeter of the triangle?<br />
<br />
<math> \textbf{(A)}\ 52\qquad\textbf{(B)}\ 57\qquad\textbf{(C)}\ 62\qquad\textbf{(D)}\ 67\qquad\textbf{(E)}\ 72 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
<br />
Mr. Patrick teaches math to 15 students. He was grading tests and found that when he graded everyone's test except Payton's, the average grade for the class was 80. after he graded Payton's test, the class average became 81. What was Payton's score on the test?<br />
<br />
<math> \textbf{(A)}\ 81\qquad\textbf{(B)}\ 85\qquad\textbf{(C)}\ 91\qquad\textbf{(D)}\ 94\qquad\textbf{(E)}\ 95 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
The sum of two positive numbers is 5 times their difference. What is the ratio of the larger number to the smaller?<br />
<br />
<math> \textbf{(A)}\ \frac54 \qquad\textbf{(B)}\ \frac32 \qquad\textbf{(C)}\ \frac95 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ \frac52 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
Amelia needs to estimate the quantity <math>\frac{a}{b} - c</math>, where <math>a, b,</math> and <math>c</math> are large positive integers. She rounds each of the integers so that the calculation will be easier to do mentally. In which of these situations will her answer necessarily be greater than the exact value of <math>\frac{a}{b} - c</math>?<br />
<br />
<math> \textbf{(A)}\ \text{She rounds all three numbers up.}\\<br />
\qquad\textbf{(B)}\ \text{She rounds } a \text{ and } b \text{ up, and she rounds } c \text{ down.}\\<br />
\qquad\textbf{(C)}\ \text{She rounds } a \text{ and } c \text{ up, and she rounds } b \text{ down.} \\<br />
\qquad\textbf{(D)}\ \text{She rounds } a \text{ up, and she rounds } b \text{ and } c \text{ down.}\\<br />
\qquad\textbf{(E)}\ \text{She rounds } c \text{ up, and she rounds } a \text{ and } b \text{ down.} </math><br />
<br />
[[2015 AMC 12A Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be <math>2 : 1</math>?<br />
<br />
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 8</math><br />
<br />
[[2015 AMC 12A Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
Two right circular cylinders have the same volume. The radius of the second cylinder is <math>10\%</math> more than the radius of the first. What is the relationship between the heights of the two cylinders?<br />
<br />
<math>\textbf{(A)}\ \text{The second height is } 10\% \text{ less than the first.} \\ \textbf{(B)}\ \text{The first height is } 10\% \text{ more than the second.}\\ \textbf{(C)}\ \text{The second height is } 21\% \text{ less than the first.} \\ \textbf{(D)}\ \text{The first height is } 21\% \text{ more than the second.}\\ \textbf{(E)}\ \text{The second height is } 80\% \text{ of the first.}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
The ratio of the length to the width of a rectangle is <math>4</math> : <math>3</math>. If the rectangle has diagonal of length <math>d</math>, then the area may be expressed as <math>kd^2</math> for some constant <math>k</math>. What is <math>k</math>?<br />
<br />
<math> \textbf{(A)}\ \frac27 \qquad\textbf{(B)}\ \frac37 \qquad\textbf{(C)}\ \frac{12}{25} \qquad\textbf{(D)}}\ \frac{16}{26} \qquad\textbf{(E)}\ \frac34</math><br />
<br />
[[2015 AMC 12A Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
A box contains 2 red marbles, 2 green marbles, and 2 yellow marbles. Carol takes 2 marbles from the box at random; then Claudia takes 2 of the remaining marbles at random; and then Cheryl takes the last 2 marbles. What is the probability that Cheryl gets 2 marbles of the same color?<br />
<br />
<math> \textbf{(A)}\ \frac{1}{10} \qquad\textbf{(B)}\ \frac16 \qquad\textbf{(C)}\ \frac15 \qquad\textbf{(D)}}\ \frac13 \qquad\textbf{(E)}\ \frac12</math><br />
<br />
[[2015 AMC 12A Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
Integers <math>x</math> and <math>y</math> with <math>x>y>0</math> satisfy <math>x+y+xy=80</math>. What is <math>x</math>?<br />
<br />
<math> \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 15 \qquad\textbf{(D)}}\ 18 \qquad\textbf{(E)}\ 26</math><br />
<br />
[[2015 AMC 12A Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
On a sheet of paper, Isabella draws a circle of radius <math>2</math>, a circle of radius <math>3</math>, and all possible lines simultaneously tangent to both circles. Isabella notices that she has drawn exactly <math>k \ge 0</math> lines. How many different values of <math>k</math> are possible?<br />
<br />
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}}\ 5\qquad\textbf{(E)}\ 6</math><br />
<br />
[[2015 AMC 12A Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
The parabolas <math>y=ax^2 - 2</math> and <math>y=4 - bx^2</math> intersect the coordinate axes in exactly four points, and these four points are the vertices of a kite of area <math>12</math>. What is <math>a+b</math>?<br />
<br />
<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 1.5\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}}\ 2.5\qquad\textbf{(E)}\ 3</math><br />
<br />
[[2015 AMC 12A Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
<br />
A league with 12 teams holds a round-robin tournament, with each team playing every other team exactly once. Games either end with one team victorious or else end in a draw. A team scores 2 points for every game it wins and 1 point for every game it draws. Which of the following is NOT a true statement about the list of 12 scores?<br />
<br />
<math> \textbf{(A)}\ \text{There must be an even number of odd scores.}\\<br />
\qquad\textbf{(B)}\ \text{There must be an even number of even scores.}\\<br />
\qquad\textbf{(C)}\ \text{There cannot be two scores of }0\text{.}\\<br />
\qquad\textbf{(D)}}\ \text{The sum of the scores must be at least }100\text{.}\\<br />
\qquad\textbf{(E)}\ \text{The highest score must be at least }12\text{.}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
What is the value of <math>a</math> for which <math>\frac{1}{\text{log}_2a} + \frac{1}{\text{log}_3a} + \frac{1}{\text{log}_4a} = 1</math>?<br />
<br />
[[2015 AMC 12A Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
What is the minimum number of digits to the right of the decimal point needed to express the fraction <math>\frac{123456789}{2^{26}\cdot 5^4}</math> as a decimal?<br />
<br />
<math> \textbf{(A)}\ 4\qquad\textbf{(B)}\ 22\qquad\textbf{(C)}\ 26\qquad\textbf{(D)}}\ 30\qquad\textbf{(E)}\ 104</math><br />
<br />
[[2015 AMC 12A Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
Tetrahedron <math>ABCD</math> has <math>AB=5,AC=3,BC=4,BD=4,AD=3,</math> and <math>CD=\frac{12}{5}\sqrt{2}</math>. What is the volume of the tetrahedron?<br />
<br />
<math> \textbf{(A)}\ 3\sqrt{2}\qquad\textbf{(B)}\ 2\sqrt{5}\qquad\textbf{(C)}\ \frac{24}{5}\qquad\textbf{(D)}}\ 3\sqrt{3}\qquad\textbf{(E)}\ \frac{24}{5}\sqrt{2}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?<br />
<br />
<math> \textbf{(A)}\ \frac{47}{256} \qquad\textbf{(B)}\ \frac{3}{16} \qquad\textbf{(C)}\ \frac{49}{256} \qquad\textbf{(D)}}\ \frac{25}{128} \qquad\textbf{(E)}\ \frac{51}{256}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
The zeros of the function <math>f(x) = x^2-ax+2a</math> are integers. What is the sum of the possible values of <math>a</math>?<br />
<br />
<math> \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 16 \qquad\textbf{(D)}}\ 17 \qquad\textbf{(E)}\ 18</math><br />
<br />
[[2015 AMC 12A Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
For some positive integers <math>p</math>, there is a quadrilateral <math>ABCD</math> with positive integer side lengths, perimeter <math>p</math>, right angles at <math>B</math> and <math>C</math>, <math>AB=2</math>, and <math>CD=AD</math>. How many different values of <math>p<2015</math> are possible?<br />
<br />
<math> \textbf{(A)}\ 30 \qquad\textbf{(B)}\ 31 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}}\ 62 \qquad\textbf{(E)}\ 63</math><br />
<br />
[[2015 AMC 12A Problems/Problem 19|Solution]]<br />
<br />
==Problem 20==<br />
<br />
Isosceles triangles <math>T</math> and <math>T'</math> are not congruent but have the same area and the same perimeter. The sides of <math>T</math> have lengths of <math>5,5,</math> and <math>8</math>, while those of <math>T'</math> have lengths of <math>a,a,</math> and <math>b</math>. Which of the following numbers is closest to <math>b</math>?<br />
<br />
<math> \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}}\ 6 \qquad\textbf{(E)}\ 8</math><br />
<br />
[[2015 AMC 12A Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
A circle of radius <math>r</math> passes through both foci of, and exactly four points on, the ellipse with equation <math>x^2+16y^2=16</math>. The set of all possible values of <math>r</math> is an interval <math>[a,b)</math>. What is <math>a+b</math>?<br />
<br />
<math> \textbf{(A)}\ 5\sqrt{2}+4 \qquad\textbf{(B)}\ \sqrt{17}+7 \qquad\textbf{(C)}\ 6\sqrt{2}+3 \qquad\textbf{(D)}}\ \sqrt{15}+8 \qquad\textbf{(E)}\ 12</math><br />
<br />
[[2015 AMC 12A Problems/Problem 21|Solution]]<br />
<br />
==Problem 22==<br />
<br />
For each positive integer <math>n</math>, let <math>S(n)</math> be the number of sequences of length <math>n</math> consisting solely of the letters <math>A</math> and <math>B</math>, with no more than three <math>A</math>s in a row and no more than three <math>B</math>s in a row. What is the remainder when <math>S(2015)</math> is divided by 12?<br />
<br />
<math> \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}}\ 8 \qquad\textbf{(E)}\ 10</math><br />
<br />
[[2015 AMC 12A Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
Let <math>S</math> be a square of side length 1. Two points are chosen independently at random on the sides of <math>S</math>. The probability that the straight-line distance between the points is at least <math>\frac12</math> is <math>\frac{a-b\pi}{c}</math>, where <math>a,b,</math> and <math>c</math> are positive integers and <math>\text{gcd}(a,b,c) = 1</math>. What is <math>a+b+c</math>?<br />
<br />
<math> \textbf{(A)}\ 59 \qquad\textbf{(B)}\ 60 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}}\ 62 \qquad\textbf{(E)}\ 63</math><br />
<br />
[[2015 AMC 12A Problems/Problem 23|Solution]]<br />
<br />
==Problem 24==<br />
<br />
Rational numbers <math>a</math> and <math>b</math> are chosen at random among all rational numbers in the interval <math>[0,2)</math> that can be written as fractions <math>\frac{n}{d}</math> where <math>n</math> and <math>d</math> are integers with <math>1 \le d \le 5</math>. What is the probability that<br />
<cmath>(\text{cos}(a\pi)+i\text{sin}(b\pi))^4</cmath><br />
is a real number?<br />
<br />
<math> \textbf{(A)}\ \frac{3}{5} \qquad\textbf{(B)}\ \frac{4}{25} \qquad\textbf{(C)}\ \frac{41}{200} \qquad\textbf{(D)}}\ \frac{6}{25} \qquad\textbf{(E)}\ \frac{13}{50}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 24|Solution]]<br />
<br />
==Problem 25==<br />
<br />
A collection of circles in the upper half-plane, all tangent to the <math>x</math>-axis, is constructed in layers as follows. Layer <math>L_0</math> consists of two circles of radii <math>70^2</math> and <math>73^2</math> that are externally tangent. For <math>k\ge1</math>, the circles in <math>\bigcup_{j=0}^{k-1}L_j</math> are ordered according to their points of tangency with the <math>x</math>-axis. For every pair of consecutive circles in this order, a new circle is constructed externally tangent to each of the two circles in the pair. Layer <math>L_k</math> consists of the <math>2^{k-1}</math> circles constructed in this way. Let <math>S=\bigcup_{j=0}^{6}L_j</math>, and for every circle <math>C</math> denote by <math>r(C)</math> its radius. What is<br />
<cmath>\sum_{C\in S} \frac{1}{\sqrt{r(C)}}?</cmath><br />
<br />
<asy><br />
import olympiad;<br />
size(350);<br />
defaultpen(linewidth(0.7));<br />
// define a bunch of arrays and starting points<br />
pair[] coord = new pair[65];<br />
int[] trav = {32,16,8,4,2,1};<br />
coord[0] = (0,73^2); coord[64] = (2*73*70,70^2);<br />
// draw the big circles and the bottom line<br />
path arc1 = arc(coord[0],coord[0].y,260,360);<br />
path arc2 = arc(coord[64],coord[64].y,175,280);<br />
fill((coord[0].x-910,coord[0].y)--arc1--cycle,gray(0.75));<br />
fill((coord[64].x+870,coord[64].y+425)--arc2--cycle,gray(0.75));<br />
draw(arc1^^arc2);<br />
draw((-930,0)--(70^2+73^2+850,0));<br />
// We now apply the findCenter function 63 times to get<br />
// the location of the centers of all 63 constructed circles.<br />
// The complicated array setup ensures that all the circles<br />
// will be taken in the right order<br />
for(int i = 0;i<=5;i=i+1)<br />
{<br />
int skip = trav[i];<br />
for(int k=skip;k<=64 - skip; k = k + 2*skip)<br />
{<br />
pair cent1 = coord[k-skip], cent2 = coord[k+skip];<br />
real r1 = cent1.y, r2 = cent2.y, rn=r1*r2/((sqrt(r1)+sqrt(r2))^2);<br />
real shiftx = cent1.x + sqrt(4*r1*rn);<br />
coord[k] = (shiftx,rn);<br />
}<br />
// Draw the remaining 63 circles<br />
}<br />
for(int i=1;i<=63;i=i+1)<br />
{<br />
filldraw(circle(coord[i],coord[i].y),gray(0.75));<br />
}</asy><br />
<br />
<br />
$ \textbf{(A)}\ \frac{286}{35} \qquad\textbf{(B)}\ \frac{583}{70} \qquad\textbf{(C)}\ \frac{715}{73} \qquad\textbf{(D)}}\ \frac{143}{14} \qquad\textbf{(E)}\ \frac{1573}{146}$<br />
<br />
[[2015 AMC 12A Problems/Problem 25|Solution]]<br />
<br />
== See also ==<br />
* [[AMC Problems and Solutions]]<br />
{{MAA Notice}}</div>Devenwarehttps://artofproblemsolving.com/wiki/index.php?title=2015_AMC_12A_Problems&diff=680092015 AMC 12A Problems2015-03-01T23:24:14Z<p>Devenware: /* Problem 6 */</p>
<hr />
<div>==Problem 1==<br />
<br />
What is the value of <math>(2^0-1+5^2-0)^{-1}\times5?</math><br />
<br />
<math> \textbf{(A)}\ -125\qquad\textbf{(B)}\ -120\qquad\textbf{(C)}\ \frac{1}{5}\qquad\textbf{(D)}\ \frac{5}{24}\qquad\textbf{(E)}\ 25 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
<br />
Two of the three sides of a triangle are 20 and 15. Which of the following numbers is not a possible perimeter of the triangle?<br />
<br />
<math> \textbf{(A)}\ 52\qquad\textbf{(B)}\ 57\qquad\textbf{(C)}\ 62\qquad\textbf{(D)}\ 67\qquad\textbf{(E)}\ 72 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
<br />
Mr. Patrick teaches math to 15 students. He was grading tests and found that when he graded everyone's test except Payton's, the average grade for the class was 80. after he graded Payton's test, the class average became 81. What was Payton's score on the test?<br />
<br />
<math> \textbf{(A)}\ 81\qquad\textbf{(B)}\ 85\qquad\textbf{(C)}\ 91\qquad\textbf{(D)}\ 94\qquad\textbf{(E)}\ 95 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
The sum of two positive numbers is 5 times their difference. What is the ratio of the larger number to the smaller?<br />
<br />
<math> \textbf{(A)}\ \frac54 \qquad\textbf{(B)}\ \frac32 \qquad\textbf{(C)}\ \frac95 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ \frac52 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
Amelia needs to estimate the quantity <math>\frac{a}{b} - c</math>, where <math>a, b,</math> and <math>c</math> are large positive integers. She rounds each of the integers so that the calculation will be easier to do mentally. In which of these situations will her answer necessarily be greater than the exact value of <math>\frac{a}{b} - c</math>?<br />
<br />
<math> \textbf{(A)}\ \text{She rounds all three numbers up.}\\<br />
\qquad\textbf{(B)}\ \text{She rounds } a \text{ and } b \text{ up, and she rounds } c \text{ down.}\\<br />
\qquad\textbf{(C)}\ \text{She rounds } a \text{ and } c \text{ up, and she rounds } b \text{ down.} \\<br />
\qquad\textbf{(D)}\ \text{She rounds } a \text{ up, and she rounds } b \text{ and } c \text{ down.}\\<br />
\qquad\textbf{(E)}\ \text{She rounds } c \text{ up, and she rounds } a \text{ and } b \text{ down.} </math><br />
<br />
[[2015 AMC 12A Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be <math>2 : 1</math>?<br />
<br />
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 8</math><br />
<br />
[[2015 AMC 12A Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
Two right circular cylinders have the same volume. The radius of the second cylinder is <math>10\%</math> more than the radius of the first. What is the relationship between the heights of the two cylinders?<br />
<br />
<math>\textbf{(A)}\ \text{The second height is } 10\% \text{ less than the first.} \\ \textbf{(B)}\ \text{The first height is } 10\% \text{ more than the second.}\\ \textbf{(C)}\ \text{The second height is } 21\% \text{ less than the first.} \\ \textbf{(D)}}\ \text{The first height is } 21\% \text{ more than the second.}\\ \textbf{(E)}\ \text{The second height is } 80\% \text{ of the first.}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
The ratio of the length to the width of a rectangle is <math>4</math> : <math>3</math>. If the rectangle has diagonal of length <math>d</math>, then the area may be expressed as <math>kd^2</math> for some constant <math>k</math>. What is <math>k</math>?<br />
<br />
<math> \textbf{(A)}\ \frac27 \qquad\textbf{(B)}\ \frac37 \qquad\textbf{(C)}\ \frac{12}{25} \qquad\textbf{(D)}}\ \frac{16}{26} \qquad\textbf{(E)}\ \frac34</math><br />
<br />
[[2015 AMC 12A Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
A box contains 2 red marbles, 2 green marbles, and 2 yellow marbles. Carol takes 2 marbles from the box at random; then Claudia takes 2 of the remaining marbles at random; and then Cheryl takes the last 2 marbles. What is the probability that Cheryl gets 2 marbles of the same color?<br />
<br />
<math> \textbf{(A)}\ \frac{1}{10} \qquad\textbf{(B)}\ \frac16 \qquad\textbf{(C)}\ \frac15 \qquad\textbf{(D)}}\ \frac13 \qquad\textbf{(E)}\ \frac12</math><br />
<br />
[[2015 AMC 12A Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
Integers <math>x</math> and <math>y</math> with <math>x>y>0</math> satisfy <math>x+y+xy=80</math>. What is <math>x</math>?<br />
<br />
<math> \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 15 \qquad\textbf{(D)}}\ 18 \qquad\textbf{(E)}\ 26</math><br />
<br />
[[2015 AMC 12A Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
On a sheet of paper, Isabella draws a circle of radius <math>2</math>, a circle of radius <math>3</math>, and all possible lines simultaneously tangent to both circles. Isabella notices that she has drawn exactly <math>k \ge 0</math> lines. How many different values of <math>k</math> are possible?<br />
<br />
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}}\ 5\qquad\textbf{(E)}\ 6</math><br />
<br />
[[2015 AMC 12A Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
The parabolas <math>y=ax^2 - 2</math> and <math>y=4 - bx^2</math> intersect the coordinate axes in exactly four points, and these four points are the vertices of a kite of area <math>12</math>. What is <math>a+b</math>?<br />
<br />
<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 1.5\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}}\ 2.5\qquad\textbf{(E)}\ 3</math><br />
<br />
[[2015 AMC 12A Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
<br />
A league with 12 teams holds a round-robin tournament, with each team playing every other team exactly once. Games either end with one team victorious or else end in a draw. A team scores 2 points for every game it wins and 1 point for every game it draws. Which of the following is NOT a true statement about the list of 12 scores?<br />
<br />
<math> \textbf{(A)}\ \text{There must be an even number of odd scores.}\\<br />
\qquad\textbf{(B)}\ \text{There must be an even number of even scores.}\\<br />
\qquad\textbf{(C)}\ \text{There cannot be two scores of }0\text{.}\\<br />
\qquad\textbf{(D)}}\ \text{The sum of the scores must be at least }100\text{.}\\<br />
\qquad\textbf{(E)}\ \text{The highest score must be at least }12\text{.}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
What is the value of <math>a</math> for which <math>\frac{1}{\text{log}_2a} + \frac{1}{\text{log}_3a} + \frac{1}{\text{log}_4a} = 1</math>?<br />
<br />
[[2015 AMC 12A Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
What is the minimum number of digits to the right of the decimal point needed to express the fraction <math>\frac{123456789}{2^{26}\cdot 5^4}</math> as a decimal?<br />
<br />
<math> \textbf{(A)}\ 4\qquad\textbf{(B)}\ 22\qquad\textbf{(C)}\ 26\qquad\textbf{(D)}}\ 30\qquad\textbf{(E)}\ 104</math><br />
<br />
[[2015 AMC 12A Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
Tetrahedron <math>ABCD</math> has <math>AB=5,AC=3,BC=4,BD=4,AD=3,</math> and <math>CD=\frac{12}{5}\sqrt{2}</math>. What is the volume of the tetrahedron?<br />
<br />
<math> \textbf{(A)}\ 3\sqrt{2}\qquad\textbf{(B)}\ 2\sqrt{5}\qquad\textbf{(C)}\ \frac{24}{5}\qquad\textbf{(D)}}\ 3\sqrt{3}\qquad\textbf{(E)}\ \frac{24}{5}\sqrt{2}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?<br />
<br />
<math> \textbf{(A)}\ \frac{47}{256} \qquad\textbf{(B)}\ \frac{3}{16} \qquad\textbf{(C)}\ \frac{49}{256} \qquad\textbf{(D)}}\ \frac{25}{128} \qquad\textbf{(E)}\ \frac{51}{256}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
The zeros of the function <math>f(x) = x^2-ax+2a</math> are integers. What is the sum of the possible values of <math>a</math>?<br />
<br />
<math> \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 16 \qquad\textbf{(D)}}\ 17 \qquad\textbf{(E)}\ 18</math><br />
<br />
[[2015 AMC 12A Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
For some positive integers <math>p</math>, there is a quadrilateral <math>ABCD</math> with positive integer side lengths, perimeter <math>p</math>, right angles at <math>B</math> and <math>C</math>, <math>AB=2</math>, and <math>CD=AD</math>. How many different values of <math>p<2015</math> are possible?<br />
<br />
<math> \textbf{(A)}\ 30 \qquad\textbf{(B)}\ 31 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}}\ 62 \qquad\textbf{(E)}\ 63</math><br />
<br />
[[2015 AMC 12A Problems/Problem 19|Solution]]<br />
<br />
==Problem 20==<br />
<br />
Isosceles triangles <math>T</math> and <math>T'</math> are not congruent but have the same area and the same perimeter. The sides of <math>T</math> have lengths of <math>5,5,</math> and <math>8</math>, while those of <math>T'</math> have lengths of <math>a,a,</math> and <math>b</math>. Which of the following numbers is closest to <math>b</math>?<br />
<br />
<math> \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}}\ 6 \qquad\textbf{(E)}\ 8</math><br />
<br />
[[2015 AMC 12A Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
A circle of radius <math>r</math> passes through both foci of, and exactly four points on, the ellipse with equation <math>x^2+16y^2=16</math>. The set of all possible values of <math>r</math> is an interval <math>[a,b)</math>. What is <math>a+b</math>?<br />
<br />
<math> \textbf{(A)}\ 5\sqrt{2}+4 \qquad\textbf{(B)}\ \sqrt{17}+7 \qquad\textbf{(C)}\ 6\sqrt{2}+3 \qquad\textbf{(D)}}\ \sqrt{15}+8 \qquad\textbf{(E)}\ 12</math><br />
<br />
[[2015 AMC 12A Problems/Problem 21|Solution]]<br />
<br />
==Problem 22==<br />
<br />
For each positive integer <math>n</math>, let <math>S(n)</math> be the number of sequences of length <math>n</math> consisting solely of the letters <math>A</math> and <math>B</math>, with no more than three <math>A</math>s in a row and no more than three <math>B</math>s in a row. What is the remainder when <math>S(2015)</math> is divided by 12?<br />
<br />
<math> \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}}\ 8 \qquad\textbf{(E)}\ 10</math><br />
<br />
[[2015 AMC 12A Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
Let <math>S</math> be a square of side length 1. Two points are chosen independently at random on the sides of <math>S</math>. The probability that the straight-line distance between the points is at least <math>\frac12</math> is <math>\frac{a-b\pi}{c}</math>, where <math>a,b,</math> and <math>c</math> are positive integers and <math>\text{gcd}(a,b,c) = 1</math>. What is <math>a+b+c</math>?<br />
<br />
<math> \textbf{(A)}\ 59 \qquad\textbf{(B)}\ 60 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}}\ 62 \qquad\textbf{(E)}\ 63</math><br />
<br />
[[2015 AMC 12A Problems/Problem 23|Solution]]<br />
<br />
==Problem 24==<br />
<br />
Rational numbers <math>a</math> and <math>b</math> are chosen at random among all rational numbers in the interval <math>[0,2)</math> that can be written as fractions <math>\frac{n}{d}</math> where <math>n</math> and <math>d</math> are integers with <math>1 \le d \le 5</math>. What is the probability that<br />
<cmath>(\text{cos}(a\pi)+i\text{sin}(b\pi))^4</cmath><br />
is a real number?<br />
<br />
<math> \textbf{(A)}\ \frac{3}{5} \qquad\textbf{(B)}\ \frac{4}{25} \qquad\textbf{(C)}\ \frac{41}{200} \qquad\textbf{(D)}}\ \frac{6}{25} \qquad\textbf{(E)}\ \frac{13}{50}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 24|Solution]]<br />
<br />
==Problem 25==<br />
<br />
A collection of circles in the upper half-plane, all tangent to the <math>x</math>-axis, is constructed in layers as follows. Layer <math>L_0</math> consists of two circles of radii <math>70^2</math> and <math>73^2</math> that are externally tangent. For <math>k\ge1</math>, the circles in <math>\bigcup_{j=0}^{k-1}L_j</math> are ordered according to their points of tangency with the <math>x</math>-axis. For every pair of consecutive circles in this order, a new circle is constructed externally tangent to each of the two circles in the pair. Layer <math>L_k</math> consists of the <math>2^{k-1}</math> circles constructed in this way. Let <math>S=\bigcup_{j=0}^{6}L_j</math>, and for every circle <math>C</math> denote by <math>r(C)</math> its radius. What is<br />
<cmath>\sum_{C\in S} \frac{1}{\sqrt{r(C)}}?</cmath><br />
<br />
<asy><br />
import olympiad;<br />
size(350);<br />
defaultpen(linewidth(0.7));<br />
// define a bunch of arrays and starting points<br />
pair[] coord = new pair[65];<br />
int[] trav = {32,16,8,4,2,1};<br />
coord[0] = (0,73^2); coord[64] = (2*73*70,70^2);<br />
// draw the big circles and the bottom line<br />
path arc1 = arc(coord[0],coord[0].y,260,360);<br />
path arc2 = arc(coord[64],coord[64].y,175,280);<br />
fill((coord[0].x-910,coord[0].y)--arc1--cycle,gray(0.75));<br />
fill((coord[64].x+870,coord[64].y+425)--arc2--cycle,gray(0.75));<br />
draw(arc1^^arc2);<br />
draw((-930,0)--(70^2+73^2+850,0));<br />
// We now apply the findCenter function 63 times to get<br />
// the location of the centers of all 63 constructed circles.<br />
// The complicated array setup ensures that all the circles<br />
// will be taken in the right order<br />
for(int i = 0;i<=5;i=i+1)<br />
{<br />
int skip = trav[i];<br />
for(int k=skip;k<=64 - skip; k = k + 2*skip)<br />
{<br />
pair cent1 = coord[k-skip], cent2 = coord[k+skip];<br />
real r1 = cent1.y, r2 = cent2.y, rn=r1*r2/((sqrt(r1)+sqrt(r2))^2);<br />
real shiftx = cent1.x + sqrt(4*r1*rn);<br />
coord[k] = (shiftx,rn);<br />
}<br />
// Draw the remaining 63 circles<br />
}<br />
for(int i=1;i<=63;i=i+1)<br />
{<br />
filldraw(circle(coord[i],coord[i].y),gray(0.75));<br />
}</asy><br />
<br />
<br />
$ \textbf{(A)}\ \frac{286}{35} \qquad\textbf{(B)}\ \frac{583}{70} \qquad\textbf{(C)}\ \frac{715}{73} \qquad\textbf{(D)}}\ \frac{143}{14} \qquad\textbf{(E)}\ \frac{1573}{146}$<br />
<br />
[[2015 AMC 12A Problems/Problem 25|Solution]]<br />
<br />
== See also ==<br />
* [[AMC Problems and Solutions]]<br />
{{MAA Notice}}</div>Devenwarehttps://artofproblemsolving.com/wiki/index.php?title=2015_AMC_12A_Problems&diff=680082015 AMC 12A Problems2015-03-01T23:23:53Z<p>Devenware: /* Problem 5 */</p>
<hr />
<div>==Problem 1==<br />
<br />
What is the value of <math>(2^0-1+5^2-0)^{-1}\times5?</math><br />
<br />
<math> \textbf{(A)}\ -125\qquad\textbf{(B)}\ -120\qquad\textbf{(C)}\ \frac{1}{5}\qquad\textbf{(D)}\ \frac{5}{24}\qquad\textbf{(E)}\ 25 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
<br />
Two of the three sides of a triangle are 20 and 15. Which of the following numbers is not a possible perimeter of the triangle?<br />
<br />
<math> \textbf{(A)}\ 52\qquad\textbf{(B)}\ 57\qquad\textbf{(C)}\ 62\qquad\textbf{(D)}\ 67\qquad\textbf{(E)}\ 72 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
<br />
Mr. Patrick teaches math to 15 students. He was grading tests and found that when he graded everyone's test except Payton's, the average grade for the class was 80. after he graded Payton's test, the class average became 81. What was Payton's score on the test?<br />
<br />
<math> \textbf{(A)}\ 81\qquad\textbf{(B)}\ 85\qquad\textbf{(C)}\ 91\qquad\textbf{(D)}\ 94\qquad\textbf{(E)}\ 95 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
The sum of two positive numbers is 5 times their difference. What is the ratio of the larger number to the smaller?<br />
<br />
<math> \textbf{(A)}\ \frac54 \qquad\textbf{(B)}\ \frac32 \qquad\textbf{(C)}\ \frac95 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ \frac52 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
Amelia needs to estimate the quantity <math>\frac{a}{b} - c</math>, where <math>a, b,</math> and <math>c</math> are large positive integers. She rounds each of the integers so that the calculation will be easier to do mentally. In which of these situations will her answer necessarily be greater than the exact value of <math>\frac{a}{b} - c</math>?<br />
<br />
<math> \textbf{(A)}\ \text{She rounds all three numbers up.}\\<br />
\qquad\textbf{(B)}\ \text{She rounds } a \text{ and } b \text{ up, and she rounds } c \text{ down.}\\<br />
\qquad\textbf{(C)}\ \text{She rounds } a \text{ and } c \text{ up, and she rounds } b \text{ down.} \\<br />
\qquad\textbf{(D)}\ \text{She rounds } a \text{ up, and she rounds } b \text{ and } c \text{ down.}\\<br />
\qquad\textbf{(E)}\ \text{She rounds } c \text{ up, and she rounds } a \text{ and } b \text{ down.} </math><br />
<br />
[[2015 AMC 12A Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be <math>2 : 1</math>?<br />
<br />
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}}\ 6 \qquad\textbf{(E)}\ 8</math><br />
<br />
[[2015 AMC 12A Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
Two right circular cylinders have the same volume. The radius of the second cylinder is <math>10\%</math> more than the radius of the first. What is the relationship between the heights of the two cylinders?<br />
<br />
<math>\textbf{(A)}\ \text{The second height is } 10\% \text{ less than the first.} \\ \textbf{(B)}\ \text{The first height is } 10\% \text{ more than the second.}\\ \textbf{(C)}\ \text{The second height is } 21\% \text{ less than the first.} \\ \textbf{(D)}}\ \text{The first height is } 21\% \text{ more than the second.}\\ \textbf{(E)}\ \text{The second height is } 80\% \text{ of the first.}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
The ratio of the length to the width of a rectangle is <math>4</math> : <math>3</math>. If the rectangle has diagonal of length <math>d</math>, then the area may be expressed as <math>kd^2</math> for some constant <math>k</math>. What is <math>k</math>?<br />
<br />
<math> \textbf{(A)}\ \frac27 \qquad\textbf{(B)}\ \frac37 \qquad\textbf{(C)}\ \frac{12}{25} \qquad\textbf{(D)}}\ \frac{16}{26} \qquad\textbf{(E)}\ \frac34</math><br />
<br />
[[2015 AMC 12A Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
A box contains 2 red marbles, 2 green marbles, and 2 yellow marbles. Carol takes 2 marbles from the box at random; then Claudia takes 2 of the remaining marbles at random; and then Cheryl takes the last 2 marbles. What is the probability that Cheryl gets 2 marbles of the same color?<br />
<br />
<math> \textbf{(A)}\ \frac{1}{10} \qquad\textbf{(B)}\ \frac16 \qquad\textbf{(C)}\ \frac15 \qquad\textbf{(D)}}\ \frac13 \qquad\textbf{(E)}\ \frac12</math><br />
<br />
[[2015 AMC 12A Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
Integers <math>x</math> and <math>y</math> with <math>x>y>0</math> satisfy <math>x+y+xy=80</math>. What is <math>x</math>?<br />
<br />
<math> \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 15 \qquad\textbf{(D)}}\ 18 \qquad\textbf{(E)}\ 26</math><br />
<br />
[[2015 AMC 12A Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
On a sheet of paper, Isabella draws a circle of radius <math>2</math>, a circle of radius <math>3</math>, and all possible lines simultaneously tangent to both circles. Isabella notices that she has drawn exactly <math>k \ge 0</math> lines. How many different values of <math>k</math> are possible?<br />
<br />
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}}\ 5\qquad\textbf{(E)}\ 6</math><br />
<br />
[[2015 AMC 12A Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
The parabolas <math>y=ax^2 - 2</math> and <math>y=4 - bx^2</math> intersect the coordinate axes in exactly four points, and these four points are the vertices of a kite of area <math>12</math>. What is <math>a+b</math>?<br />
<br />
<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 1.5\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}}\ 2.5\qquad\textbf{(E)}\ 3</math><br />
<br />
[[2015 AMC 12A Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
<br />
A league with 12 teams holds a round-robin tournament, with each team playing every other team exactly once. Games either end with one team victorious or else end in a draw. A team scores 2 points for every game it wins and 1 point for every game it draws. Which of the following is NOT a true statement about the list of 12 scores?<br />
<br />
<math> \textbf{(A)}\ \text{There must be an even number of odd scores.}\\<br />
\qquad\textbf{(B)}\ \text{There must be an even number of even scores.}\\<br />
\qquad\textbf{(C)}\ \text{There cannot be two scores of }0\text{.}\\<br />
\qquad\textbf{(D)}}\ \text{The sum of the scores must be at least }100\text{.}\\<br />
\qquad\textbf{(E)}\ \text{The highest score must be at least }12\text{.}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
What is the value of <math>a</math> for which <math>\frac{1}{\text{log}_2a} + \frac{1}{\text{log}_3a} + \frac{1}{\text{log}_4a} = 1</math>?<br />
<br />
[[2015 AMC 12A Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
What is the minimum number of digits to the right of the decimal point needed to express the fraction <math>\frac{123456789}{2^{26}\cdot 5^4}</math> as a decimal?<br />
<br />
<math> \textbf{(A)}\ 4\qquad\textbf{(B)}\ 22\qquad\textbf{(C)}\ 26\qquad\textbf{(D)}}\ 30\qquad\textbf{(E)}\ 104</math><br />
<br />
[[2015 AMC 12A Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
Tetrahedron <math>ABCD</math> has <math>AB=5,AC=3,BC=4,BD=4,AD=3,</math> and <math>CD=\frac{12}{5}\sqrt{2}</math>. What is the volume of the tetrahedron?<br />
<br />
<math> \textbf{(A)}\ 3\sqrt{2}\qquad\textbf{(B)}\ 2\sqrt{5}\qquad\textbf{(C)}\ \frac{24}{5}\qquad\textbf{(D)}}\ 3\sqrt{3}\qquad\textbf{(E)}\ \frac{24}{5}\sqrt{2}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?<br />
<br />
<math> \textbf{(A)}\ \frac{47}{256} \qquad\textbf{(B)}\ \frac{3}{16} \qquad\textbf{(C)}\ \frac{49}{256} \qquad\textbf{(D)}}\ \frac{25}{128} \qquad\textbf{(E)}\ \frac{51}{256}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
The zeros of the function <math>f(x) = x^2-ax+2a</math> are integers. What is the sum of the possible values of <math>a</math>?<br />
<br />
<math> \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 16 \qquad\textbf{(D)}}\ 17 \qquad\textbf{(E)}\ 18</math><br />
<br />
[[2015 AMC 12A Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
For some positive integers <math>p</math>, there is a quadrilateral <math>ABCD</math> with positive integer side lengths, perimeter <math>p</math>, right angles at <math>B</math> and <math>C</math>, <math>AB=2</math>, and <math>CD=AD</math>. How many different values of <math>p<2015</math> are possible?<br />
<br />
<math> \textbf{(A)}\ 30 \qquad\textbf{(B)}\ 31 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}}\ 62 \qquad\textbf{(E)}\ 63</math><br />
<br />
[[2015 AMC 12A Problems/Problem 19|Solution]]<br />
<br />
==Problem 20==<br />
<br />
Isosceles triangles <math>T</math> and <math>T'</math> are not congruent but have the same area and the same perimeter. The sides of <math>T</math> have lengths of <math>5,5,</math> and <math>8</math>, while those of <math>T'</math> have lengths of <math>a,a,</math> and <math>b</math>. Which of the following numbers is closest to <math>b</math>?<br />
<br />
<math> \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}}\ 6 \qquad\textbf{(E)}\ 8</math><br />
<br />
[[2015 AMC 12A Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
A circle of radius <math>r</math> passes through both foci of, and exactly four points on, the ellipse with equation <math>x^2+16y^2=16</math>. The set of all possible values of <math>r</math> is an interval <math>[a,b)</math>. What is <math>a+b</math>?<br />
<br />
<math> \textbf{(A)}\ 5\sqrt{2}+4 \qquad\textbf{(B)}\ \sqrt{17}+7 \qquad\textbf{(C)}\ 6\sqrt{2}+3 \qquad\textbf{(D)}}\ \sqrt{15}+8 \qquad\textbf{(E)}\ 12</math><br />
<br />
[[2015 AMC 12A Problems/Problem 21|Solution]]<br />
<br />
==Problem 22==<br />
<br />
For each positive integer <math>n</math>, let <math>S(n)</math> be the number of sequences of length <math>n</math> consisting solely of the letters <math>A</math> and <math>B</math>, with no more than three <math>A</math>s in a row and no more than three <math>B</math>s in a row. What is the remainder when <math>S(2015)</math> is divided by 12?<br />
<br />
<math> \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}}\ 8 \qquad\textbf{(E)}\ 10</math><br />
<br />
[[2015 AMC 12A Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
Let <math>S</math> be a square of side length 1. Two points are chosen independently at random on the sides of <math>S</math>. The probability that the straight-line distance between the points is at least <math>\frac12</math> is <math>\frac{a-b\pi}{c}</math>, where <math>a,b,</math> and <math>c</math> are positive integers and <math>\text{gcd}(a,b,c) = 1</math>. What is <math>a+b+c</math>?<br />
<br />
<math> \textbf{(A)}\ 59 \qquad\textbf{(B)}\ 60 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}}\ 62 \qquad\textbf{(E)}\ 63</math><br />
<br />
[[2015 AMC 12A Problems/Problem 23|Solution]]<br />
<br />
==Problem 24==<br />
<br />
Rational numbers <math>a</math> and <math>b</math> are chosen at random among all rational numbers in the interval <math>[0,2)</math> that can be written as fractions <math>\frac{n}{d}</math> where <math>n</math> and <math>d</math> are integers with <math>1 \le d \le 5</math>. What is the probability that<br />
<cmath>(\text{cos}(a\pi)+i\text{sin}(b\pi))^4</cmath><br />
is a real number?<br />
<br />
<math> \textbf{(A)}\ \frac{3}{5} \qquad\textbf{(B)}\ \frac{4}{25} \qquad\textbf{(C)}\ \frac{41}{200} \qquad\textbf{(D)}}\ \frac{6}{25} \qquad\textbf{(E)}\ \frac{13}{50}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 24|Solution]]<br />
<br />
==Problem 25==<br />
<br />
A collection of circles in the upper half-plane, all tangent to the <math>x</math>-axis, is constructed in layers as follows. Layer <math>L_0</math> consists of two circles of radii <math>70^2</math> and <math>73^2</math> that are externally tangent. For <math>k\ge1</math>, the circles in <math>\bigcup_{j=0}^{k-1}L_j</math> are ordered according to their points of tangency with the <math>x</math>-axis. For every pair of consecutive circles in this order, a new circle is constructed externally tangent to each of the two circles in the pair. Layer <math>L_k</math> consists of the <math>2^{k-1}</math> circles constructed in this way. Let <math>S=\bigcup_{j=0}^{6}L_j</math>, and for every circle <math>C</math> denote by <math>r(C)</math> its radius. What is<br />
<cmath>\sum_{C\in S} \frac{1}{\sqrt{r(C)}}?</cmath><br />
<br />
<asy><br />
import olympiad;<br />
size(350);<br />
defaultpen(linewidth(0.7));<br />
// define a bunch of arrays and starting points<br />
pair[] coord = new pair[65];<br />
int[] trav = {32,16,8,4,2,1};<br />
coord[0] = (0,73^2); coord[64] = (2*73*70,70^2);<br />
// draw the big circles and the bottom line<br />
path arc1 = arc(coord[0],coord[0].y,260,360);<br />
path arc2 = arc(coord[64],coord[64].y,175,280);<br />
fill((coord[0].x-910,coord[0].y)--arc1--cycle,gray(0.75));<br />
fill((coord[64].x+870,coord[64].y+425)--arc2--cycle,gray(0.75));<br />
draw(arc1^^arc2);<br />
draw((-930,0)--(70^2+73^2+850,0));<br />
// We now apply the findCenter function 63 times to get<br />
// the location of the centers of all 63 constructed circles.<br />
// The complicated array setup ensures that all the circles<br />
// will be taken in the right order<br />
for(int i = 0;i<=5;i=i+1)<br />
{<br />
int skip = trav[i];<br />
for(int k=skip;k<=64 - skip; k = k + 2*skip)<br />
{<br />
pair cent1 = coord[k-skip], cent2 = coord[k+skip];<br />
real r1 = cent1.y, r2 = cent2.y, rn=r1*r2/((sqrt(r1)+sqrt(r2))^2);<br />
real shiftx = cent1.x + sqrt(4*r1*rn);<br />
coord[k] = (shiftx,rn);<br />
}<br />
// Draw the remaining 63 circles<br />
}<br />
for(int i=1;i<=63;i=i+1)<br />
{<br />
filldraw(circle(coord[i],coord[i].y),gray(0.75));<br />
}</asy><br />
<br />
<br />
$ \textbf{(A)}\ \frac{286}{35} \qquad\textbf{(B)}\ \frac{583}{70} \qquad\textbf{(C)}\ \frac{715}{73} \qquad\textbf{(D)}}\ \frac{143}{14} \qquad\textbf{(E)}\ \frac{1573}{146}$<br />
<br />
[[2015 AMC 12A Problems/Problem 25|Solution]]<br />
<br />
== See also ==<br />
* [[AMC Problems and Solutions]]<br />
{{MAA Notice}}</div>Devenwarehttps://artofproblemsolving.com/wiki/index.php?title=2015_AMC_12A_Problems&diff=680072015 AMC 12A Problems2015-03-01T23:23:19Z<p>Devenware: /* Problem 4 */</p>
<hr />
<div>==Problem 1==<br />
<br />
What is the value of <math>(2^0-1+5^2-0)^{-1}\times5?</math><br />
<br />
<math> \textbf{(A)}\ -125\qquad\textbf{(B)}\ -120\qquad\textbf{(C)}\ \frac{1}{5}\qquad\textbf{(D)}\ \frac{5}{24}\qquad\textbf{(E)}\ 25 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
<br />
Two of the three sides of a triangle are 20 and 15. Which of the following numbers is not a possible perimeter of the triangle?<br />
<br />
<math> \textbf{(A)}\ 52\qquad\textbf{(B)}\ 57\qquad\textbf{(C)}\ 62\qquad\textbf{(D)}\ 67\qquad\textbf{(E)}\ 72 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
<br />
Mr. Patrick teaches math to 15 students. He was grading tests and found that when he graded everyone's test except Payton's, the average grade for the class was 80. after he graded Payton's test, the class average became 81. What was Payton's score on the test?<br />
<br />
<math> \textbf{(A)}\ 81\qquad\textbf{(B)}\ 85\qquad\textbf{(C)}\ 91\qquad\textbf{(D)}\ 94\qquad\textbf{(E)}\ 95 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
The sum of two positive numbers is 5 times their difference. What is the ratio of the larger number to the smaller?<br />
<br />
<math> \textbf{(A)}\ \frac54 \qquad\textbf{(B)}\ \frac32 \qquad\textbf{(C)}\ \frac95 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ \frac52 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
Amelia needs to estimate the quantity <math>\frac{a}{b} - c</math>, where <math>a, b,</math> and <math>c</math> are large positive integers. She rounds each of the integers so that the calculation will be easier to do mentally. In which of these situations will her answer necessarily be greater than the exact value of <math>\frac{a}{b} - c</math>?<br />
<br />
<math> \textbf{(A)}\ \text{She rounds all three numbers up.}\\<br />
\qquad\textbf{(B)}\ \text{She rounds } a \text{ and } b \text{ up, and she rounds } c \text{ down.}\\<br />
\qquad\textbf{(C)}\ \text{She rounds } a \text{ and } c \text{ up, and she rounds } b \text{ down.} \\<br />
\qquad\textbf{(D)}}\ \text{She rounds } a \text{ up, and she rounds } b \text{ and } c \text{ down.}\\<br />
\qquad\textbf{(E)}\ \text{She rounds } c \text{ up, and she rounds } a \text{ and } b \text{ down.} </math><br />
<br />
[[2015 AMC 12A Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be <math>2 : 1</math>?<br />
<br />
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}}\ 6 \qquad\textbf{(E)}\ 8</math><br />
<br />
[[2015 AMC 12A Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
Two right circular cylinders have the same volume. The radius of the second cylinder is <math>10\%</math> more than the radius of the first. What is the relationship between the heights of the two cylinders?<br />
<br />
<math>\textbf{(A)}\ \text{The second height is } 10\% \text{ less than the first.} \\ \textbf{(B)}\ \text{The first height is } 10\% \text{ more than the second.}\\ \textbf{(C)}\ \text{The second height is } 21\% \text{ less than the first.} \\ \textbf{(D)}}\ \text{The first height is } 21\% \text{ more than the second.}\\ \textbf{(E)}\ \text{The second height is } 80\% \text{ of the first.}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
The ratio of the length to the width of a rectangle is <math>4</math> : <math>3</math>. If the rectangle has diagonal of length <math>d</math>, then the area may be expressed as <math>kd^2</math> for some constant <math>k</math>. What is <math>k</math>?<br />
<br />
<math> \textbf{(A)}\ \frac27 \qquad\textbf{(B)}\ \frac37 \qquad\textbf{(C)}\ \frac{12}{25} \qquad\textbf{(D)}}\ \frac{16}{26} \qquad\textbf{(E)}\ \frac34</math><br />
<br />
[[2015 AMC 12A Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
A box contains 2 red marbles, 2 green marbles, and 2 yellow marbles. Carol takes 2 marbles from the box at random; then Claudia takes 2 of the remaining marbles at random; and then Cheryl takes the last 2 marbles. What is the probability that Cheryl gets 2 marbles of the same color?<br />
<br />
<math> \textbf{(A)}\ \frac{1}{10} \qquad\textbf{(B)}\ \frac16 \qquad\textbf{(C)}\ \frac15 \qquad\textbf{(D)}}\ \frac13 \qquad\textbf{(E)}\ \frac12</math><br />
<br />
[[2015 AMC 12A Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
Integers <math>x</math> and <math>y</math> with <math>x>y>0</math> satisfy <math>x+y+xy=80</math>. What is <math>x</math>?<br />
<br />
<math> \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 15 \qquad\textbf{(D)}}\ 18 \qquad\textbf{(E)}\ 26</math><br />
<br />
[[2015 AMC 12A Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
On a sheet of paper, Isabella draws a circle of radius <math>2</math>, a circle of radius <math>3</math>, and all possible lines simultaneously tangent to both circles. Isabella notices that she has drawn exactly <math>k \ge 0</math> lines. How many different values of <math>k</math> are possible?<br />
<br />
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}}\ 5\qquad\textbf{(E)}\ 6</math><br />
<br />
[[2015 AMC 12A Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
The parabolas <math>y=ax^2 - 2</math> and <math>y=4 - bx^2</math> intersect the coordinate axes in exactly four points, and these four points are the vertices of a kite of area <math>12</math>. What is <math>a+b</math>?<br />
<br />
<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 1.5\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}}\ 2.5\qquad\textbf{(E)}\ 3</math><br />
<br />
[[2015 AMC 12A Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
<br />
A league with 12 teams holds a round-robin tournament, with each team playing every other team exactly once. Games either end with one team victorious or else end in a draw. A team scores 2 points for every game it wins and 1 point for every game it draws. Which of the following is NOT a true statement about the list of 12 scores?<br />
<br />
<math> \textbf{(A)}\ \text{There must be an even number of odd scores.}\\<br />
\qquad\textbf{(B)}\ \text{There must be an even number of even scores.}\\<br />
\qquad\textbf{(C)}\ \text{There cannot be two scores of }0\text{.}\\<br />
\qquad\textbf{(D)}}\ \text{The sum of the scores must be at least }100\text{.}\\<br />
\qquad\textbf{(E)}\ \text{The highest score must be at least }12\text{.}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
What is the value of <math>a</math> for which <math>\frac{1}{\text{log}_2a} + \frac{1}{\text{log}_3a} + \frac{1}{\text{log}_4a} = 1</math>?<br />
<br />
[[2015 AMC 12A Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
What is the minimum number of digits to the right of the decimal point needed to express the fraction <math>\frac{123456789}{2^{26}\cdot 5^4}</math> as a decimal?<br />
<br />
<math> \textbf{(A)}\ 4\qquad\textbf{(B)}\ 22\qquad\textbf{(C)}\ 26\qquad\textbf{(D)}}\ 30\qquad\textbf{(E)}\ 104</math><br />
<br />
[[2015 AMC 12A Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
Tetrahedron <math>ABCD</math> has <math>AB=5,AC=3,BC=4,BD=4,AD=3,</math> and <math>CD=\frac{12}{5}\sqrt{2}</math>. What is the volume of the tetrahedron?<br />
<br />
<math> \textbf{(A)}\ 3\sqrt{2}\qquad\textbf{(B)}\ 2\sqrt{5}\qquad\textbf{(C)}\ \frac{24}{5}\qquad\textbf{(D)}}\ 3\sqrt{3}\qquad\textbf{(E)}\ \frac{24}{5}\sqrt{2}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?<br />
<br />
<math> \textbf{(A)}\ \frac{47}{256} \qquad\textbf{(B)}\ \frac{3}{16} \qquad\textbf{(C)}\ \frac{49}{256} \qquad\textbf{(D)}}\ \frac{25}{128} \qquad\textbf{(E)}\ \frac{51}{256}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
The zeros of the function <math>f(x) = x^2-ax+2a</math> are integers. What is the sum of the possible values of <math>a</math>?<br />
<br />
<math> \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 16 \qquad\textbf{(D)}}\ 17 \qquad\textbf{(E)}\ 18</math><br />
<br />
[[2015 AMC 12A Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
For some positive integers <math>p</math>, there is a quadrilateral <math>ABCD</math> with positive integer side lengths, perimeter <math>p</math>, right angles at <math>B</math> and <math>C</math>, <math>AB=2</math>, and <math>CD=AD</math>. How many different values of <math>p<2015</math> are possible?<br />
<br />
<math> \textbf{(A)}\ 30 \qquad\textbf{(B)}\ 31 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}}\ 62 \qquad\textbf{(E)}\ 63</math><br />
<br />
[[2015 AMC 12A Problems/Problem 19|Solution]]<br />
<br />
==Problem 20==<br />
<br />
Isosceles triangles <math>T</math> and <math>T'</math> are not congruent but have the same area and the same perimeter. The sides of <math>T</math> have lengths of <math>5,5,</math> and <math>8</math>, while those of <math>T'</math> have lengths of <math>a,a,</math> and <math>b</math>. Which of the following numbers is closest to <math>b</math>?<br />
<br />
<math> \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}}\ 6 \qquad\textbf{(E)}\ 8</math><br />
<br />
[[2015 AMC 12A Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
A circle of radius <math>r</math> passes through both foci of, and exactly four points on, the ellipse with equation <math>x^2+16y^2=16</math>. The set of all possible values of <math>r</math> is an interval <math>[a,b)</math>. What is <math>a+b</math>?<br />
<br />
<math> \textbf{(A)}\ 5\sqrt{2}+4 \qquad\textbf{(B)}\ \sqrt{17}+7 \qquad\textbf{(C)}\ 6\sqrt{2}+3 \qquad\textbf{(D)}}\ \sqrt{15}+8 \qquad\textbf{(E)}\ 12</math><br />
<br />
[[2015 AMC 12A Problems/Problem 21|Solution]]<br />
<br />
==Problem 22==<br />
<br />
For each positive integer <math>n</math>, let <math>S(n)</math> be the number of sequences of length <math>n</math> consisting solely of the letters <math>A</math> and <math>B</math>, with no more than three <math>A</math>s in a row and no more than three <math>B</math>s in a row. What is the remainder when <math>S(2015)</math> is divided by 12?<br />
<br />
<math> \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}}\ 8 \qquad\textbf{(E)}\ 10</math><br />
<br />
[[2015 AMC 12A Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
Let <math>S</math> be a square of side length 1. Two points are chosen independently at random on the sides of <math>S</math>. The probability that the straight-line distance between the points is at least <math>\frac12</math> is <math>\frac{a-b\pi}{c}</math>, where <math>a,b,</math> and <math>c</math> are positive integers and <math>\text{gcd}(a,b,c) = 1</math>. What is <math>a+b+c</math>?<br />
<br />
<math> \textbf{(A)}\ 59 \qquad\textbf{(B)}\ 60 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}}\ 62 \qquad\textbf{(E)}\ 63</math><br />
<br />
[[2015 AMC 12A Problems/Problem 23|Solution]]<br />
<br />
==Problem 24==<br />
<br />
Rational numbers <math>a</math> and <math>b</math> are chosen at random among all rational numbers in the interval <math>[0,2)</math> that can be written as fractions <math>\frac{n}{d}</math> where <math>n</math> and <math>d</math> are integers with <math>1 \le d \le 5</math>. What is the probability that<br />
<cmath>(\text{cos}(a\pi)+i\text{sin}(b\pi))^4</cmath><br />
is a real number?<br />
<br />
<math> \textbf{(A)}\ \frac{3}{5} \qquad\textbf{(B)}\ \frac{4}{25} \qquad\textbf{(C)}\ \frac{41}{200} \qquad\textbf{(D)}}\ \frac{6}{25} \qquad\textbf{(E)}\ \frac{13}{50}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 24|Solution]]<br />
<br />
==Problem 25==<br />
<br />
A collection of circles in the upper half-plane, all tangent to the <math>x</math>-axis, is constructed in layers as follows. Layer <math>L_0</math> consists of two circles of radii <math>70^2</math> and <math>73^2</math> that are externally tangent. For <math>k\ge1</math>, the circles in <math>\bigcup_{j=0}^{k-1}L_j</math> are ordered according to their points of tangency with the <math>x</math>-axis. For every pair of consecutive circles in this order, a new circle is constructed externally tangent to each of the two circles in the pair. Layer <math>L_k</math> consists of the <math>2^{k-1}</math> circles constructed in this way. Let <math>S=\bigcup_{j=0}^{6}L_j</math>, and for every circle <math>C</math> denote by <math>r(C)</math> its radius. What is<br />
<cmath>\sum_{C\in S} \frac{1}{\sqrt{r(C)}}?</cmath><br />
<br />
<asy><br />
import olympiad;<br />
size(350);<br />
defaultpen(linewidth(0.7));<br />
// define a bunch of arrays and starting points<br />
pair[] coord = new pair[65];<br />
int[] trav = {32,16,8,4,2,1};<br />
coord[0] = (0,73^2); coord[64] = (2*73*70,70^2);<br />
// draw the big circles and the bottom line<br />
path arc1 = arc(coord[0],coord[0].y,260,360);<br />
path arc2 = arc(coord[64],coord[64].y,175,280);<br />
fill((coord[0].x-910,coord[0].y)--arc1--cycle,gray(0.75));<br />
fill((coord[64].x+870,coord[64].y+425)--arc2--cycle,gray(0.75));<br />
draw(arc1^^arc2);<br />
draw((-930,0)--(70^2+73^2+850,0));<br />
// We now apply the findCenter function 63 times to get<br />
// the location of the centers of all 63 constructed circles.<br />
// The complicated array setup ensures that all the circles<br />
// will be taken in the right order<br />
for(int i = 0;i<=5;i=i+1)<br />
{<br />
int skip = trav[i];<br />
for(int k=skip;k<=64 - skip; k = k + 2*skip)<br />
{<br />
pair cent1 = coord[k-skip], cent2 = coord[k+skip];<br />
real r1 = cent1.y, r2 = cent2.y, rn=r1*r2/((sqrt(r1)+sqrt(r2))^2);<br />
real shiftx = cent1.x + sqrt(4*r1*rn);<br />
coord[k] = (shiftx,rn);<br />
}<br />
// Draw the remaining 63 circles<br />
}<br />
for(int i=1;i<=63;i=i+1)<br />
{<br />
filldraw(circle(coord[i],coord[i].y),gray(0.75));<br />
}</asy><br />
<br />
<br />
$ \textbf{(A)}\ \frac{286}{35} \qquad\textbf{(B)}\ \frac{583}{70} \qquad\textbf{(C)}\ \frac{715}{73} \qquad\textbf{(D)}}\ \frac{143}{14} \qquad\textbf{(E)}\ \frac{1573}{146}$<br />
<br />
[[2015 AMC 12A Problems/Problem 25|Solution]]<br />
<br />
== See also ==<br />
* [[AMC Problems and Solutions]]<br />
{{MAA Notice}}</div>Devenwarehttps://artofproblemsolving.com/wiki/index.php?title=2015_AMC_12A_Problems&diff=680042015 AMC 12A Problems2015-03-01T23:15:20Z<p>Devenware: /* Problem 1 */</p>
<hr />
<div>==Problem 1==<br />
<br />
What is the value of <math>(2^0-1+5^2-0)^{-1}\times5?</math><br />
<br />
<math> \textbf{(A)}\ -125\qquad\textbf{(B)}\ -120\qquad\textbf{(C)}\ \frac{1}{5}\qquad\textbf{(D)}\ \frac{5}{24}\qquad\textbf{(E)}\ 25 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
<br />
Two of the three sides of a triangle are 20 and 15. Which of the following numbers is not a possible perimeter of the triangle?<br />
<br />
<math> \textbf{(A)}\ 52\qquad\textbf{(B)}\ 57\qquad\textbf{(C)}\ 62\qquad\textbf{(D)}\ 67\qquad\textbf{(E)}\ 72 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
<br />
Mr. Patrick teaches math to 15 students. He was grading tests and found that when he graded everyone's test except Payton's, the average grade for the class was 80. after he graded Payton's test, the class average became 81. What was Payton's score on the test?<br />
<br />
<math> \textbf{(A)}\ 81\qquad\textbf{(B)}\ 85\qquad\textbf{(C)}\ 91\qquad\textbf{(D)}\ 94\qquad\textbf{(E)}\ 95 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
The sum of two positive numbers is 5 times their difference. What is the ratio of the larger number to the smaller?<br />
<br />
<math> \textbf{(A)}\ \frac54 \qquad\textbf{(B)}\ \frac32 \qquad\textbf{(C)}\ \frac95 \qquad\textbf{(D)}}\ 2 \qquad\textbf{(E)}\ \frac52 </math><br />
<br />
[[2015 AMC 12A Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
Amelia needs to estimate the quantity <math>\frac{a}{b} - c</math>, where <math>a, b,</math> and <math>c</math> are large positive integers. She rounds each of the integers so that the calculation will be easier to do mentally. In which of these situations will her answer necessarily be greater than the exact value of <math>\frac{a}{b} - c</math>?<br />
<br />
<math> \textbf{(A)}\ \text{She rounds all three numbers up.}\\<br />
\qquad\textbf{(B)}\ \text{She rounds } a \text{ and } b \text{ up, and she rounds } c \text{ down.}\\<br />
\qquad\textbf{(C)}\ \text{She rounds } a \text{ and } c \text{ up, and she rounds } b \text{ down.} \\<br />
\qquad\textbf{(D)}}\ \text{She rounds } a \text{ up, and she rounds } b \text{ and } c \text{ down.}\\<br />
\qquad\textbf{(E)}\ \text{She rounds } c \text{ up, and she rounds } a \text{ and } b \text{ down.} </math><br />
<br />
[[2015 AMC 12A Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be <math>2 : 1</math>?<br />
<br />
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}}\ 6 \qquad\textbf{(E)}\ 8</math><br />
<br />
[[2015 AMC 12A Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
Two right circular cylinders have the same volume. The radius of the second cylinder is <math>10\%</math> more than the radius of the first. What is the relationship between the heights of the two cylinders?<br />
<br />
<math>\textbf{(A)}\ \text{The second height is } 10\% \text{ less than the first.} \\ \textbf{(B)}\ \text{The first height is } 10\% \text{ more than the second.}\\ \textbf{(C)}\ \text{The second height is } 21\% \text{ less than the first.} \\ \textbf{(D)}}\ \text{The first height is } 21\% \text{ more than the second.}\\ \textbf{(E)}\ \text{The second height is } 80\% \text{ of the first.}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
The ratio of the length to the width of a rectangle is <math>4</math> : <math>3</math>. If the rectangle has diagonal of length <math>d</math>, then the area may be expressed as <math>kd^2</math> for some constant <math>k</math>. What is <math>k</math>?<br />
<br />
<math> \textbf{(A)}\ \frac27 \qquad\textbf{(B)}\ \frac37 \qquad\textbf{(C)}\ \frac{12}{25} \qquad\textbf{(D)}}\ \frac{16}{26} \qquad\textbf{(E)}\ \frac34</math><br />
<br />
[[2015 AMC 12A Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
A box contains 2 red marbles, 2 green marbles, and 2 yellow marbles. Carol takes 2 marbles from the box at random; then Claudia takes 2 of the remaining marbles at random; and then Cheryl takes the last 2 marbles. What is the probability that Cheryl gets 2 marbles of the same color?<br />
<br />
<math> \textbf{(A)}\ \frac{1}{10} \qquad\textbf{(B)}\ \frac16 \qquad\textbf{(C)}\ \frac15 \qquad\textbf{(D)}}\ \frac13 \qquad\textbf{(E)}\ \frac12</math><br />
<br />
[[2015 AMC 12A Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
Integers <math>x</math> and <math>y</math> with <math>x>y>0</math> satisfy <math>x+y+xy=80</math>. What is <math>x</math>?<br />
<br />
<math> \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 15 \qquad\textbf{(D)}}\ 18 \qquad\textbf{(E)}\ 26</math><br />
<br />
[[2015 AMC 12A Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
On a sheet of paper, Isabella draws a circle of radius <math>2</math>, a circle of radius <math>3</math>, and all possible lines simultaneously tangent to both circles. Isabella notices that she has drawn exactly <math>k \ge 0</math> lines. How many different values of <math>k</math> are possible?<br />
<br />
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}}\ 5\qquad\textbf{(E)}\ 6</math><br />
<br />
[[2015 AMC 12A Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
The parabolas <math>y=ax^2 - 2</math> and <math>y=4 - bx^2</math> intersect the coordinate axes in exactly four points, and these four points are the vertices of a kite of area <math>12</math>. What is <math>a+b</math>?<br />
<br />
<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 1.5\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}}\ 2.5\qquad\textbf{(E)}\ 3</math><br />
<br />
[[2015 AMC 12A Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
<br />
A league with 12 teams holds a round-robin tournament, with each team playing every other team exactly once. Games either end with one team victorious or else end in a draw. A team scores 2 points for every game it wins and 1 point for every game it draws. Which of the following is NOT a true statement about the list of 12 scores?<br />
<br />
<math> \textbf{(A)}\ \text{There must be an even number of odd scores.}\\<br />
\qquad\textbf{(B)}\ \text{There must be an even number of even scores.}\\<br />
\qquad\textbf{(C)}\ \text{There cannot be two scores of }0\text{.}\\<br />
\qquad\textbf{(D)}}\ \text{The sum of the scores must be at least }100\text{.}\\<br />
\qquad\textbf{(E)}\ \text{The highest score must be at least }12\text{.}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
What is the value of <math>a</math> for which <math>\frac{1}{\text{log}_2a} + \frac{1}{\text{log}_3a} + \frac{1}{\text{log}_4a} = 1</math>?<br />
<br />
[[2015 AMC 12A Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
What is the minimum number of digits to the right of the decimal point needed to express the fraction <math>\frac{123456789}{2^{26}\cdot 5^4}</math> as a decimal?<br />
<br />
<math> \textbf{(A)}\ 4\qquad\textbf{(B)}\ 22\qquad\textbf{(C)}\ 26\qquad\textbf{(D)}}\ 30\qquad\textbf{(E)}\ 104</math><br />
<br />
[[2015 AMC 12A Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
Tetrahedron <math>ABCD</math> has <math>AB=5,AC=3,BC=4,BD=4,AD=3,</math> and <math>CD=\frac{12}{5}\sqrt{2}</math>. What is the volume of the tetrahedron?<br />
<br />
<math> \textbf{(A)}\ 3\sqrt{2}\qquad\textbf{(B)}\ 2\sqrt{5}\qquad\textbf{(C)}\ \frac{24}{5}\qquad\textbf{(D)}}\ 3\sqrt{3}\qquad\textbf{(E)}\ \frac{24}{5}\sqrt{2}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?<br />
<br />
<math> \textbf{(A)}\ \frac{47}{256} \qquad\textbf{(B)}\ \frac{3}{16} \qquad\textbf{(C)}\ \frac{49}{256} \qquad\textbf{(D)}}\ \frac{25}{128} \qquad\textbf{(E)}\ \frac{51}{256}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
The zeros of the function <math>f(x) = x^2-ax+2a</math> are integers. What is the sum of the possible values of <math>a</math>?<br />
<br />
<math> \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 16 \qquad\textbf{(D)}}\ 17 \qquad\textbf{(E)}\ 18</math><br />
<br />
[[2015 AMC 12A Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
For some positive integers <math>p</math>, there is a quadrilateral <math>ABCD</math> with positive integer side lengths, perimeter <math>p</math>, right angles at <math>B</math> and <math>C</math>, <math>AB=2</math>, and <math>CD=AD</math>. How many different values of <math>p<2015</math> are possible?<br />
<br />
<math> \textbf{(A)}\ 30 \qquad\textbf{(B)}\ 31 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}}\ 62 \qquad\textbf{(E)}\ 63</math><br />
<br />
[[2015 AMC 12A Problems/Problem 19|Solution]]<br />
<br />
==Problem 20==<br />
<br />
Isosceles triangles <math>T</math> and <math>T'</math> are not congruent but have the same area and the same perimeter. The sides of <math>T</math> have lengths of <math>5,5,</math> and <math>8</math>, while those of <math>T'</math> have lengths of <math>a,a,</math> and <math>b</math>. Which of the following numbers is closest to <math>b</math>?<br />
<br />
<math> \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}}\ 6 \qquad\textbf{(E)}\ 8</math><br />
<br />
[[2015 AMC 12A Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
A circle of radius <math>r</math> passes through both foci of, and exactly four points on, the ellipse with equation <math>x^2+16y^2=16</math>. The set of all possible values of <math>r</math> is an interval <math>[a,b)</math>. What is <math>a+b</math>?<br />
<br />
<math> \textbf{(A)}\ 5\sqrt{2}+4 \qquad\textbf{(B)}\ \sqrt{17}+7 \qquad\textbf{(C)}\ 6\sqrt{2}+3 \qquad\textbf{(D)}}\ \sqrt{15}+8 \qquad\textbf{(E)}\ 12</math><br />
<br />
[[2015 AMC 12A Problems/Problem 21|Solution]]<br />
<br />
==Problem 22==<br />
<br />
For each positive integer <math>n</math>, let <math>S(n)</math> be the number of sequences of length <math>n</math> consisting solely of the letters <math>A</math> and <math>B</math>, with no more than three <math>A</math>s in a row and no more than three <math>B</math>s in a row. What is the remainder when <math>S(2015)</math> is divided by 12?<br />
<br />
<math> \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}}\ 8 \qquad\textbf{(E)}\ 10</math><br />
<br />
[[2015 AMC 12A Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
Let <math>S</math> be a square of side length 1. Two points are chosen independently at random on the sides of <math>S</math>. The probability that the straight-line distance between the points is at least <math>\frac12</math> is <math>\frac{a-b\pi}{c}</math>, where <math>a,b,</math> and <math>c</math> are positive integers and <math>\text{gcd}(a,b,c) = 1</math>. What is <math>a+b+c</math>?<br />
<br />
<math> \textbf{(A)}\ 59 \qquad\textbf{(B)}\ 60 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}}\ 62 \qquad\textbf{(E)}\ 63</math><br />
<br />
[[2015 AMC 12A Problems/Problem 23|Solution]]<br />
<br />
==Problem 24==<br />
<br />
Rational numbers <math>a</math> and <math>b</math> are chosen at random among all rational numbers in the interval <math>[0,2)</math> that can be written as fractions <math>\frac{n}{d}</math> where <math>n</math> and <math>d</math> are integers with <math>1 \le d \le 5</math>. What is the probability that<br />
<cmath>(\text{cos}(a\pi)+i\text{sin}(b\pi))^4</cmath><br />
is a real number?<br />
<br />
<math> \textbf{(A)}\ \frac{3}{5} \qquad\textbf{(B)}\ \frac{4}{25} \qquad\textbf{(C)}\ \frac{41}{200} \qquad\textbf{(D)}}\ \frac{6}{25} \qquad\textbf{(E)}\ \frac{13}{50}</math><br />
<br />
[[2015 AMC 12A Problems/Problem 24|Solution]]<br />
<br />
==Problem 25==<br />
<br />
A collection of circles in the upper half-plane, all tangent to the <math>x</math>-axis, is constructed in layers as follows. Layer <math>L_0</math> consists of two circles of radii <math>70^2</math> and <math>73^2</math> that are externally tangent. For <math>k\ge1</math>, the circles in <math>\bigcup_{j=0}^{k-1}L_j</math> are ordered according to their points of tangency with the <math>x</math>-axis. For every pair of consecutive circles in this order, a new circle is constructed externally tangent to each of the two circles in the pair. Layer <math>L_k</math> consists of the <math>2^{k-1}</math> circles constructed in this way. Let <math>S=\bigcup_{j=0}^{6}L_j</math>, and for every circle <math>C</math> denote by <math>r(C)</math> its radius. What is<br />
<cmath>\sum_{C\in S} \frac{1}{\sqrt{r(C)}}?</cmath><br />
<br />
<asy><br />
import olympiad;<br />
size(350);<br />
defaultpen(linewidth(0.7));<br />
// define a bunch of arrays and starting points<br />
pair[] coord = new pair[65];<br />
int[] trav = {32,16,8,4,2,1};<br />
coord[0] = (0,73^2); coord[64] = (2*73*70,70^2);<br />
// draw the big circles and the bottom line<br />
path arc1 = arc(coord[0],coord[0].y,260,360);<br />
path arc2 = arc(coord[64],coord[64].y,175,280);<br />
fill((coord[0].x-910,coord[0].y)--arc1--cycle,gray(0.75));<br />
fill((coord[64].x+870,coord[64].y+425)--arc2--cycle,gray(0.75));<br />
draw(arc1^^arc2);<br />
draw((-930,0)--(70^2+73^2+850,0));<br />
// We now apply the findCenter function 63 times to get<br />
// the location of the centers of all 63 constructed circles.<br />
// The complicated array setup ensures that all the circles<br />
// will be taken in the right order<br />
for(int i = 0;i<=5;i=i+1)<br />
{<br />
int skip = trav[i];<br />
for(int k=skip;k<=64 - skip; k = k + 2*skip)<br />
{<br />
pair cent1 = coord[k-skip], cent2 = coord[k+skip];<br />
real r1 = cent1.y, r2 = cent2.y, rn=r1*r2/((sqrt(r1)+sqrt(r2))^2);<br />
real shiftx = cent1.x + sqrt(4*r1*rn);<br />
coord[k] = (shiftx,rn);<br />
}<br />
// Draw the remaining 63 circles<br />
}<br />
for(int i=1;i<=63;i=i+1)<br />
{<br />
filldraw(circle(coord[i],coord[i].y),gray(0.75));<br />
}</asy><br />
<br />
<br />
$ \textbf{(A)}\ \frac{286}{35} \qquad\textbf{(B)}\ \frac{583}{70} \qquad\textbf{(C)}\ \frac{715}{73} \qquad\textbf{(D)}}\ \frac{143}{14} \qquad\textbf{(E)}\ \frac{1573}{146}$<br />
<br />
[[2015 AMC 12A Problems/Problem 25|Solution]]<br />
<br />
== See also ==<br />
* [[AMC Problems and Solutions]]<br />
{{MAA Notice}}</div>Devenware