https://artofproblemsolving.com/wiki/api.php?action=feedcontributions&user=Sssuperm1ch4el&feedformat=atomAoPS Wiki - User contributions [en]2024-03-29T10:56:29ZUser contributionsMediaWiki 1.31.1https://artofproblemsolving.com/wiki/index.php?title=2016_AMC_12B_Problems/Problem_9&diff=768222016 AMC 12B Problems/Problem 92016-02-21T20:04:54Z<p>Sssuperm1ch4el: /* Solution */</p>
<hr />
<div>==Problem==<br />
<br />
Carl decided to in his rectangular garden. He bought <math>20</math> fence posts, placed one on each of the four corners, and spaced out the rest evenly along the edges of the garden, leaving exactly <math>4</math> yards between neighboring posts. The longer side of his garden, including the corners, has twice as many posts as the shorter side, including the corners. What is the area, in square yards, of Carl’s garden?<br />
<br />
<math>\textbf{(A)}\ 256\qquad\textbf{(B)}\ 336\qquad\textbf{(C)}\ 384\qquad\textbf{(D)}\ 448\qquad\textbf{(E)}\ 512</math><br />
<br />
==Solution==<br />
To start, use algebra to determine the number of posts on each side. You have (the long sides count for <math>2</math> because there are twice as many) <math>6x = 20 + 4</math> (each corner is double counted so you must add <math>4</math>) Making the shorter end have <math>4</math>, and the longer end have <math>8</math>. <math>((8-1)*4)*((4-1)*4) = 28*12 = 336</math>. Therefore, the answer is <math>\boxed{\textbf{(B)}\ 336}</math><br />
<br />
==See Also==<br />
{{AMC12 box|year=2016|ab=B|num-b=8|num-a=10}}<br />
{{MAA Notice}}</div>Sssuperm1ch4elhttps://artofproblemsolving.com/wiki/index.php?title=2016_AMC_12B_Problems/Problem_7&diff=768172016 AMC 12B Problems/Problem 72016-02-21T20:01:12Z<p>Sssuperm1ch4el: /* Solution */</p>
<hr />
<div>==Problem==<br />
<br />
Josh writes the numbers <math>1,2,3,\dots,99,100</math>. He marks out <math>1</math>, skips the next number <math>(2)</math>, marks out <math>3</math>, and continues skipping and marking out the next number to the end of the list. Then he goes back to the start of his list, marks out the first remaining number <math>(2)</math>, skips the next number <math>(4)</math>, marks out <math>6</math>, skips <math>8</math>, marks out <math>10</math>, and so on to the end. Josh continues in this manner until only one number remains. What is that number?<br />
<br />
<math>\textbf{(A)}\ 13 \qquad<br />
\textbf{(B)}\ 32 \qquad<br />
\textbf{(C)}\ 56 \qquad<br />
\textbf{(D)}\ 64 \qquad<br />
\textbf{(E)}\ 96</math><br />
<br />
==Solution==<br />
Following the pattern, you are crossing out...<br />
<br />
Time 1: Every non-multiple of <math>2</math><br />
<br />
Time 2: Every non-multiple of <math>4</math><br />
<br />
Time 3: Every non-multiple of <math>8</math><br />
<br />
Following this pattern, you are left with every multiple of <math>64</math> which is only <math>\boxed{\textbf{(D)}64}</math>.<br />
<br />
==See Also==<br />
{{AMC12 box|year=2016|ab=B|num-b=6|num-a=8}}<br />
{{MAA Notice}}</div>Sssuperm1ch4elhttps://artofproblemsolving.com/wiki/index.php?title=2016_AMC_12B_Problems/Problem_7&diff=768152016 AMC 12B Problems/Problem 72016-02-21T20:00:13Z<p>Sssuperm1ch4el: </p>
<hr />
<div>==Problem==<br />
<br />
Josh writes the numbers <math>1,2,3,\dots,99,100</math>. He marks out <math>1</math>, skips the next number <math>(2)</math>, marks out <math>3</math>, and continues skipping and marking out the next number to the end of the list. Then he goes back to the start of his list, marks out the first remaining number <math>(2)</math>, skips the next number <math>(4)</math>, marks out <math>6</math>, skips <math>8</math>, marks out <math>10</math>, and so on to the end. Josh continues in this manner until only one number remains. What is that number?<br />
<br />
<math>\textbf{(A)}\ 13 \qquad<br />
\textbf{(B)}\ 32 \qquad<br />
\textbf{(C)}\ 56 \qquad<br />
\textbf{(D)}\ 64 \qquad<br />
\textbf{(E)}\ 96</math><br />
<br />
==Solution==<br />
Following the pattern, you are crossing out...<br />
<br />
Time 1: Every non-multiple of <math>2</math><br />
<br />
Time 2: Every non-multiple of <math>4</math><br />
<br />
Time 3: Every non-multiple of <math>8</math><br />
<br />
Following this pattern, you are left with every multiple of <math>64</math> which is only <math>64</math>.<br />
<br />
==See Also==<br />
{{AMC12 box|year=2016|ab=B|num-b=6|num-a=8}}<br />
{{MAA Notice}}</div>Sssuperm1ch4elhttps://artofproblemsolving.com/wiki/index.php?title=2016_AMC_12B_Problems/Problem_6&diff=767542016 AMC 12B Problems/Problem 62016-02-21T18:22:37Z<p>Sssuperm1ch4el: /* Solution */</p>
<hr />
<div>==Problem==<br />
<br />
All three vertices of <math>\bigtriangleup ABC</math> lie on the parabola defined by <math>y=x^2</math>, with <math>A</math> at the origin and <math>\overline{BC}</math> parallel to the <math>x</math>-axis. The area of the triangle is <math>64</math>. What is the length of <math>BC</math>? <br />
<br />
<math>\textbf{(A)}\ 4\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 16</math><br />
<br />
==Solution==<br />
Plotting points <math>B</math> and <math>C</math> on the graph shows that they are at <math>\left( -x,x^2\right)</math> and <math>\left( x,x^2\right)</math>, which is isosceles. By setting up the triangle angle formula you get: <math>64=\frac{1}{2}*2x*x^2 = 64=x^3</math> Making <math>*4</math>, and the length of <math>BC</math> is <math>2x</math>, so the answer is <math>\boxed{\textbf{(D)} 8}</math>.<br />
<br />
==See Also==<br />
{{AMC12 box|year=2016|ab=B|num-b=5|num-a=7}}<br />
{{MAA Notice}}</div>Sssuperm1ch4elhttps://artofproblemsolving.com/wiki/index.php?title=2016_AMC_12B_Problems/Problem_6&diff=767392016 AMC 12B Problems/Problem 62016-02-21T18:00:38Z<p>Sssuperm1ch4el: </p>
<hr />
<div>==Problem==<br />
<br />
All three vertices of <math>\bigtriangleup ABC</math> lie on the parabola defined by <math>y=x^2</math>, with <math>A</math> at the origin and <math>\overline{BC}</math> parallel to the <math>x</math>-axis. The area of the triangle is <math>64</math>. What is the length of <math>BC</math>? <br />
<br />
<math>\textbf{(A)}\ 4\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 16</math><br />
<br />
==Solution==<br />
Plotting points <math>B</math> and <math>C</math> on the graph shows that they are at <math>\left( -x,x^2\right)</math> and <math>\left( x,x^2\right)</math>, which is isosceles. By setting up the triangle angle formula you get: <math>64=\frac{1}{2}*2x*x^2 = 64=x^3</math> Making <math>*4</math>, and the length of <math>BC</math> is <math>2x</math>, so the answer is <math>8</math>.<br />
<br />
==See Also==<br />
{{AMC12 box|year=2016|ab=B|num-b=5|num-a=7}}<br />
{{MAA Notice}}</div>Sssuperm1ch4elhttps://artofproblemsolving.com/wiki/index.php?title=2016_AMC_12A_Problems/Problem_11&diff=754602016 AMC 12A Problems/Problem 112016-02-04T20:10:04Z<p>Sssuperm1ch4el: Created page with "==Problem== Each of the <math>100</math> students in a certain summer camp can either sing, dance, or act. Some students have more than one talent, but no student has all thr..."</p>
<hr />
<div>==Problem==<br />
<br />
Each of the <math>100</math> students in a certain summer camp can either sing, dance, or act. Some students have more than one talent, but no student has all three talents. There are <math>42</math> students who cannot sing, <math>65</math> students who cannot dance, and <math>29</math> students who cannot act. How many students have two of these talents? <br />
<br />
<math>\textbf{(A)}\ 16\qquad\textbf{(B)}\ 25\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 49\qquad\textbf{(E)}\ 64</math> <br />
<br />
==Solution==<br />
<br />
<br />
<br />
==See Also==<br />
{{AMC12 box|year=2016|ab=A|num-b=24|num-a=26}}<br />
{{MAA Notice}}</div>Sssuperm1ch4elhttps://artofproblemsolving.com/wiki/index.php?title=2016_AMC_12A_Problems&diff=754412016 AMC 12A Problems2016-02-04T18:41:32Z<p>Sssuperm1ch4el: /* Problem 24 */</p>
<hr />
<div>{{AMC12 Problems|year=2016|ab=A}}<br />
<br />
==Problem 1==<br />
<br />
What is the value of <math>\frac{11!-10!}{9!}</math>?<br />
<br />
<math>\textbf{(A)}\ 99\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 110\qquad\textbf{(D)}\ 121\qquad\textbf{(E)}\ 132</math><br />
<br />
[[2016 AMC 12A Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
<br />
For what value of <math>x</math> does <math>10^x\cdot100^{2x}=1000^5</math>?<br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5</math><br />
<br />
[[2016 AMC 12A Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
The remainder can be defined for all real numbers <math>x</math> and <math>y</math> with <math>y \neq 0</math> by <cmath>\text{rem} (x ,y)=x-y\left \lfloor \frac{x}{y} \right \rfloor</cmath>where <math>\left \lfloor \tfrac{x}{y} \right \rfloor</math> denotes the greatest integer less than or equal to <math>\tfrac{x}{y}</math>. What is the value of <math>\text{rem} (\tfrac{3}{8}, -\tfrac{2}{5} )</math>?<br />
<br />
<math>\textbf{(A) } -\frac{3}{8} \qquad \textbf{(B) } -\frac{1}{40} \qquad \textbf{(C) } 0 \qquad \textbf{(D) } \frac{3}{8} \qquad \textbf{(E) } \frac{31}{40}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
The mean, median, and mode of the <math>7</math> data values <math>60, 100, x, 40, 50, 200, 90</math> are all equal to <math>x</math>. What is the value of <math>x</math>?<br />
<br />
<math>\textbf{(A)}\ 50\qquad\textbf{(B)}\ 60\qquad\textbf{(C)}\ 75\qquad\textbf{(D)}\ 90\qquad\textbf{(E)}\ 100</math><br />
<br />
[[2016 AMC 12A Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
Goldbach's conjecture states that every even integer greater than 2 can be written as the sum of two prime numbers (for example, <math>2016=13+2003</math>). So far, no one has been able to prove that the conjecture is true, and no one has found a counterexample to show that the conjecture is false. What would a counterexample consist of?<br />
<br />
<math> \textbf{(A)}\ \text{an odd integer greater than } 2 \text{ that can be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(B)}\ \text{an odd integer greater than } 2 \text{ that cannot be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(C)}\ \text{an even integer greater than } 2 \text{ that can be written as the sum of two numbers that are not prime}\\<br />
\qquad\textbf{(D)}\ \text{an even integer greater than } 2 \text{ that can be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(E)}\ \text{an even integer greater than } 2 \text{ that cannot be written as the sum of two prime numbers}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
A triangular array of <math>2016</math> coins has <math>1</math> coin in the first row, <math>2</math> coins in the second row, <math>3</math> coins in the third row, and so on up to <math>N</math> coins in the <math>N</math>th row. What is the sum of the digits of <math>N</math> ?<br />
<br />
<math>\textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 10</math><br />
<br />
[[2016 AMC 12A Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
Which of these describes the graph of <math>x^2(x+y+1)=y^2(x+y+1)</math> ?<br />
<br />
<math> \textbf{(A)}\ \text{two parallel lines}\\<br />
\qquad\textbf{(B)}\ \text{two intersecting lines}\\<br />
\qquad\textbf{(C)}\ \text{three lines that all pass through a common point}\\<br />
\qquad\textbf{(D)}\ \text{three lines that do not all pass through a comment point}\\<br />
\qquad\textbf{(E)}\ \text{a line and a parabola}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
What is the area of the shaded region of the given <math>8\times 5</math> rectangle?<br />
<br />
<asy><br />
<br />
size(6cm);<br />
defaultpen(fontsize(9pt));<br />
draw((0,0)--(8,0)--(8,5)--(0,5)--cycle);<br />
filldraw((7,0)--(8,0)--(8,1)--(0,4)--(0,5)--(1,5)--cycle,gray(0.8));<br />
<br />
label("$1$",(1/2,5),dir(90));<br />
label("$7$",(9/2,5),dir(90));<br />
<br />
label("$1$",(8,1/2),dir(0));<br />
label("$4$",(8,3),dir(0));<br />
<br />
label("$1$",(15/2,0),dir(270));<br />
label("$7$",(7/2,0),dir(270));<br />
<br />
label("$1$",(0,9/2),dir(180));<br />
label("$4$",(0,2),dir(180));<br />
<br />
</asy><br />
<br />
<math>\textbf{(A)}\ 4\dfrac{3}{4}\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 5\dfrac{1}{4}\qquad\textbf{(D)}\ 6\dfrac{1}{2}\qquad\textbf{(E)}\ 8</math><br />
<br />
[[2016 AMC 12A Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
The five small shaded squares inside this unit square are congruent and have disjoint interiors. The midpoint of each side of the middle square coincides with one of the vertices of the other four small squares as shown. The common side length is <math>\tfrac{a-\sqrt{2}}{b}</math>, where <math>a</math> and <math>b</math> are positive integers. What is <math>a+b</math> ?<br />
<br />
<pre style="color: gray">TODO: Diagram</pre><br />
<br />
<math>\textbf{(A)}\ 7\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 11</math><br />
<br />
[[2016 AMC 12A Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
Five friends sat in a movie theater in a row containing <math>5</math> seats, numbered <math>1</math> to <math>5</math> from left to right. (The directions "left" and "right" are from the point of view of the people as they sit in the seats.) During the movie Ada went to the lobby to get some popcorn. When she returned, she found that Bea had moved two seats to the right, Ceci had moved one seat to the left, and Dee and Edie had switched seats, leaving an end seat for Ada. In which seat had Ada been sitting before she got up?<br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5</math><br />
<br />
[[2016 AMC 12A Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
Each of the 100 students in a certain summer camp can either sing, dance, or act. Some students have more than one talent, but no student has all three talents. There are 42 students who cannot sing, 65 students who cannot dance, and 29 students who cannot act. How many students have two of these talents?<br />
<br />
<math>\textbf{(A)}\ 16\qquad\textbf{(B)}\ 25\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 49\qquad\textbf{(E)}\ 64</math><br />
<br />
[[2016 AMC 12A Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
In <math>\triangle ABC</math>, <math>AB = 6</math>, <math>BC = 7</math>, and <math>CA = 8</math>. Point <math>D</math> lies on <math>\overline{BC}</math>, and <math>\overline{AD}</math> bisects <math>\angle BAC</math>. Point <math>E</math> lies on <math>\overline{AC}</math>, and <math>\overline{BE}</math> bisects <math>\angle ABC</math>. The bisectors intersect at <math>F</math>. What is the ratio <math>AF</math> : <math>FD</math>?<br />
<br />
<pre style="color: gray">TODO: Diagram</pre><br />
<br />
<math>\textbf{(A)}\ 3:2\qquad\textbf{(B)}\ 5:3\qquad\textbf{(C)}\ 2:1\qquad\textbf{(D)}\ 7:3\qquad\textbf{(E)}\ 5:2</math><br />
<br />
[[2016 AMC 12A Problems/Problem 12|Solution]]<br />
<br />
<br />
==Problem 13==<br />
<br />
Let <math>N</math> be a positive multiple of <math>5</math>. One red ball and <math>N</math> green balls are arranged in a line in random order. Let <math>P(N)</math> be the probability that at least <math>\tfrac{3}{5}</math> of the green balls are on the same side of the red ball. Observe that <math>P(5)=1</math> and that <math>P(N)</math> approaches <math>\tfrac{4}{5}</math> as <math>N</math> grows large. What is the sum of the digits of the least value of <math>N</math> such that <math>P(N) < \tfrac{321}{400}</math>?<br />
<br />
<math>\textbf{(A)}\ 12\qquad\textbf{(B)}\ 14\qquad\textbf{(C)}\ 16\qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 20</math><br />
<br />
[[2016 AMC 12A Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
Each vertex of a cube is to be labeled with an integer from 1 through 8, 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 />
[[2016 AMC 12A Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
Circles with centers <math>P, Q</math> and <math>R</math>, having radii <math>1, 2</math> and <math>3</math>, respectively, lie on the same side of line <math>l</math> and are tangent to <math>l</math> at <math>P', Q'</math> and <math>R'</math>, respectively, with <math>Q'</math> between <math>P'</math> and <math>R'</math>. The circle with center <math>Q</math> is externally tangent to each of the other two circles. What is the area of triangle <math>PQR</math>?<br />
<br />
<math>\textbf{(A) } 0\qquad \textbf{(B) } \sqrt{\frac{2}{3}}\qquad\textbf{(C) } 1\qquad\textbf{(D) } \sqrt{6}-\sqrt{2}\qquad\textbf{(E) }\sqrt{\frac{3}{2}}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
The graphs of <math>y=\log_3 x, y=\log_x 3, y=\log_\frac{1}{3} x,</math> and <math>y=\log_x \dfrac{1}{3}</math> are plotted on the same set of axes. How many points in the plane with positive <math>x</math>-coordinates lie on two or more of the graphs? <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 />
[[2016 AMC 12A Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Let <math>ABCD</math> be a square. Let <math>E, F, G</math> and <math>H</math> be the centers, respectively, of equilateral triangles with bases <math>\overline{AB}, \overline{BC}, \overline{CD},</math> and <math>\overline{DA},</math> each exterior to the square. What is the ratio of the area of square <math>EFGH</math> to the area of square <math>ABCD</math>? <br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ \frac{2+\sqrt{3}}{3} \qquad\textbf{(C)}\ \sqrt{2} \qquad\textbf{(D)}\ \frac{\sqrt{2}+\sqrt{3}}{2} \qquad\textbf{(E)}\ \sqrt{3}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
For some positive integer <math>n,</math> the number <math>110n^3</math> has <math>110</math> positive integer divisors, including <math>1</math> and the number <math>110n^3.</math> How many positive integer divisors does the number <math>81n^4</math> have? <br />
<br />
<math>\textbf{(A)}\ 110\qquad\textbf{(B)}\ 191\qquad\textbf{(C)}\ 261\qquad\textbf{(D)}\ 325\qquad\textbf{(E)}\ 425</math><br />
<br />
[[2016 AMC 12A Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
Jerry starts at <math>0</math> on the real number line. He tosses a fair coin <math>8</math> times. When he gets heads, he moves <math>1</math> unit in the positive direction; when he gets tails, he moves <math>1</math> unit in the negative direction. The probability that he reaches <math>4</math> at some time during this process <math>\frac{a}{b},</math> where <math>a</math> and <math>b</math> are relatively prime positive integers. What is <math>a + b?</math> (For example, he succeeds if his sequence of tosses is <math>HTHHHHHH.</math>)<br />
<br />
<math>\textbf{(A)}\ 69\qquad\textbf{(B)}\ 151\qquad\textbf{(C)}\ 257\qquad\textbf{(D)}\ 293\qquad\textbf{(E)}\ 313</math><br />
<br />
[[2016 AMC 12A Problems/Problem 19|Solution]]<br />
<br />
<br />
==Problem 20==<br />
<br />
A binary operation <math>\ \diamondsuit </math> has the properties that <math>a\ \diamondsuit\ (b\ \diamondsuit\ c) = (a\ \diamondsuit\ b)\ \cdot\ c</math> and that <math>a\ \diamondsuit\ a = 1</math> for all nonzero real numbers <math>a, b</math> and <math>c.</math> (Here the dot <math>\ \cdot</math> represents the usual multiplication operation.) The solution to the equation <math>2016\ \diamondsuit\ (6\ \diamondsuit\ x) = 100</math> can be written as <math>\frac{p}{q},</math> where <math>p</math> and <math>q</math> are relativelt prime positive integers. What is <math>p + q?</math> <br />
<br />
<math>\textbf{(A)}\ 109\qquad\textbf{(B)}\ 201\qquad\textbf{(C)}\ 301\qquad\textbf{(D)}\ 3049\qquad\textbf{(E)}\ 33,601</math><br />
<br />
[[2016 AMC 12A Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
A quadrilateral is inscribed in a circle of radius <math>200\sqrt{2}.</math> Three of the sides of this quadrilateral have length <math>200.</math> What is the length of its fourth side? <br />
<br />
<math>\textbf{(A)}\ 200\qquad\textbf{(B)}\ 200\sqrt{2} \qquad\textbf{(C)}\ 200\sqrt{3} \qquad\textbf{(D)}\ 300\sqrt{2} \qquad\textbf{(E)}\ 500</math><br />
<br />
[[2016 AMC 12A Problems/Problem 21|Solution]]<br />
<br />
==Problem 22==<br />
<br />
How many ordered triples <math>(x,y,z)</math> of positive integers satisfy <math>\text{lcm}(x,y) = 72, \text{lcm}(x,z) = 600</math> and <math>\text{lcm}(y,z)=900</math>?<br />
<br />
<math>\textbf{(A)}\ 15\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 64</math><br />
<br />
[[2016 AMC 12A Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
Three numbers in the interval <math>\left[0,1\right]</math> are chosen independently and at random. What is the probability that the chosen numbers are the side lengths of a triangle with positive area.<br />
<br />
<math>\textbf{(A)}\ \dfrac{1}{6}\qquad\textbf{(B)}\ \dfrac{1}{3}\qquad\textbf{(C)}\ \dfrac{1}{2}\qquad\textbf{(D)}\ \dfrac{2}{3}\qquad\textbf{(E)}\ \dfrac{5}{6}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 23|Solution]]<br />
<br />
==Problem 24==<br />
<br />
There is a smallest positive real number <math>a</math> such that there exists a positive real number <math>b</math> such that all the roots of the polynomial <math>x^3-ax^2+bx-a</math> are real. In fact, for this value of <math>a</math> the value of <math>b</math> is unique. What is the value of <math>b?</math><br />
<br />
<math>\textbf{(A)}\ 8\qquad\textbf{(B)}\ 9\qquad\textbf{(C)}\ 10\qquad\textbf{(D)}\ 11\qquad\textbf{(E)}\ 12</math><br />
<br />
[[2016 AMC 12A Problems/Problem 24|Solution]]<br />
<br />
==Problem 25==<br />
<br />
Let <math>k</math> be a positive integer. Bernardo and Silvia take turns writing and erasing numbers on a blackboard as follows: Bernardo starts by writing the smallest perfect square with <math>k+1</math> digits. Every time Bernardo writes a number, Silvia erases the last <math>k</math> digits of it. Bernardo then writes the next perfect square, Silvia erases the last <math>k</math> digits of it, and this process continues until the last two numbers that remain on the board differ by at least 2. Let <math>f(k)</math> be the smallest positive integer not written on the board. For example, if <math>k = 1</math>, then the numbers that Bernardo writes are <math>16, 25, 36, 49, 64</math>, and the numbers showing on the board after Silvia erases are <math>1, 2, 3, 4,</math> and <math>6</math>, and thus <math>f(1) = 5</math>. What is the sum of the digits of <math>f(2) + f(4)+ f(6) + ... + f(2016)</math>?<br />
<br />
<math>\textbf{(A)}\ 7986\qquad\textbf{(B)}\ 8002\qquad\textbf{(C)}\ 8030\qquad\textbf{(D)}\ 8048\qquad\textbf{(E)}\ 8064</math><br />
<br />
[[2016 AMC 12A Problems/Problem 25|Solution]]<br />
<br />
{{MAA Notice}}</div>Sssuperm1ch4elhttps://artofproblemsolving.com/wiki/index.php?title=2016_AMC_12A_Problems&diff=754402016 AMC 12A Problems2016-02-04T18:37:23Z<p>Sssuperm1ch4el: /* Problem 23 */</p>
<hr />
<div>{{AMC12 Problems|year=2016|ab=A}}<br />
<br />
==Problem 1==<br />
<br />
What is the value of <math>\frac{11!-10!}{9!}</math>?<br />
<br />
<math>\textbf{(A)}\ 99\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 110\qquad\textbf{(D)}\ 121\qquad\textbf{(E)}\ 132</math><br />
<br />
[[2016 AMC 12A Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
<br />
For what value of <math>x</math> does <math>10^x\cdot100^{2x}=1000^5</math>?<br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5</math><br />
<br />
[[2016 AMC 12A Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
The remainder can be defined for all real numbers <math>x</math> and <math>y</math> with <math>y \neq 0</math> by <cmath>\text{rem} (x ,y)=x-y\left \lfloor \frac{x}{y} \right \rfloor</cmath>where <math>\left \lfloor \tfrac{x}{y} \right \rfloor</math> denotes the greatest integer less than or equal to <math>\tfrac{x}{y}</math>. What is the value of <math>\text{rem} (\tfrac{3}{8}, -\tfrac{2}{5} )</math>?<br />
<br />
<math>\textbf{(A) } -\frac{3}{8} \qquad \textbf{(B) } -\frac{1}{40} \qquad \textbf{(C) } 0 \qquad \textbf{(D) } \frac{3}{8} \qquad \textbf{(E) } \frac{31}{40}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
The mean, median, and mode of the <math>7</math> data values <math>60, 100, x, 40, 50, 200, 90</math> are all equal to <math>x</math>. What is the value of <math>x</math>?<br />
<br />
<math>\textbf{(A)}\ 50\qquad\textbf{(B)}\ 60\qquad\textbf{(C)}\ 75\qquad\textbf{(D)}\ 90\qquad\textbf{(E)}\ 100</math><br />
<br />
[[2016 AMC 12A Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
Goldbach's conjecture states that every even integer greater than 2 can be written as the sum of two prime numbers (for example, <math>2016=13+2003</math>). So far, no one has been able to prove that the conjecture is true, and no one has found a counterexample to show that the conjecture is false. What would a counterexample consist of?<br />
<br />
<math> \textbf{(A)}\ \text{an odd integer greater than } 2 \text{ that can be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(B)}\ \text{an odd integer greater than } 2 \text{ that cannot be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(C)}\ \text{an even integer greater than } 2 \text{ that can be written as the sum of two numbers that are not prime}\\<br />
\qquad\textbf{(D)}\ \text{an even integer greater than } 2 \text{ that can be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(E)}\ \text{an even integer greater than } 2 \text{ that cannot be written as the sum of two prime numbers}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
A triangular array of <math>2016</math> coins has <math>1</math> coin in the first row, <math>2</math> coins in the second row, <math>3</math> coins in the third row, and so on up to <math>N</math> coins in the <math>N</math>th row. What is the sum of the digits of <math>N</math> ?<br />
<br />
<math>\textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 10</math><br />
<br />
[[2016 AMC 12A Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
Which of these describes the graph of <math>x^2(x+y+1)=y^2(x+y+1)</math> ?<br />
<br />
<math> \textbf{(A)}\ \text{two parallel lines}\\<br />
\qquad\textbf{(B)}\ \text{two intersecting lines}\\<br />
\qquad\textbf{(C)}\ \text{three lines that all pass through a common point}\\<br />
\qquad\textbf{(D)}\ \text{three lines that do not all pass through a comment point}\\<br />
\qquad\textbf{(E)}\ \text{a line and a parabola}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
What is the area of the shaded region of the given <math>8\times 5</math> rectangle?<br />
<br />
<asy><br />
<br />
size(6cm);<br />
defaultpen(fontsize(9pt));<br />
draw((0,0)--(8,0)--(8,5)--(0,5)--cycle);<br />
filldraw((7,0)--(8,0)--(8,1)--(0,4)--(0,5)--(1,5)--cycle,gray(0.8));<br />
<br />
label("$1$",(1/2,5),dir(90));<br />
label("$7$",(9/2,5),dir(90));<br />
<br />
label("$1$",(8,1/2),dir(0));<br />
label("$4$",(8,3),dir(0));<br />
<br />
label("$1$",(15/2,0),dir(270));<br />
label("$7$",(7/2,0),dir(270));<br />
<br />
label("$1$",(0,9/2),dir(180));<br />
label("$4$",(0,2),dir(180));<br />
<br />
</asy><br />
<br />
<math>\textbf{(A)}\ 4\dfrac{3}{4}\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 5\dfrac{1}{4}\qquad\textbf{(D)}\ 6\dfrac{1}{2}\qquad\textbf{(E)}\ 8</math><br />
<br />
[[2016 AMC 12A Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
The five small shaded squares inside this unit square are congruent and have disjoint interiors. The midpoint of each side of the middle square coincides with one of the vertices of the other four small squares as shown. The common side length is <math>\tfrac{a-\sqrt{2}}{b}</math>, where <math>a</math> and <math>b</math> are positive integers. What is <math>a+b</math> ?<br />
<br />
<pre style="color: gray">TODO: Diagram</pre><br />
<br />
<math>\textbf{(A)}\ 7\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 11</math><br />
<br />
[[2016 AMC 12A Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
Five friends sat in a movie theater in a row containing <math>5</math> seats, numbered <math>1</math> to <math>5</math> from left to right. (The directions "left" and "right" are from the point of view of the people as they sit in the seats.) During the movie Ada went to the lobby to get some popcorn. When she returned, she found that Bea had moved two seats to the right, Ceci had moved one seat to the left, and Dee and Edie had switched seats, leaving an end seat for Ada. In which seat had Ada been sitting before she got up?<br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5</math><br />
<br />
[[2016 AMC 12A Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
Each of the 100 students in a certain summer camp can either sing, dance, or act. Some students have more than one talent, but no student has all three talents. There are 42 students who cannot sing, 65 students who cannot dance, and 29 students who cannot act. How many students have two of these talents?<br />
<br />
<math>\textbf{(A)}\ 16\qquad\textbf{(B)}\ 25\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 49\qquad\textbf{(E)}\ 64</math><br />
<br />
[[2016 AMC 12A Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
In <math>\triangle ABC</math>, <math>AB = 6</math>, <math>BC = 7</math>, and <math>CA = 8</math>. Point <math>D</math> lies on <math>\overline{BC}</math>, and <math>\overline{AD}</math> bisects <math>\angle BAC</math>. Point <math>E</math> lies on <math>\overline{AC}</math>, and <math>\overline{BE}</math> bisects <math>\angle ABC</math>. The bisectors intersect at <math>F</math>. What is the ratio <math>AF</math> : <math>FD</math>?<br />
<br />
<pre style="color: gray">TODO: Diagram</pre><br />
<br />
<math>\textbf{(A)}\ 3:2\qquad\textbf{(B)}\ 5:3\qquad\textbf{(C)}\ 2:1\qquad\textbf{(D)}\ 7:3\qquad\textbf{(E)}\ 5:2</math><br />
<br />
[[2016 AMC 12A Problems/Problem 12|Solution]]<br />
<br />
<br />
==Problem 13==<br />
<br />
Let <math>N</math> be a positive multiple of <math>5</math>. One red ball and <math>N</math> green balls are arranged in a line in random order. Let <math>P(N)</math> be the probability that at least <math>\tfrac{3}{5}</math> of the green balls are on the same side of the red ball. Observe that <math>P(5)=1</math> and that <math>P(N)</math> approaches <math>\tfrac{4}{5}</math> as <math>N</math> grows large. What is the sum of the digits of the least value of <math>N</math> such that <math>P(N) < \tfrac{321}{400}</math>?<br />
<br />
<math>\textbf{(A)}\ 12\qquad\textbf{(B)}\ 14\qquad\textbf{(C)}\ 16\qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 20</math><br />
<br />
[[2016 AMC 12A Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
Each vertex of a cube is to be labeled with an integer from 1 through 8, 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 />
[[2016 AMC 12A Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
Circles with centers <math>P, Q</math> and <math>R</math>, having radii <math>1, 2</math> and <math>3</math>, respectively, lie on the same side of line <math>l</math> and are tangent to <math>l</math> at <math>P', Q'</math> and <math>R'</math>, respectively, with <math>Q'</math> between <math>P'</math> and <math>R'</math>. The circle with center <math>Q</math> is externally tangent to each of the other two circles. What is the area of triangle <math>PQR</math>?<br />
<br />
<math>\textbf{(A) } 0\qquad \textbf{(B) } \sqrt{\frac{2}{3}}\qquad\textbf{(C) } 1\qquad\textbf{(D) } \sqrt{6}-\sqrt{2}\qquad\textbf{(E) }\sqrt{\frac{3}{2}}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
The graphs of <math>y=\log_3 x, y=\log_x 3, y=\log_\frac{1}{3} x,</math> and <math>y=\log_x \dfrac{1}{3}</math> are plotted on the same set of axes. How many points in the plane with positive <math>x</math>-coordinates lie on two or more of the graphs? <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 />
[[2016 AMC 12A Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Let <math>ABCD</math> be a square. Let <math>E, F, G</math> and <math>H</math> be the centers, respectively, of equilateral triangles with bases <math>\overline{AB}, \overline{BC}, \overline{CD},</math> and <math>\overline{DA},</math> each exterior to the square. What is the ratio of the area of square <math>EFGH</math> to the area of square <math>ABCD</math>? <br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ \frac{2+\sqrt{3}}{3} \qquad\textbf{(C)}\ \sqrt{2} \qquad\textbf{(D)}\ \frac{\sqrt{2}+\sqrt{3}}{2} \qquad\textbf{(E)}\ \sqrt{3}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
For some positive integer <math>n,</math> the number <math>110n^3</math> has <math>110</math> positive integer divisors, including <math>1</math> and the number <math>110n^3.</math> How many positive integer divisors does the number <math>81n^4</math> have? <br />
<br />
<math>\textbf{(A)}\ 110\qquad\textbf{(B)}\ 191\qquad\textbf{(C)}\ 261\qquad\textbf{(D)}\ 325\qquad\textbf{(E)}\ 425</math><br />
<br />
[[2016 AMC 12A Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
Jerry starts at <math>0</math> on the real number line. He tosses a fair coin <math>8</math> times. When he gets heads, he moves <math>1</math> unit in the positive direction; when he gets tails, he moves <math>1</math> unit in the negative direction. The probability that he reaches <math>4</math> at some time during this process <math>\frac{a}{b},</math> where <math>a</math> and <math>b</math> are relatively prime positive integers. What is <math>a + b?</math> (For example, he succeeds if his sequence of tosses is <math>HTHHHHHH.</math>)<br />
<br />
<math>\textbf{(A)}\ 69\qquad\textbf{(B)}\ 151\qquad\textbf{(C)}\ 257\qquad\textbf{(D)}\ 293\qquad\textbf{(E)}\ 313</math><br />
<br />
[[2016 AMC 12A Problems/Problem 19|Solution]]<br />
<br />
<br />
==Problem 20==<br />
<br />
A binary operation <math>\ \diamondsuit </math> has the properties that <math>a\ \diamondsuit\ (b\ \diamondsuit\ c) = (a\ \diamondsuit\ b)\ \cdot\ c</math> and that <math>a\ \diamondsuit\ a = 1</math> for all nonzero real numbers <math>a, b</math> and <math>c.</math> (Here the dot <math>\ \cdot</math> represents the usual multiplication operation.) The solution to the equation <math>2016\ \diamondsuit\ (6\ \diamondsuit\ x) = 100</math> can be written as <math>\frac{p}{q},</math> where <math>p</math> and <math>q</math> are relativelt prime positive integers. What is <math>p + q?</math> <br />
<br />
<math>\textbf{(A)}\ 109\qquad\textbf{(B)}\ 201\qquad\textbf{(C)}\ 301\qquad\textbf{(D)}\ 3049\qquad\textbf{(E)}\ 33,601</math><br />
<br />
[[2016 AMC 12A Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
A quadrilateral is inscribed in a circle of radius <math>200\sqrt{2}.</math> Three of the sides of this quadrilateral have length <math>200.</math> What is the length of its fourth side? <br />
<br />
<math>\textbf{(A)}\ 200\qquad\textbf{(B)}\ 200\sqrt{2} \qquad\textbf{(C)}\ 200\sqrt{3} \qquad\textbf{(D)}\ 300\sqrt{2} \qquad\textbf{(E)}\ 500</math><br />
<br />
[[2016 AMC 12A Problems/Problem 21|Solution]]<br />
<br />
==Problem 22==<br />
<br />
How many ordered triples <math>(x,y,z)</math> of positive integers satisfy <math>\text{lcm}(x,y) = 72, \text{lcm}(x,z) = 600</math> and <math>\text{lcm}(y,z)=900</math>?<br />
<br />
<math>\textbf{(A)}\ 15\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 64</math><br />
<br />
[[2016 AMC 12A Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
Three numbers in the interval <math>\left[0,1\right]</math> are chosen independently and at random. What is the probability that the chosen numbers are the side lengths of a triangle with positive area.<br />
<br />
<math>\textbf{(A)}\ \dfrac{1}{6}\qquad\textbf{(B)}\ \dfrac{1}{3}\qquad\textbf{(C)}\ \dfrac{1}{2}\qquad\textbf{(D)}\ \dfrac{2}{3}\qquad\textbf{(E)}\ \dfrac{5}{6}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 23|Solution]]<br />
<br />
==Problem 24==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 24|Solution]]<br />
<br />
<br />
==Problem 25==<br />
<br />
Let <math>k</math> be a positive integer. Bernardo and Silvia take turns writing and erasing numbers on a blackboard as follows: Bernardo starts by writing the smallest perfect square with <math>k+1</math> digits. Every time Bernardo writes a number, Silvia erases the last <math>k</math> digits of it. Bernardo then writes the next perfect square, Silvia erases the last <math>k</math> digits of it, and this process continues until the last two numbers that remain on the board differ by at least 2. Let <math>f(k)</math> be the smallest positive integer not written on the board. For example, if <math>k = 1</math>, then the numbers that Bernardo writes are <math>16, 25, 36, 49, 64</math>, and the numbers showing on the board after Silvia erases are <math>1, 2, 3, 4,</math> and <math>6</math>, and thus <math>f(1) = 5</math>. What is the sum of the digits of <math>f(2) + f(4)+ f(6) + ... + f(2016)</math>?<br />
<br />
<math>\textbf{(A)}\ 7986\qquad\textbf{(B)}\ 8002\qquad\textbf{(C)}\ 8030\qquad\textbf{(D)}\ 8048\qquad\textbf{(E)}\ 8064</math><br />
<br />
[[2016 AMC 12A Problems/Problem 25|Solution]]<br />
<br />
{{MAA Notice}}</div>Sssuperm1ch4elhttps://artofproblemsolving.com/wiki/index.php?title=2016_AMC_12A_Problems&diff=754202016 AMC 12A Problems2016-02-04T17:22:31Z<p>Sssuperm1ch4el: /* Problem 21 */</p>
<hr />
<div>{{AMC12 Problems|year=2016|ab=A}}<br />
<br />
==Problem 1==<br />
<br />
What is the value of <math>\frac{11!-10!}{9!}</math>?<br />
<br />
<math>\textbf{(A)}\ 99\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 110\qquad\textbf{(D)}\ 121\qquad\textbf{(E)}\ 132</math><br />
<br />
[[2016 AMC 12A Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
<br />
For what value of <math>x</math> does <math>10^x\cdot100^{2x}=1000^5</math>?<br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5</math><br />
<br />
[[2016 AMC 12A Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
The remainder can be defined for all real numbers <math>x</math> and <math>y</math> with <math>y \neq 0</math> by <cmath>\text{rem} (x ,y)=x-y\left \lfloor \frac{x}{y} \right \rfloor</cmath>where <math>\left \lfloor \tfrac{x}{y} \right \rfloor</math> denotes the greatest integer less than or equal to <math>\tfrac{x}{y}</math>. What is the value of <math>\text{rem} (\tfrac{3}{8}, -\tfrac{2}{5} )</math>?<br />
<br />
<math>\textbf{(A) } -\frac{3}{8} \qquad \textbf{(B) } -\frac{1}{40} \qquad \textbf{(C) } 0 \qquad \textbf{(D) } \frac{3}{8} \qquad \textbf{(E) } \frac{31}{40}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
The mean, median, and mode of the <math>7</math> data values <math>60, 100, x, 40, 50, 200, 90</math> are all equal to <math>x</math>. What is the value of <math>x</math>?<br />
<br />
<math>\textbf{(A)}\ 50\qquad\textbf{(B)}\ 60\qquad\textbf{(C)}\ 75\qquad\textbf{(D)}\ 90\qquad\textbf{(E)}\ 100</math><br />
<br />
[[2016 AMC 12A Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
Goldbach's conjecture states that every even integer greater than 2 can be written as the sum of two prime numbers (for example, <math>2016=13+2003</math>). So far, no one has been able to prove that the conjecture is true, and no one has found a counterexample to show that the conjecture is false. What would a counterexample consist of?<br />
<br />
<math> \textbf{(A)}\ \text{an odd integer greater than } 2 \text{ that can be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(B)}\ \text{an odd integer greater than } 2 \text{ that cannot be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(C)}\ \text{an even integer greater than } 2 \text{ that can be written as the sum of two numbers that are not prime}\\<br />
\qquad\textbf{(D)}\ \text{an even integer greater than } 2 \text{ that can be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(E)}\ \text{an even integer greater than } 2 \text{ that cannot be written as the sum of two prime numbers}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
A triangular array of <math>2016</math> coins has <math>1</math> coin in the first row, <math>2</math> coins in the second row, <math>3</math> coins in the third row, and so on up to <math>N</math> coins in the <math>N</math>th row. What is the sum of the digits of <math>N</math> ?<br />
<br />
<math>\textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 10</math><br />
<br />
[[2016 AMC 12A Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
Which of these describes the graph of <math>x^2(x+y+1)=y^2(x+y+1)</math> ?<br />
<br />
<math> \textbf{(A)}\ \text{two parallel lines}\\<br />
\qquad\textbf{(B)}\ \text{two intersecting lines}\\<br />
\qquad\textbf{(C)}\ \text{three lines that all pass through a common point}\\<br />
\qquad\textbf{(D)}\ \text{three lines that do not all pass through a comment point}\\<br />
\qquad\textbf{(E)}\ \text{a line and a parabola}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
What is the area of the shaded reigon of the given <math>8\times 5</math> rectangle?<br />
<br />
<asy><br />
<br />
size(6cm);<br />
defaultpen(fontsize(9pt));<br />
draw((0,0)--(8,0)--(8,5)--(0,5)--cycle);<br />
filldraw((7,0)--(8,0)--(8,1)--(0,4)--(0,5)--(1,5)--cycle,gray(0.8));<br />
<br />
label("$1$",(1/2,5),dir(90));<br />
label("$7$",(9/2,5),dir(90));<br />
<br />
label("$1$",(8,1/2),dir(0));<br />
label("$4$",(8,3),dir(0));<br />
<br />
label("$1$",(15/2,0),dir(270));<br />
label("$7$",(7/2,0),dir(270));<br />
<br />
label("$1$",(0,9/2),dir(180));<br />
label("$4$",(0,2),dir(180));<br />
<br />
</asy><br />
<br />
<math>\textbf{(A)}\ 4\dfrac{3}{4}\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 5\dfrac{1}{4}\qquad\textbf{(D)}\ 6\dfrac{1}{2}\qquad\textbf{(E)}\ 8</math><br />
<br />
[[2016 AMC 12A Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
The five small shaded squares inside this unit square are congruent and have disjoint interiors. The midpoint of each side of the middle square coincides with one of the vertices of the other four small squares as shown. The common side length is <math>\tfrac{a-\sqrt{2}}{b}</math>, where <math>a</math> and <math>b</math> are positive integers. What is <math>a+b</math> ?<br />
<br />
<pre style="color: gray">TODO: Diagram</pre><br />
<br />
<math>\textbf{(A)}\ 7\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 11</math><br />
<br />
[[2016 AMC 12A Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
Five friends sat in a movie theater in a row containing <math>5</math> seats, numbered <math>1</math> to <math>5</math> from left to right. (The directions "left" and "right" are from the point of view of the people as they sit in the seats.) During the movie Ada went to the lobby to get some popcorn. When she returned, she found that Bea had moved two seats to the right, Ceci had moved one seat to the left, and Dee and Edie had switched seats, leaving an end seat for Ada. In which seat had Ada been sitting before she got up?<br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5</math><br />
<br />
[[2016 AMC 12A Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
Each of the 100 students in a certain summer camp can either sing, dance, or act. Some students have more than one talent, but no student has all three talents. There are 42 students who cannot sing, 65 students who cannot dance, and 29 students who cannot act. How many students have two of these talents?<br />
<br />
<math>\textbf{(A)}\ 16\qquad\textbf{(B)}\ 25\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 49\qquad\textbf{(E)}\ 64</math><br />
<br />
[[2016 AMC 12A Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
In <math>\triangle ABC</math>, <math>AB = 6</math>, <math>BC = 7</math>, and <math>CA = 8</math>. Point <math>D</math> lies on <math>\overline{BC}</math>, and <math>\overline{AD}</math> bisects <math>\angle BAC</math>. Point <math>E</math> lies on <math>\overline{AC}</math>, and <math>\overline{BE}</math> bisects <math>\angle ABC</math>. The bisectors intersect at <math>F</math>. What is the ratio <math>AF</math> : <math>FD</math>?<br />
<br />
<pre style="color: gray">TODO: Diagram</pre><br />
<br />
<math>\textbf{(A)}\ 3:2\qquad\textbf{(B)}\ 5:3\qquad\textbf{(C)}\ 2:1\qquad\textbf{(D)}\ 7:3\qquad\textbf{(E)}\ 5:2</math><br />
<br />
[[2016 AMC 12A Problems/Problem 12|Solution]]<br />
<br />
<br />
==Problem 13==<br />
<br />
Let <math>N</math> be a positive multiple of <math>5</math>. One red ball and <math>N</math> green balls are arranged in a line in random order. Let <math>P(N)</math> be the probability that at least <math>\tfrac{3}{5}</math> of the green balls are on the same side of the red ball. Observe that <math>P(5)=1</math> and that <math>P(N)</math> approaches <math>\tfrac{4}{5}</math> as <math>N</math> grows large. What is the sum of the digits of the least value of <math>N</math> such that <math>P(N) < \tfrac{321}{400}</math>?<br />
<br />
<math>\textbf{(A)}\ 12\qquad\textbf{(B)}\ 14\qquad\textbf{(C)}\ 16\qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 20</math><br />
<br />
[[2016 AMC 12A Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
Each vertex of a cube is to be labeled with an integer from 1 through 8, 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 />
[[2016 AMC 12A Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
Circles with centers <math>P, Q</math> and <math>R</math>, having radii <math>1, 2</math> and <math>3</math>, respectively, lie on the same side of line <math>l</math> and are tangent to <math>l</math> at <math>P', Q'</math> and <math>R'</math>, respectively, with <math>Q'</math> between <math>P'</math> and <math>R'</math>. The circle with center <math>Q</math> is externally tangent to each of the other two circles. What is the area of triangle <math>PQR</math>?<br />
<br />
<math>\textbf{(A) } 0\qquad \textbf{(B) } \sqrt{\frac{2}{3}}\qquad\textbf{(C) } 1\qquad\textbf{(D) } \sqrt{6}-\sqrt{2}\qquad\textbf{(E) }\sqrt{\frac{3}{2}}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
The graphs of <math>y=\log_3 x, y=\log_x 3, y=\log_\frac{1}{3} x,</math> and <math>y=\log_x \dfrac{1}{3}</math> are plotted on the same set of axes. How many points in the plane with positive <math>x</math>-coordinates lie on two or more of the graphs? <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 />
[[2016 AMC 12A Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Let <math>ABCD</math> be a square. Let <math>E, F, G</math> and <math>H</math> be the centers, respectively, of equilateral triangles with bases <math>\overline{AB}, \overline{BC}, \overline{CD},</math> and <math>\overline{DA},</math> each exterior to the square. What is rge ration of the area of square <math>EFGH</math> to the area of square <math>ABCD</math>? <br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ \frac{2+\sqrt{3}}{3} \qquad\textbf{(C)}\ \sqrt{2} \qquad\textbf{(D)}\ \frac{\sqrt{2}+\sqrt{3}}{2} \qquad\textbf{(E)}\ \sqrt{3}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
For some positive integer <math>n,</math> the number <math>110n^3</math> has <math>110</math> positive integer divisors, including <math>1</math> and the number <math>110n^3.</math> How many positive integer divisors does the number <math>81n^4</math> have? <br />
<br />
<math>\textbf{(A)}\ 110\qquad\textbf{(B)}\ 191\qquad\textbf{(C)}\ 261\qquad\textbf{(D)}\ 325\qquad\textbf{(E)}\ 425</math><br />
<br />
[[2016 AMC 12A Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
Jerry starts at <math>0</math> on the real number line. He tosses a fair coin <math>8</math> times. When he gets heads, he moves <math>1</math> unit in the positive direction; when he gets tails, he moves <math>1</math> unit in the negative direction. The probability that he reaches <math>4</math> at some time during this process <math>\frac{a}{b},</math> where <math>a</math> and <math>b</math> are relatively prime positive integers. What is <math>a + b?</math> (For example, he succeeds if his sequence of tosses is <math>HTHHHHHH.</math>)<br />
<br />
<math>\textbf{(A)}\ 69\qquad\textbf{(B)}\ 151\qquad\textbf{(C)}\ 257\qquad\textbf{(D)}\ 293\qquad\textbf{(E)}\ 313</math><br />
<br />
[[2016 AMC 12A Problems/Problem 19|Solution]]<br />
<br />
<br />
==Problem 20==<br />
<br />
A binary operation <math>\ \diamondsuit </math> has the properties that <math>a\ \diamondsuit\ (b\ \diamondsuit\ c) = (a\ \diamondsuit\ b)\ \cdot\ c</math> and that <math>a\ \diamondsuit\ a = 1</math> for all nonzero real numbers <math>a, b</math> and <math>c.</math> (Here the dot <math>\ \cdot</math> represents the usual multiplication operation.) The solution to the equation <math>2016\ \diamondsuit\ (6\ \diamondsuit\ x) = 100</math> can be written as <math>\frac{p}{q},</math> where <math>p</math> and <math>q</math> are relativelt prime positive integers. What is <math>p + q?</math> <br />
<br />
<math>\textbf{(A)}\ 109\qquad\textbf{(B)}\ 201\qquad\textbf{(C)}\ 301\qquad\textbf{(D)}\ 3049\qquad\textbf{(E)}\ 33,601</math><br />
<br />
[[2016 AMC 12A Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
A quadrilateral is inscribed in a circle of radius <math>200\sqrt{2}.</math> Three of the sides of this quadrilateral have length <math>200.</math> What is the length of its fourth side? <br />
<br />
<math>\textbf{(A)}\ 200\qquad\textbf{(B)}\ 200\sqrt{2} \qquad\textbf{(C)}\ 200\sqrt{3} \qquad\textbf{(D)}\ 300\sqrt{2} \qquad\textbf{(E)}\ 500</math><br />
<br />
[[2016 AMC 12A Problems/Problem 21|Solution]]<br />
<br />
==Problem 22==<br />
<br />
How many ordered triples <math>(x,y,z)</math> of positive integers satisfy <math>\text{lcm}(x,y) = 72, \text{lcm}(x,z) = 600</math> and <math>\text{lcm}(y,z)=900</math>?<br />
<br />
<math>\textbf{(A)}\ 15\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 64</math><br />
<br />
[[2016 AMC 12A Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 23|Solution]]<br />
<br />
<br />
==Problem 24==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 24|Solution]]<br />
<br />
<br />
==Problem 25==<br />
<br />
Let <math>k</math> be a positive integer. Bernardo and Silvia take turns writing and erasing numbers on a blackboard as follows: Bernardo starts by writing the smallest perfect square with <math>k+1</math> digits. Every time Bernardo writes a number, Silvia erases the last <math>k</math> digits of it. Bernardo then writes the next perfect square, Silvia erases the last <math>k</math> digits of it, and this process continues until the last two numbers that remain on the board differ by at least 2. Let <math>f(k)</math> be the smallest positive integer not written on the board. For example, if <math>k = 1</math>, then the numbers that Bernardo writes are <math>16, 25, 36, 49, 64</math>, and the numbers showing on the board after Silvia erases are <math>1, 2, 3, 4,</math> and <math>6</math>, and thus <math>f(1) = 5</math>. What is the sum of the digits of <math>f(2) + f(4)+ f(6) + ... + f(2016)</math>?<br />
<br />
<math>\textbf{(A)}\ 7986\qquad\textbf{(B)}\ 8002\qquad\textbf{(C)}\ 8030\qquad\textbf{(D)}\ 8048\qquad\textbf{(E)}\ 8064</math><br />
<br />
[[2016 AMC 12A Problems/Problem 25|Solution]]<br />
<br />
{{MAA Notice}}</div>Sssuperm1ch4elhttps://artofproblemsolving.com/wiki/index.php?title=2016_AMC_12A_Problems&diff=754172016 AMC 12A Problems2016-02-04T17:19:56Z<p>Sssuperm1ch4el: /* Problem 20 */</p>
<hr />
<div>{{AMC12 Problems|year=2016|ab=A}}<br />
<br />
==Problem 1==<br />
<br />
What is the value of <math>\frac{11!-10!}{9!}</math>?<br />
<br />
<math>\textbf{(A)}\ 99\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 110\qquad\textbf{(D)}\ 121\qquad\textbf{(E)}\ 132</math><br />
<br />
[[2016 AMC 12A Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
<br />
For what value of <math>x</math> does <math>10^x\cdot100^{2x}=1000^5</math>?<br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5</math><br />
<br />
[[2016 AMC 12A Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
The remainder can be defined for all real numbers <math>x</math> and <math>y</math> with <math>y \neq 0</math> by <cmath>\text{rem} (x ,y)=x-y\left \lfloor \frac{x}{y} \right \rfloor</cmath>where <math>\left \lfloor \tfrac{x}{y} \right \rfloor</math> denotes the greatest integer less than or equal to <math>\tfrac{x}{y}</math>. What is the value of <math>\text{rem} (\tfrac{3}{8}, -\tfrac{2}{5} )</math>?<br />
<br />
<math>\textbf{(A) } -\frac{3}{8} \qquad \textbf{(B) } -\frac{1}{40} \qquad \textbf{(C) } 0 \qquad \textbf{(D) } \frac{3}{8} \qquad \textbf{(E) } \frac{31}{40}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
The mean, median, and mode of the <math>7</math> data values <math>60, 100, x, 40, 50, 200, 90</math> are all equal to <math>x</math>. What is the value of <math>x</math>?<br />
<br />
<math>\textbf{(A)}\ 50\qquad\textbf{(B)}\ 60\qquad\textbf{(C)}\ 75\qquad\textbf{(D)}\ 90\qquad\textbf{(E)}\ 100</math><br />
<br />
[[2016 AMC 12A Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
Goldbach's conjecture states that every even integer greater than 2 can be written as the sum of two prime numbers (for example, <math>2016=13+2003</math>). So far, no one has been able to prove that the conjecture is true, and no one has found a counterexample to show that the conjecture is false. What would a counterexample consist of?<br />
<br />
<math> \textbf{(A)}\ \text{an odd integer greater than } 2 \text{ that can be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(B)}\ \text{an odd integer greater than } 2 \text{ that cannot be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(C)}\ \text{an even integer greater than } 2 \text{ that can be written as the sum of two numbers that are not prime}\\<br />
\qquad\textbf{(D)}\ \text{an even integer greater than } 2 \text{ that can be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(E)}\ \text{an even integer greater than } 2 \text{ that cannot be written as the sum of two prime numbers}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
A triangular array of <math>2016</math> coins has <math>1</math> coin in the first row, <math>2</math> coins in the second row, <math>3</math> coins in the third row, and so on up to <math>N</math> coins in the <math>N</math>th row. What is the sum of the digits of <math>N</math> ?<br />
<br />
<math>\textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 10</math><br />
<br />
[[2016 AMC 12A Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
Which of these describes the graph of <math>x^2(x+y+1)=y^2(x+y+1)</math> ?<br />
<br />
<math> \textbf{(A)}\ \text{two parallel lines}\\<br />
\qquad\textbf{(B)}\ \text{two intersecting lines}\\<br />
\qquad\textbf{(C)}\ \text{three lines that all pass through a common point}\\<br />
\qquad\textbf{(D)}\ \text{three lines that do not all pass through a comment point}\\<br />
\qquad\textbf{(E)}\ \text{a line and a parabola}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
What is the area of the shaded reigon of the given <math>8\times 5</math> rectangle?<br />
<br />
<asy><br />
<br />
size(6cm);<br />
defaultpen(fontsize(9pt));<br />
draw((0,0)--(8,0)--(8,5)--(0,5)--cycle);<br />
filldraw((7,0)--(8,0)--(8,1)--(0,4)--(0,5)--(1,5)--cycle,gray(0.8));<br />
<br />
label("$1$",(1/2,5),dir(90));<br />
label("$7$",(9/2,5),dir(90));<br />
<br />
label("$1$",(8,1/2),dir(0));<br />
label("$4$",(8,3),dir(0));<br />
<br />
label("$1$",(15/2,0),dir(270));<br />
label("$7$",(7/2,0),dir(270));<br />
<br />
label("$1$",(0,9/2),dir(180));<br />
label("$4$",(0,2),dir(180));<br />
<br />
</asy><br />
<br />
<math>\textbf{(A)}\ 4\dfrac{3}{4}\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 5\dfrac{1}{4}\qquad\textbf{(D)}\ 6\dfrac{1}{2}\qquad\textbf{(E)}\ 8</math><br />
<br />
[[2016 AMC 12A Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
The five small shaded squares inside this unit square are congruent and have disjoint interiors. The midpoint of each side of the middle square coincides with one of the vertices of the other four small squares as shown. The common side length is <math>\tfrac{a-\sqrt{2}}{b}</math>, where <math>a</math> and <math>b</math> are positive integers. What is <math>a+b</math> ?<br />
<br />
<pre style="color: gray">TODO: Diagram</pre><br />
<br />
<math>\textbf{(A)}\ 7\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 11</math><br />
<br />
[[2016 AMC 12A Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
Five friends sat in a movie theater in a row containing <math>5</math> seats, numbered <math>1</math> to <math>5</math> from left to right. (The directions "left" and "right" are from the point of view of the people as they sit in the seats.) During the movie Ada went to the lobby to get some popcorn. When she returned, she found that Bea had moved two seats to the right, Ceci had moved one seat to the left, and Dee and Edie had switched seats, leaving an end seat for Ada. In which seat had Ada been sitting before she got up?<br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5</math><br />
<br />
[[2016 AMC 12A Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
Each of the 100 students in a certain summer camp can either sing, dance, or act. Some students have more than one talent, but no student has all three talents. There are 42 students who cannot sing, 65 students who cannot dance, and 29 students who cannot act. How many students have two of these talents?<br />
<br />
<math>\textbf{(A)}\ 16\qquad\textbf{(B)}\ 25\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 49\qquad\textbf{(E)}\ 64</math><br />
<br />
[[2016 AMC 12A Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
In <math>\triangle ABC</math>, <math>AB = 6</math>, <math>BC = 7</math>, and <math>CA = 8</math>. Point <math>D</math> lies on <math>\overline{BC}</math>, and <math>\overline{AD}</math> bisects <math>\angle BAC</math>. Point <math>E</math> lies on <math>\overline{AC}</math>, and <math>\overline{BE}</math> bisects <math>\angle ABC</math>. The bisectors intersect at <math>F</math>. What is the ratio <math>AF</math> : <math>FD</math>?<br />
<br />
<pre style="color: gray">TODO: Diagram</pre><br />
<br />
<math>\textbf{(A)}\ 3:2\qquad\textbf{(B)}\ 5:3\qquad\textbf{(C)}\ 2:1\qquad\textbf{(D)}\ 7:3\qquad\textbf{(E)}\ 5:2</math><br />
<br />
[[2016 AMC 12A Problems/Problem 12|Solution]]<br />
<br />
<br />
==Problem 13==<br />
<br />
Let <math>N</math> be a positive multiple of <math>5</math>. One red ball and <math>N</math> green balls are arranged in a line in random order. Let <math>P(N)</math> be the probability that at least <math>\tfrac{3}{5}</math> of the green balls are on the same side of the red ball. Observe that <math>P(5)=1</math> and that <math>P(N)</math> approaches <math>\tfrac{4}{5}</math> as <math>N</math> grows large. What is the sum of the digits of the least value of <math>N</math> such that <math>P(N) < \tfrac{321}{400}</math>?<br />
<br />
<math>\textbf{(A)}\ 12\qquad\textbf{(B)}\ 14\qquad\textbf{(C)}\ 16\qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 20</math><br />
<br />
[[2016 AMC 12A Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
Each vertex of a cube is to be labeled with an integer from 1 through 8, 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 />
[[2016 AMC 12A Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
Circles with centers <math>P, Q</math> and <math>R</math>, having radii <math>1, 2</math> and <math>3</math>, respectively, lie on the same side of line <math>l</math> and are tangent to <math>l</math> at <math>P', Q'</math> and <math>R'</math>, respectively, with <math>Q'</math> between <math>P'</math> and <math>R'</math>. The circle with center <math>Q</math> is externally tangent to each of the other two circles. What is the area of triangle <math>PQR</math>?<br />
<br />
<math>\textbf{(A) } 0\qquad \textbf{(B) } \sqrt{\frac{2}{3}}\qquad\textbf{(C) } 1\qquad\textbf{(D) } \sqrt{6}-\sqrt{2}\qquad\textbf{(E) }\sqrt{\frac{3}{2}}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
The graphs of <math>y=\log_3 x, y=\log_x 3, y=\log_\frac{1}{3} x,</math> and <math>y=\log_x \dfrac{1}{3}</math> are plotted on the same set of axes. How many points in the plane with positive <math>x</math>-coordinates lie on two or more of the graphs? <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 />
[[2016 AMC 12A Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Let <math>ABCD</math> be a square. Let <math>E, F, G</math> and <math>H</math> be the centers, respectively, of equilateral triangles with bases <math>\overline{AB}, \overline{BC}, \overline{CD},</math> and <math>\overline{DA},</math> each exterior to the square. What is rge ration of the area of square <math>EFGH</math> to the area of square <math>ABCD</math>? <br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ \frac{2+\sqrt{3}}{3} \qquad\textbf{(C)}\ \sqrt{2} \qquad\textbf{(D)}\ \frac{\sqrt{2}+\sqrt{3}}{2} \qquad\textbf{(E)}\ \sqrt{3}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
For some positive integer <math>n,</math> the number <math>110n^3</math> has <math>110</math> positive integer divisors, including <math>1</math> and the number <math>110n^3.</math> How many positive integer divisors does the number <math>81n^4</math> have? <br />
<br />
<math>\textbf{(A)}\ 110\qquad\textbf{(B)}\ 191\qquad\textbf{(C)}\ 261\qquad\textbf{(D)}\ 325\qquad\textbf{(E)}\ 425</math><br />
<br />
[[2016 AMC 12A Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
Jerry starts at <math>0</math> on the real number line. He tosses a fair coin <math>8</math> times. When he gets heads, he moves <math>1</math> unit in the positive direction; when he gets tails, he moves <math>1</math> unit in the negative direction. The probability that he reaches <math>4</math> at some time during this process <math>\frac{a}{b},</math> where <math>a</math> and <math>b</math> are relatively prime positive integers. What is <math>a + b?</math> (For example, he succeeds if his sequence of tosses is <math>HTHHHHHH.</math>)<br />
<br />
<math>\textbf{(A)}\ 69\qquad\textbf{(B)}\ 151\qquad\textbf{(C)}\ 257\qquad\textbf{(D)}\ 293\qquad\textbf{(E)}\ 313</math><br />
<br />
[[2016 AMC 12A Problems/Problem 19|Solution]]<br />
<br />
<br />
==Problem 20==<br />
<br />
A binary operation <math>\ \diamondsuit </math> has the properties that <math>a\ \diamondsuit\ (b\ \diamondsuit\ c) = (a\ \diamondsuit\ b)\ \cdot\ c</math> and that <math>a\ \diamondsuit\ a = 1</math> for all nonzero real numbers <math>a, b</math> and <math>c.</math> (Here the dot <math>\ \cdot</math> represents the usual multiplication operation.) The solution to the equation <math>2016\ \diamondsuit\ (6\ \diamondsuit\ x) = 100</math> can be written as <math>\frac{p}{q},</math> where <math>p</math> and <math>q</math> are relativelt prime positive integers. What is <math>p + q?</math> <br />
<br />
<math>\textbf{(A)}\ 109\qquad\textbf{(B)}\ 201\qquad\textbf{(C)}\ 301\qquad\textbf{(D)}\ 3049\qquad\textbf{(E)}\ 33,601</math><br />
<br />
[[2016 AMC 12A Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 21|Solution]]<br />
<br />
<br />
==Problem 22==<br />
<br />
How many ordered triples <math>(x,y,z)</math> of positive integers satisfy <math>\text{lcm}(x,y) = 72, \text{lcm}(x,z) = 600</math> and <math>\text{lcm}(y,z)=900</math>?<br />
<br />
<math>\textbf{(A)}\ 15\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 64</math><br />
<br />
[[2016 AMC 12A Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 23|Solution]]<br />
<br />
<br />
==Problem 24==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 24|Solution]]<br />
<br />
<br />
==Problem 25==<br />
<br />
Let <math>k</math> be a positive integer. Bernardo and Silvia take turns writing and erasing numbers on a blackboard as follows: Bernardo starts by writing the smallest perfect square with <math>k+1</math> digits. Every time Bernardo writes a number, Silvia erases the last <math>k</math> digits of it. Bernardo then writes the next perfect square, Silvia erases the last <math>k</math> digits of it, and this process continues until the last two numbers that remain on the board differ by at least 2. Let <math>f(k)</math> be the smallest positive integer not written on the board. For example, if <math>k = 1</math>, then the numbers that Bernardo writes are <math>16, 25, 36, 49, 64</math>, and the numbers showing on the board after Silvia erases are <math>1, 2, 3, 4,</math> and <math>6</math>, and thus <math>f(1) = 5</math>. What is the sum of the digits of <math>f(2) + f(4)+ f(6) + ... + f(2016)</math>?<br />
<br />
<math>\textbf{(A)}\ 7986\qquad\textbf{(B)}\ 8002\qquad\textbf{(C)}\ 8030\qquad\textbf{(D)}\ 8048\qquad\textbf{(E)}\ 8064</math><br />
<br />
[[2016 AMC 12A Problems/Problem 25|Solution]]<br />
<br />
{{MAA Notice}}</div>Sssuperm1ch4elhttps://artofproblemsolving.com/wiki/index.php?title=2016_AMC_12A_Problems&diff=754152016 AMC 12A Problems2016-02-04T17:18:47Z<p>Sssuperm1ch4el: /* Problem 20 */</p>
<hr />
<div>{{AMC12 Problems|year=2016|ab=A}}<br />
<br />
==Problem 1==<br />
<br />
What is the value of <math>\frac{11!-10!}{9!}</math>?<br />
<br />
<math>\textbf{(A)}\ 99\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 110\qquad\textbf{(D)}\ 121\qquad\textbf{(E)}\ 132</math><br />
<br />
[[2016 AMC 12A Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
<br />
For what value of <math>x</math> does <math>10^x\cdot100^{2x}=1000^5</math>?<br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5</math><br />
<br />
[[2016 AMC 12A Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
The remainder can be defined for all real numbers <math>x</math> and <math>y</math> with <math>y \neq 0</math> by <cmath>\text{rem} (x ,y)=x-y\left \lfloor \frac{x}{y} \right \rfloor</cmath>where <math>\left \lfloor \tfrac{x}{y} \right \rfloor</math> denotes the greatest integer less than or equal to <math>\tfrac{x}{y}</math>. What is the value of <math>\text{rem} (\tfrac{3}{8}, -\tfrac{2}{5} )</math>?<br />
<br />
<math>\textbf{(A) } -\frac{3}{8} \qquad \textbf{(B) } -\frac{1}{40} \qquad \textbf{(C) } 0 \qquad \textbf{(D) } \frac{3}{8} \qquad \textbf{(E) } \frac{31}{40}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
The mean, median, and mode of the <math>7</math> data values <math>60, 100, x, 40, 50, 200, 90</math> are all equal to <math>x</math>. What is the value of <math>x</math>?<br />
<br />
<math>\textbf{(A)}\ 50\qquad\textbf{(B)}\ 60\qquad\textbf{(C)}\ 75\qquad\textbf{(D)}\ 90\qquad\textbf{(E)}\ 100</math><br />
<br />
[[2016 AMC 12A Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
Goldbach's conjecture states that every even integer greater than 2 can be written as the sum of two prime numbers (for example, <math>2016=13+2003</math>). So far, no one has been able to prove that the conjecture is true, and no one has found a counterexample to show that the conjecture is false. What would a counterexample consist of?<br />
<br />
<math> \textbf{(A)}\ \text{an odd integer greater than } 2 \text{ that can be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(B)}\ \text{an odd integer greater than } 2 \text{ that cannot be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(C)}\ \text{an even integer greater than } 2 \text{ that can be written as the sum of two numbers that are not prime}\\<br />
\qquad\textbf{(D)}\ \text{an even integer greater than } 2 \text{ that can be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(E)}\ \text{an even integer greater than } 2 \text{ that cannot be written as the sum of two prime numbers}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
A triangular array of <math>2016</math> coins has <math>1</math> coin in the first row, <math>2</math> coins in the second row, <math>3</math> coins in the third row, and so on up to <math>N</math> coins in the <math>N</math>th row. What is the sum of the digits of <math>N</math> ?<br />
<br />
<math>\textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 10</math><br />
<br />
[[2016 AMC 12A Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
Which of these describes the graph of <math>x^2(x+y+1)=y^2(x+y+1)</math> ?<br />
<br />
<math> \textbf{(A)}\ \text{two parallel lines}\\<br />
\qquad\textbf{(B)}\ \text{two intersecting lines}\\<br />
\qquad\textbf{(C)}\ \text{three lines that all pass through a common point}\\<br />
\qquad\textbf{(D)}\ \text{three lines that do not all pass through a comment point}\\<br />
\qquad\textbf{(E)}\ \text{a line and a parabola}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
What is the area of the shaded reigon of the given <math>8\times 5</math> rectangle?<br />
<br />
<asy><br />
<br />
size(6cm);<br />
defaultpen(fontsize(9pt));<br />
draw((0,0)--(8,0)--(8,5)--(0,5)--cycle);<br />
filldraw((7,0)--(8,0)--(8,1)--(0,4)--(0,5)--(1,5)--cycle,gray(0.8));<br />
<br />
label("$1$",(1/2,5),dir(90));<br />
label("$7$",(9/2,5),dir(90));<br />
<br />
label("$1$",(8,1/2),dir(0));<br />
label("$4$",(8,3),dir(0));<br />
<br />
label("$1$",(15/2,0),dir(270));<br />
label("$7$",(7/2,0),dir(270));<br />
<br />
label("$1$",(0,9/2),dir(180));<br />
label("$4$",(0,2),dir(180));<br />
<br />
</asy><br />
<br />
<math>\textbf{(A)}\ 4\dfrac{3}{4}\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 5\dfrac{1}{4}\qquad\textbf{(D)}\ 6\dfrac{1}{2}\qquad\textbf{(E)}\ 8</math><br />
<br />
[[2016 AMC 12A Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
The five small shaded squares inside this unit square are congruent and have disjoint interiors. The midpoint of each side of the middle square coincides with one of the vertices of the other four small squares as shown. The common side length is <math>\tfrac{a-\sqrt{2}}{b}</math>, where <math>a</math> and <math>b</math> are positive integers. What is <math>a+b</math> ?<br />
<br />
<pre style="color: gray">TODO: Diagram</pre><br />
<br />
<math>\textbf{(A)}\ 7\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 11</math><br />
<br />
[[2016 AMC 12A Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
Five friends sat in a movie theater in a row containing <math>5</math> seats, numbered <math>1</math> to <math>5</math> from left to right. (The directions "left" and "right" are from the point of view of the people as they sit in the seats.) During the movie Ada went to the lobby to get some popcorn. When she returned, she found that Bea had moved two seats to the right, Ceci had moved one seat to the left, and Dee and Edie had switched seats, leaving an end seat for Ada. In which seat had Ada been sitting before she got up?<br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5</math><br />
<br />
[[2016 AMC 12A Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
Each of the 100 students in a certain summer camp can either sing, dance, or act. Some students have more than one talent, but no student has all three talents. There are 42 students who cannot sing, 65 students who cannot dance, and 29 students who cannot act. How many students have two of these talents?<br />
<br />
<math>\textbf{(A)}\ 16\qquad\textbf{(B)}\ 25\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 49\qquad\textbf{(E)}\ 64</math><br />
<br />
[[2016 AMC 12A Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
In <math>\triangle ABC</math>, <math>AB = 6</math>, <math>BC = 7</math>, and <math>CA = 8</math>. Point <math>D</math> lies on <math>\overline{BC}</math>, and <math>\overline{AD}</math> bisects <math>\angle BAC</math>. Point <math>E</math> lies on <math>\overline{AC}</math>, and <math>\overline{BE}</math> bisects <math>\angle ABC</math>. The bisectors intersect at <math>F</math>. What is the ratio <math>AF</math> : <math>FD</math>?<br />
<br />
<pre style="color: gray">TODO: Diagram</pre><br />
<br />
<math>\textbf{(A)}\ 3:2\qquad\textbf{(B)}\ 5:3\qquad\textbf{(C)}\ 2:1\qquad\textbf{(D)}\ 7:3\qquad\textbf{(E)}\ 5:2</math><br />
<br />
[[2016 AMC 12A Problems/Problem 12|Solution]]<br />
<br />
<br />
==Problem 13==<br />
<br />
Let <math>N</math> be a positive multiple of <math>5</math>. One red ball and <math>N</math> green balls are arranged in a line in random order. Let <math>P(N)</math> be the probability that at least <math>\tfrac{3}{5}</math> of the green balls are on the same side of the red ball. Observe that <math>P(5)=1</math> and that <math>P(N)</math> approaches <math>\tfrac{4}{5}</math> as <math>N</math> grows large. What is the sum of the digits of the least value of <math>N</math> such that <math>P(N) < \tfrac{321}{400}</math>?<br />
<br />
<math>\textbf{(A)}\ 12\qquad\textbf{(B)}\ 14\qquad\textbf{(C)}\ 16\qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 20</math><br />
<br />
[[2016 AMC 12A Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
Each vertex of a cube is to be labeled with an integer from 1 through 8, 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 />
[[2016 AMC 12A Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
Circles with centers <math>P, Q</math> and <math>R</math>, having radii <math>1, 2</math> and <math>3</math>, respectively, lie on the same side of line <math>l</math> and are tangent to <math>l</math> at <math>P', Q'</math> and <math>R'</math>, respectively, with <math>Q'</math> between <math>P'</math> and <math>R'</math>. The circle with center <math>Q</math> is externally tangent to each of the other two circles. What is the area of triangle <math>PQR</math>?<br />
<br />
<math>\textbf{(A) } 0\qquad \textbf{(B) } \sqrt{\frac{2}{3}}\qquad\textbf{(C) } 1\qquad\textbf{(D) } \sqrt{6}-\sqrt{2}\qquad\textbf{(E) }\sqrt{\frac{3}{2}}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
The graphs of <math>y=\log_3 x, y=\log_x 3, y=\log_\frac{1}{3} x,</math> and <math>y=\log_x \dfrac{1}{3}</math> are plotted on the same set of axes. How many points in the plane with positive <math>x</math>-coordinates lie on two or more of the graphs? <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 />
[[2016 AMC 12A Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Let <math>ABCD</math> be a square. Let <math>E, F, G</math> and <math>H</math> be the centers, respectively, of equilateral triangles with bases <math>\overline{AB}, \overline{BC}, \overline{CD},</math> and <math>\overline{DA},</math> each exterior to the square. What is rge ration of the area of square <math>EFGH</math> to the area of square <math>ABCD</math>? <br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ \frac{2+\sqrt{3}}{3} \qquad\textbf{(C)}\ \sqrt{2} \qquad\textbf{(D)}\ \frac{\sqrt{2}+\sqrt{3}}{2} \qquad\textbf{(E)}\ \sqrt{3}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
For some positive integer <math>n,</math> the number <math>110n^3</math> has <math>110</math> positive integer divisors, including <math>1</math> and the number <math>110n^3.</math> How many positive integer divisors does the number <math>81n^4</math> have? <br />
<br />
<math>\textbf{(A)}\ 110\qquad\textbf{(B)}\ 191\qquad\textbf{(C)}\ 261\qquad\textbf{(D)}\ 325\qquad\textbf{(E)}\ 425</math><br />
<br />
[[2016 AMC 12A Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
Jerry starts at <math>0</math> on the real number line. He tosses a fair coin <math>8</math> times. When he gets heads, he moves <math>1</math> unit in the positive direction; when he gets tails, he moves <math>1</math> unit in the negative direction. The probability that he reaches <math>4</math> at some time during this process <math>\frac{a}{b},</math> where <math>a</math> and <math>b</math> are relatively prime positive integers. What is <math>a + b?</math> (For example, he succeeds if his sequence of tosses is <math>HTHHHHHH.</math>)<br />
<br />
<math>\textbf{(A)}\ 69\qquad\textbf{(B)}\ 151\qquad\textbf{(C)}\ 257\qquad\textbf{(D)}\ 293\qquad\textbf{(E)}\ 313</math><br />
<br />
[[2016 AMC 12A Problems/Problem 19|Solution]]<br />
<br />
<br />
==Problem 20==<br />
<br />
A binary operation <math>\ \diamondsuit\$ has the properties that </math>a\ \diamondsuit\ (b\ \diamondsuit\ c) = (a\ \diamondsuit\ b)\ \cdot\ c<math> and that </math>a\ \diamondsuit\ a = 1<math> for all nonzero real numbers </math>a, b<math> and </math>c.<math> (Here the dot </math>\ \cdot\$ represents the usual multiplication operation.) The solution to the equation <math>2016\ \diamondsuit\ (6\ \diamondsuit\ x) = 100</math> can be written as <math>\frac{p}{q},</math> where <math>p</math> and <math>q</math> are relativelt prime positive integers. What is <math>p + q?</math> <br />
<br />
<math>\textbf{(A)}\ 109\qquad\textbf{(B)}\ 201\qquad\textbf{(C)}\ 301\qquad\textbf{(D)}\ 3049\qquad\textbf{(E)}\ 33,601</math><br />
<br />
[[2016 AMC 12A Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 21|Solution]]<br />
<br />
<br />
==Problem 22==<br />
<br />
How many ordered triples <math>(x,y,z)</math> of positive integers satisfy <math>\text{lcm}(x,y) = 72, \text{lcm}(x,z) = 600</math> and <math>\text{lcm}(y,z)=900</math>?<br />
<br />
<math>\textbf{(A)}\ 15\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 64</math><br />
<br />
[[2016 AMC 12A Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 23|Solution]]<br />
<br />
<br />
==Problem 24==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 24|Solution]]<br />
<br />
<br />
==Problem 25==<br />
<br />
Let <math>k</math> be a positive integer. Bernardo and Silvia take turns writing and erasing numbers on a blackboard as follows: Bernardo starts by writing the smallest perfect square with <math>k+1</math> digits. Every time Bernardo writes a number, Silvia erases the last <math>k</math> digits of it. Bernardo then writes the next perfect square, Silvia erases the last <math>k</math> digits of it, and this process continues until the last two numbers that remain on the board differ by at least 2. Let <math>f(k)</math> be the smallest positive integer not written on the board. For example, if <math>k = 1</math>, then the numbers that Bernardo writes are <math>16, 25, 36, 49, 64</math>, and the numbers showing on the board after Silvia erases are <math>1, 2, 3, 4,</math> and <math>6</math>, and thus <math>f(1) = 5</math>. What is the sum of the digits of <math>f(2) + f(4)+ f(6) + ... + f(2016)</math>?<br />
<br />
<math>\textbf{(A)}\ 7986\qquad\textbf{(B)}\ 8002\qquad\textbf{(C)}\ 8030\qquad\textbf{(D)}\ 8048\qquad\textbf{(E)}\ 8064</math><br />
<br />
[[2016 AMC 12A Problems/Problem 25|Solution]]<br />
<br />
{{MAA Notice}}</div>Sssuperm1ch4elhttps://artofproblemsolving.com/wiki/index.php?title=2016_AMC_12A_Problems&diff=754092016 AMC 12A Problems2016-02-04T17:13:50Z<p>Sssuperm1ch4el: /* Problem 20 */</p>
<hr />
<div>{{AMC12 Problems|year=2016|ab=A}}<br />
<br />
==Problem 1==<br />
<br />
What is the value of <math>\frac{11!-10!}{9!}</math>?<br />
<br />
<math>\textbf{(A)}\ 99\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 110\qquad\textbf{(D)}\ 121\qquad\textbf{(E)}\ 132</math><br />
<br />
[[2016 AMC 12A Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
<br />
For what value of <math>x</math> does <math>10^x\cdot100^{2x}=1000^5</math>?<br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5</math><br />
<br />
[[2016 AMC 12A Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
The remainder can be defined for all real numbers <math>x</math> and <math>y</math> with <math>y \neq 0</math> by <cmath>\text{rem} (x ,y)=x-y\left \lfloor \frac{x}{y} \right \rfloor</cmath>where <math>\left \lfloor \tfrac{x}{y} \right \rfloor</math> denotes the greatest integer less than or equal to <math>\tfrac{x}{y}</math>. What is the value of <math>\text{rem} (\tfrac{3}{8}, -\tfrac{2}{5} )</math>?<br />
<br />
<math>\textbf{(A) } -\frac{3}{8} \qquad \textbf{(B) } -\frac{1}{40} \qquad \textbf{(C) } 0 \qquad \textbf{(D) } \frac{3}{8} \qquad \textbf{(E) } \frac{31}{40}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
The mean, median, and mode of the <math>7</math> data values <math>60, 100, x, 40, 50, 200, 90</math> are all equal to <math>x</math>. What is the value of <math>x</math>?<br />
<br />
<math>\textbf{(A)}\ 50\qquad\textbf{(B)}\ 60\qquad\textbf{(C)}\ 75\qquad\textbf{(D)}\ 90\qquad\textbf{(E)}\ 100</math><br />
<br />
[[2016 AMC 12A Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
Goldbach's conjecture states that every even integer greater than 2 can be written as the sum of two prime numbers (for example, <math>2016=13+2003</math>). So far, no one has been able to prove that the conjecture is true, and no one has found a counterexample to show that the conjecture is false. What would a counterexample consist of?<br />
<br />
<math> \textbf{(A)}\ \text{an odd integer greater than } 2 \text{ that can be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(B)}\ \text{an odd integer greater than } 2 \text{ that cannot be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(C)}\ \text{an even integer greater than } 2 \text{ that can be written as the sum of two numbers that are not prime}\\<br />
\qquad\textbf{(D)}\ \text{an even integer greater than } 2 \text{ that can be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(E)}\ \text{an even integer greater than } 2 \text{ that cannot be written as the sum of two prime numbers}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
A triangular array of <math>2016</math> coins has <math>1</math> coin in the first row, <math>2</math> coins in the second row, <math>3</math> coins in the third row, and so on up to <math>N</math> coins in the <math>N</math>th row. What is the sum of the digits of <math>N</math> ?<br />
<br />
<math>\textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 10</math><br />
<br />
[[2016 AMC 12A Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
Which of these describes the graph of <math>x^2(x+y+1)=y^2(x+y+1)</math> ?<br />
<br />
<math> \textbf{(A)}\ \text{two parallel lines}\\<br />
\qquad\textbf{(B)}\ \text{two intersecting lines}\\<br />
\qquad\textbf{(C)}\ \text{three lines that all pass through a common point}\\<br />
\qquad\textbf{(D)}\ \text{three lines that do not all pass through a comment point}\\<br />
\qquad\textbf{(E)}\ \text{a line and a parabola}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
What is the area of the shaded reigon of the given <math>8\times 5</math> rectangle?<br />
<br />
<asy><br />
<br />
size(6cm);<br />
defaultpen(fontsize(9pt));<br />
draw((0,0)--(8,0)--(8,5)--(0,5)--cycle);<br />
filldraw((7,0)--(8,0)--(8,1)--(0,4)--(0,5)--(1,5)--cycle,gray(0.8));<br />
<br />
label("$1$",(1/2,5),dir(90));<br />
label("$7$",(9/2,5),dir(90));<br />
<br />
label("$1$",(8,1/2),dir(0));<br />
label("$4$",(8,3),dir(0));<br />
<br />
label("$1$",(15/2,0),dir(270));<br />
label("$7$",(7/2,0),dir(270));<br />
<br />
label("$1$",(0,9/2),dir(180));<br />
label("$4$",(0,2),dir(180));<br />
<br />
</asy><br />
<br />
<math>\textbf{(A)}\ 4\dfrac{3}{4}\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 5\dfrac{1}{4}\qquad\textbf{(D)}\ 6\dfrac{1}{2}\qquad\textbf{(E)}\ 8</math><br />
<br />
[[2016 AMC 12A Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
The five small shaded squares inside this unit square are congruent and have disjoint interiors. The midpoint of each side of the middle square coincides with one of the vertices of the other four small squares as shown. The common side length is <math>\tfrac{a-\sqrt{2}}{b}</math>, where <math>a</math> and <math>b</math> are positive integers. What is <math>a+b</math> ?<br />
<br />
<pre style="color: gray">TODO: Diagram</pre><br />
<br />
<math>\textbf{(A)}\ 7\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 11</math><br />
<br />
[[2016 AMC 12A Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
Five friends sat in a movie theater in a row containing <math>5</math> seats, numbered <math>1</math> to <math>5</math> from left to right. (The directions "left" and "right" are from the point of view of the people as they sit in the seats.) During the movie Ada went to the lobby to get some popcorn. When she returned, she found that Bea had moved two seats to the right, Ceci had moved one seat to the left, and Dee and Edie had switched seats, leaving an end seat for Ada. In which seat had Ada been sitting before she got up?<br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5</math><br />
<br />
[[2016 AMC 12A Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
Each of the 100 students in a certain summer camp can either sing, dance, or act. Some students have more than one talent, but no student has all three talents. There are 42 students who cannot sing, 65 students who cannot dance, and 29 students who cannot act. How many students have two of these talents?<br />
<br />
<math>\textbf{(A)}\ 16\qquad\textbf{(B)}\ 25\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 49\qquad\textbf{(E)}\ 64</math><br />
<br />
[[2016 AMC 12A Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
In <math>\triangle ABC</math>, <math>AB = 6</math>, <math>BC = 7</math>, and <math>CA = 8</math>. Point <math>D</math> lies on <math>\overline{BC}</math>, and <math>\overline{AD}</math> bisects <math>\angle BAC</math>. Point <math>E</math> lies on <math>\overline{AC}</math>, and <math>\overline{BE}</math> bisects <math>\angle ABC</math>. The bisectors intersect at <math>F</math>. What is the ratio <math>AF</math> : <math>FD</math>?<br />
<br />
<pre style="color: gray">TODO: Diagram</pre><br />
<br />
<math>\textbf{(A)}\ 3:2\qquad\textbf{(B)}\ 5:3\qquad\textbf{(C)}\ 2:1\qquad\textbf{(D)}\ 7:3\qquad\textbf{(E)}\ 5:2</math><br />
<br />
[[2016 AMC 12A Problems/Problem 12|Solution]]<br />
<br />
<br />
==Problem 13==<br />
<br />
Let <math>N</math> be a positive multiple of <math>5</math>. One red ball and <math>N</math> green balls are arranged in a line in random order. Let <math>P(N)</math> be the probability that at least <math>\tfrac{3}{5}</math> of the green balls are on the same side of the red ball. Observe that <math>P(5)=1</math> and that <math>P(N)</math> approaches <math>\tfrac{4}{5}</math> as <math>N</math> grows large. What is the sum of the digits of the least value of <math>N</math> such that <math>P(N) < \tfrac{321}{400}</math>?<br />
<br />
<math>\textbf{(A)}\ 12\qquad\textbf{(B)}\ 14\qquad\textbf{(C)}\ 16\qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 20</math><br />
<br />
[[2016 AMC 12A Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
Each vertex of a cube is to be labeled with an integer from 1 through 8, 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 />
[[2016 AMC 12A Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
Circles with centers <math>P, Q</math> and <math>R</math>, having radii <math>1, 2</math> and <math>3</math>, respectively, lie on the same side of line <math>l</math> and are tangent to <math>l</math> at <math>P', Q'</math> and <math>R'</math>, respectively, with <math>Q'</math> between <math>P'</math> and <math>R'</math>. The circle with center <math>Q</math> is externally tangent to each of the other two circles. What is the area of triangle <math>PQR</math>?<br />
<br />
<math>\textbf{(A) } 0\qquad \textbf{(B) } \sqrt{\frac{2}{3}}\qquad\textbf{(C) } 1\qquad\textbf{(D) } \sqrt{6}-\sqrt{2}\qquad\textbf{(E) }\sqrt{\frac{3}{2}}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
The graphs of <math>y=\log_3 x, y=\log_x 3, y=\log_\frac{1}{3} x,</math> and <math>y=\log_x \dfrac{1}{3}</math> are plotted on the same set of axes. How many points in the plane with positive <math>x</math>-coordinates lie on two or more of the graphs? <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 />
[[2016 AMC 12A Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Let <math>ABCD</math> be a square. Let <math>E, F, G</math> and <math>H</math> be the centers, respectively, of equilateral triangles with bases <math>\overline{AB}, \overline{BC}, \overline{CD},</math> and <math>\overline{DA},</math> each exterior to the square. What is rge ration of the area of square <math>EFGH</math> to the area of square <math>ABCD</math>? <br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ \frac{2+\sqrt{3}}{3} \qquad\textbf{(C)}\ \sqrt{2} \qquad\textbf{(D)}\ \frac{\sqrt{2}+\sqrt{3}}{2} \qquad\textbf{(E)}\ \sqrt{3}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
For some positive integer <math>n,</math> the number <math>110n^3</math> has <math>110</math> positive integer divisors, including <math>1</math> and the number <math>110n^3.</math> How many positive integer divisors does the number <math>81n^4</math> have? <br />
<br />
<math>\textbf{(A)}\ 110\qquad\textbf{(B)}\ 191\qquad\textbf{(C)}\ 261\qquad\textbf{(D)}\ 325\qquad\textbf{(E)}\ 425</math><br />
<br />
[[2016 AMC 12A Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
Jerry starts at <math>0</math> on the real number line. He tosses a fair coin <math>8</math> times. When he gets heads, he moves <math>1</math> unit in the positive direction; when he gets tails, he moves <math>1</math> unit in the negative direction. The probability that he reaches <math>4</math> at some time during this process <math>\frac{a}{b},</math> where <math>a</math> and <math>b</math> are relatively prime positive integers. What is <math>a + b?</math> (For example, he succeeds if his sequence of tosses is <math>HTHHHHHH.</math>)<br />
<br />
<math>\textbf{(A)}\ 69\qquad\textbf{(B)}\ 151\qquad\textbf{(C)}\ 257\qquad\textbf{(D)}\ 293\qquad\textbf{(E)}\ 313</math><br />
<br />
[[2016 AMC 12A Problems/Problem 19|Solution]]<br />
<br />
<br />
==Problem 20==<br />
<br />
A binary operation <math>\ \diamondsuit</math> has the properties that <math>a\ \diamondsuit\ (b\ \diamondsuit\ c) = (a\ \diamondsuit\ b)\ \cdot\ c</math> and that <math>a\ \diamondsuit\ a = 1</math> for all nonzero real numbers <math>a, b</math> and <math>c.</math> (Here the dot <math>\ \cdot</math> represents the usual multiplication operation.) The solution to the equation <math>2016\ \diamondsuit\ (6\ \diamondsuit\ x) = 100</math> can be written as <math>\frac{p}{q},</math> where <math>p</math> and <math>q</math> are relativelt prime positive integers. What is <math>p + q?</math> <br />
<br />
<math>\textbf{(A)}\ 109\qquad\textbf{(B)}\ 201\qquad\textbf{(C)}\ 301\qquad\textbf{(D)}\ 3049\qquad\textbf{(E)}\ 33,601</math><br />
<br />
[[2016 AMC 12A Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 21|Solution]]<br />
<br />
<br />
==Problem 22==<br />
<br />
How many ordered triples <math>(x,y,z)</math> of positive integers satisfy <math>\text{lcm}(x,y) = 72, \text{lcm}(x,z) = 600</math> and <math>\text{lcm}(y,z)=900</math>?<br />
<br />
<math>\textbf{(A)}\ 15\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 64</math><br />
<br />
[[2016 AMC 12A Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 23|Solution]]<br />
<br />
<br />
==Problem 24==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 24|Solution]]<br />
<br />
<br />
==Problem 25==<br />
<br />
Let <math>k</math> be a positive integer. Bernardo and Silvia take turns writing and erasing numbers on a blackboard as follows: Bernardo starts by writing the smallest perfect square with <math>k+1</math> digits. Every time Bernardo writes a number, Silvia erases the last <math>k</math> digits of it. Bernardo then writes the next perfect square, Silvia erases the last <math>k</math> digits of it, and this process continues until the last two numbers that remain on the board differ by at least 2. Let <math>f(k)</math> be the smallest positive integer not written on the board. For example, if <math>k = 1</math>, then the numbers that Bernardo writes are <math>16, 25, 36, 49, 64</math>, and the numbers showing on the board after Silvia erases are <math>1, 2, 3, 4,</math> and <math>6</math>, and thus <math>f(1) = 5</math>. What is the sum of the digits of <math>f(2) + f(4)+ f(6) + ... + f(2016)</math>?<br />
<br />
<math>\textbf{(A)}\ 7986\qquad\textbf{(B)}\ 8002\qquad\textbf{(C)}\ 8030\qquad\textbf{(D)}\ 8048\qquad\textbf{(E)}\ 8064</math><br />
<br />
[[2016 AMC 12A Problems/Problem 25|Solution]]<br />
<br />
{{MAA Notice}}</div>Sssuperm1ch4elhttps://artofproblemsolving.com/wiki/index.php?title=2016_AMC_12A_Problems&diff=754072016 AMC 12A Problems2016-02-04T17:13:00Z<p>Sssuperm1ch4el: /* Problem 20 */</p>
<hr />
<div>{{AMC12 Problems|year=2016|ab=A}}<br />
<br />
==Problem 1==<br />
<br />
What is the value of <math>\frac{11!-10!}{9!}</math>?<br />
<br />
<math>\textbf{(A)}\ 99\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 110\qquad\textbf{(D)}\ 121\qquad\textbf{(E)}\ 132</math><br />
<br />
[[2016 AMC 12A Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
<br />
For what value of <math>x</math> does <math>10^x\cdot100^{2x}=1000^5</math>?<br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5</math><br />
<br />
[[2016 AMC 12A Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
The remainder can be defined for all real numbers <math>x</math> and <math>y</math> with <math>y \neq 0</math> by <cmath>\text{rem} (x ,y)=x-y\left \lfloor \frac{x}{y} \right \rfloor</cmath>where <math>\left \lfloor \tfrac{x}{y} \right \rfloor</math> denotes the greatest integer less than or equal to <math>\tfrac{x}{y}</math>. What is the value of <math>\text{rem} (\tfrac{3}{8}, -\tfrac{2}{5} )</math>?<br />
<br />
<math>\textbf{(A) } -\frac{3}{8} \qquad \textbf{(B) } -\frac{1}{40} \qquad \textbf{(C) } 0 \qquad \textbf{(D) } \frac{3}{8} \qquad \textbf{(E) } \frac{31}{40}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
The mean, median, and mode of the <math>7</math> data values <math>60, 100, x, 40, 50, 200, 90</math> are all equal to <math>x</math>. What is the value of <math>x</math>?<br />
<br />
<math>\textbf{(A)}\ 50\qquad\textbf{(B)}\ 60\qquad\textbf{(C)}\ 75\qquad\textbf{(D)}\ 90\qquad\textbf{(E)}\ 100</math><br />
<br />
[[2016 AMC 12A Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
Goldbach's conjecture states that every even integer greater than 2 can be written as the sum of two prime numbers (for example, <math>2016=13+2003</math>). So far, no one has been able to prove that the conjecture is true, and no one has found a counterexample to show that the conjecture is false. What would a counterexample consist of?<br />
<br />
<math> \textbf{(A)}\ \text{an odd integer greater than } 2 \text{ that can be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(B)}\ \text{an odd integer greater than } 2 \text{ that cannot be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(C)}\ \text{an even integer greater than } 2 \text{ that can be written as the sum of two numbers that are not prime}\\<br />
\qquad\textbf{(D)}\ \text{an even integer greater than } 2 \text{ that can be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(E)}\ \text{an even integer greater than } 2 \text{ that cannot be written as the sum of two prime numbers}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
A triangular array of <math>2016</math> coins has <math>1</math> coin in the first row, <math>2</math> coins in the second row, <math>3</math> coins in the third row, and so on up to <math>N</math> coins in the <math>N</math>th row. What is the sum of the digits of <math>N</math> ?<br />
<br />
<math>\textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 10</math><br />
<br />
[[2016 AMC 12A Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
Which of these describes the graph of <math>x^2(x+y+1)=y^2(x+y+1)</math> ?<br />
<br />
<math> \textbf{(A)}\ \text{two parallel lines}\\<br />
\qquad\textbf{(B)}\ \text{two intersecting lines}\\<br />
\qquad\textbf{(C)}\ \text{three lines that all pass through a common point}\\<br />
\qquad\textbf{(D)}\ \text{three lines that do not all pass through a comment point}\\<br />
\qquad\textbf{(E)}\ \text{a line and a parabola}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
What is the area of the shaded reigon of the given <math>8\times 5</math> rectangle?<br />
<br />
<asy><br />
<br />
size(6cm);<br />
defaultpen(fontsize(9pt));<br />
draw((0,0)--(8,0)--(8,5)--(0,5)--cycle);<br />
filldraw((7,0)--(8,0)--(8,1)--(0,4)--(0,5)--(1,5)--cycle,gray(0.8));<br />
<br />
label("$1$",(1/2,5),dir(90));<br />
label("$7$",(9/2,5),dir(90));<br />
<br />
label("$1$",(8,1/2),dir(0));<br />
label("$4$",(8,3),dir(0));<br />
<br />
label("$1$",(15/2,0),dir(270));<br />
label("$7$",(7/2,0),dir(270));<br />
<br />
label("$1$",(0,9/2),dir(180));<br />
label("$4$",(0,2),dir(180));<br />
<br />
</asy><br />
<br />
<math>\textbf{(A)}\ 4\dfrac{3}{4}\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 5\dfrac{1}{4}\qquad\textbf{(D)}\ 6\dfrac{1}{2}\qquad\textbf{(E)}\ 8</math><br />
<br />
[[2016 AMC 12A Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
The five small shaded squares inside this unit square are congruent and have disjoint interiors. The midpoint of each side of the middle square coincides with one of the vertices of the other four small squares as shown. The common side length is <math>\tfrac{a-\sqrt{2}}{b}</math>, where <math>a</math> and <math>b</math> are positive integers. What is <math>a+b</math> ?<br />
<br />
<pre style="color: gray">TODO: Diagram</pre><br />
<br />
<math>\textbf{(A)}\ 7\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 11</math><br />
<br />
[[2016 AMC 12A Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
Five friends sat in a movie theater in a row containing <math>5</math> seats, numbered <math>1</math> to <math>5</math> from left to right. (The directions "left" and "right" are from the point of view of the people as they sit in the seats.) During the movie Ada went to the lobby to get some popcorn. When she returned, she found that Bea had moved two seats to the right, Ceci had moved one seat to the left, and Dee and Edie had switched seats, leaving an end seat for Ada. In which seat had Ada been sitting before she got up?<br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5</math><br />
<br />
[[2016 AMC 12A Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
Each of the 100 students in a certain summer camp can either sing, dance, or act. Some students have more than one talent, but no student has all three talents. There are 42 students who cannot sing, 65 students who cannot dance, and 29 students who cannot act. How many students have two of these talents?<br />
<br />
<math>\textbf{(A)}\ 16\qquad\textbf{(B)}\ 25\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 49\qquad\textbf{(E)}\ 64</math><br />
<br />
[[2016 AMC 12A Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
In <math>\triangle ABC</math>, <math>AB = 6</math>, <math>BC = 7</math>, and <math>CA = 8</math>. Point <math>D</math> lies on <math>\overline{BC}</math>, and <math>\overline{AD}</math> bisects <math>\angle BAC</math>. Point <math>E</math> lies on <math>\overline{AC}</math>, and <math>\overline{BE}</math> bisects <math>\angle ABC</math>. The bisectors intersect at <math>F</math>. What is the ratio <math>AF</math> : <math>FD</math>?<br />
<br />
<pre style="color: gray">TODO: Diagram</pre><br />
<br />
<math>\textbf{(A)}\ 3:2\qquad\textbf{(B)}\ 5:3\qquad\textbf{(C)}\ 2:1\qquad\textbf{(D)}\ 7:3\qquad\textbf{(E)}\ 5:2</math><br />
<br />
[[2016 AMC 12A Problems/Problem 12|Solution]]<br />
<br />
<br />
==Problem 13==<br />
<br />
Let <math>N</math> be a positive multiple of <math>5</math>. One red ball and <math>N</math> green balls are arranged in a line in random order. Let <math>P(N)</math> be the probability that at least <math>\tfrac{3}{5}</math> of the green balls are on the same side of the red ball. Observe that <math>P(5)=1</math> and that <math>P(N)</math> approaches <math>\tfrac{4}{5}</math> as <math>N</math> grows large. What is the sum of the digits of the least value of <math>N</math> such that <math>P(N) < \tfrac{321}{400}</math>?<br />
<br />
<math>\textbf{(A)}\ 12\qquad\textbf{(B)}\ 14\qquad\textbf{(C)}\ 16\qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 20</math><br />
<br />
[[2016 AMC 12A Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
Each vertex of a cube is to be labeled with an integer from 1 through 8, 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 />
[[2016 AMC 12A Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
Circles with centers <math>P, Q</math> and <math>R</math>, having radii <math>1, 2</math> and <math>3</math>, respectively, lie on the same side of line <math>l</math> and are tangent to <math>l</math> at <math>P', Q'</math> and <math>R'</math>, respectively, with <math>Q'</math> between <math>P'</math> and <math>R'</math>. The circle with center <math>Q</math> is externally tangent to each of the other two circles. What is the area of triangle <math>PQR</math>?<br />
<br />
<math>\textbf{(A) } 0\qquad \textbf{(B) } \sqrt{\frac{2}{3}}\qquad\textbf{(C) } 1\qquad\textbf{(D) } \sqrt{6}-\sqrt{2}\qquad\textbf{(E) }\sqrt{\frac{3}{2}}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
The graphs of <math>y=\log_3 x, y=\log_x 3, y=\log_\frac{1}{3} x,</math> and <math>y=\log_x \dfrac{1}{3}</math> are plotted on the same set of axes. How many points in the plane with positive <math>x</math>-coordinates lie on two or more of the graphs? <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 />
[[2016 AMC 12A Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Let <math>ABCD</math> be a square. Let <math>E, F, G</math> and <math>H</math> be the centers, respectively, of equilateral triangles with bases <math>\overline{AB}, \overline{BC}, \overline{CD},</math> and <math>\overline{DA},</math> each exterior to the square. What is rge ration of the area of square <math>EFGH</math> to the area of square <math>ABCD</math>? <br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ \frac{2+\sqrt{3}}{3} \qquad\textbf{(C)}\ \sqrt{2} \qquad\textbf{(D)}\ \frac{\sqrt{2}+\sqrt{3}}{2} \qquad\textbf{(E)}\ \sqrt{3}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
For some positive integer <math>n,</math> the number <math>110n^3</math> has <math>110</math> positive integer divisors, including <math>1</math> and the number <math>110n^3.</math> How many positive integer divisors does the number <math>81n^4</math> have? <br />
<br />
<math>\textbf{(A)}\ 110\qquad\textbf{(B)}\ 191\qquad\textbf{(C)}\ 261\qquad\textbf{(D)}\ 325\qquad\textbf{(E)}\ 425</math><br />
<br />
[[2016 AMC 12A Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
Jerry starts at <math>0</math> on the real number line. He tosses a fair coin <math>8</math> times. When he gets heads, he moves <math>1</math> unit in the positive direction; when he gets tails, he moves <math>1</math> unit in the negative direction. The probability that he reaches <math>4</math> at some time during this process <math>\frac{a}{b},</math> where <math>a</math> and <math>b</math> are relatively prime positive integers. What is <math>a + b?</math> (For example, he succeeds if his sequence of tosses is <math>HTHHHHHH.</math>)<br />
<br />
<math>\textbf{(A)}\ 69\qquad\textbf{(B)}\ 151\qquad\textbf{(C)}\ 257\qquad\textbf{(D)}\ 293\qquad\textbf{(E)}\ 313</math><br />
<br />
[[2016 AMC 12A Problems/Problem 19|Solution]]<br />
<br />
<br />
==Problem 20==<br />
<br />
A binary operation <math>\ \diamondsuit\ \$ has the properties that </math>a\ \diamondsuit\ (b\ \diamondsuit\ c) = (a\ \diamondsuit\ b)\ \cdot\ c<math> and that </math>a\ \diamondsuit\ a = 1<math> for all nonzero real numbers </math>a, b<math> and </math>c.<math> (Here the dot </math>\ \cdot<math> represents the usual multiplication operation.) The solution to the equation </math>2016\ \diamondsuit\ (6\ \diamondsuit\ x) = 100<math> can be written as </math>\frac{p}{q},<math> where </math>p<math> and </math>q<math> are relativelt prime positive integers. What is </math>p + q?<math> <br />
<br />
</math>\textbf{(A)}\ 109\qquad\textbf{(B)}\ 201\qquad\textbf{(C)}\ 301\qquad\textbf{(D)}\ 3049\qquad\textbf{(E)}\ 33,601$<br />
<br />
[[2016 AMC 12A Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 21|Solution]]<br />
<br />
<br />
==Problem 22==<br />
<br />
How many ordered triples <math>(x,y,z)</math> of positive integers satisfy <math>\text{lcm}(x,y) = 72, \text{lcm}(x,z) = 600</math> and <math>\text{lcm}(y,z)=900</math>?<br />
<br />
<math>\textbf{(A)}\ 15\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 64</math><br />
<br />
[[2016 AMC 12A Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 23|Solution]]<br />
<br />
<br />
==Problem 24==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 24|Solution]]<br />
<br />
<br />
==Problem 25==<br />
<br />
Let <math>k</math> be a positive integer. Bernardo and Silvia take turns writing and erasing numbers on a blackboard as follows: Bernardo starts by writing the smallest perfect square with <math>k+1</math> digits. Every time Bernardo writes a number, Silvia erases the last <math>k</math> digits of it. Bernardo then writes the next perfect square, Silvia erases the last <math>k</math> digits of it, and this process continues until the last two numbers that remain on the board differ by at least 2. Let <math>f(k)</math> be the smallest positive integer not written on the board. For example, if <math>k = 1</math>, then the numbers that Bernardo writes are <math>16, 25, 36, 49, 64</math>, and the numbers showing on the board after Silvia erases are <math>1, 2, 3, 4,</math> and <math>6</math>, and thus <math>f(1) = 5</math>. What is the sum of the digits of <math>f(2) + f(4)+ f(6) + ... + f(2016)</math>?<br />
<br />
<math>\textbf{(A)}\ 7986\qquad\textbf{(B)}\ 8002\qquad\textbf{(C)}\ 8030\qquad\textbf{(D)}\ 8048\qquad\textbf{(E)}\ 8064</math><br />
<br />
[[2016 AMC 12A Problems/Problem 25|Solution]]<br />
<br />
{{MAA Notice}}</div>Sssuperm1ch4elhttps://artofproblemsolving.com/wiki/index.php?title=2016_AMC_12A_Problems&diff=754052016 AMC 12A Problems2016-02-04T17:12:08Z<p>Sssuperm1ch4el: /* Problem 20 */</p>
<hr />
<div>{{AMC12 Problems|year=2016|ab=A}}<br />
<br />
==Problem 1==<br />
<br />
What is the value of <math>\frac{11!-10!}{9!}</math>?<br />
<br />
<math>\textbf{(A)}\ 99\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 110\qquad\textbf{(D)}\ 121\qquad\textbf{(E)}\ 132</math><br />
<br />
[[2016 AMC 12A Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
<br />
For what value of <math>x</math> does <math>10^x\cdot100^{2x}=1000^5</math>?<br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5</math><br />
<br />
[[2016 AMC 12A Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
The remainder can be defined for all real numbers <math>x</math> and <math>y</math> with <math>y \neq 0</math> by <cmath>\text{rem} (x ,y)=x-y\left \lfloor \frac{x}{y} \right \rfloor</cmath>where <math>\left \lfloor \tfrac{x}{y} \right \rfloor</math> denotes the greatest integer less than or equal to <math>\tfrac{x}{y}</math>. What is the value of <math>\text{rem} (\tfrac{3}{8}, -\tfrac{2}{5} )</math>?<br />
<br />
<math>\textbf{(A) } -\frac{3}{8} \qquad \textbf{(B) } -\frac{1}{40} \qquad \textbf{(C) } 0 \qquad \textbf{(D) } \frac{3}{8} \qquad \textbf{(E) } \frac{31}{40}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
The mean, median, and mode of the <math>7</math> data values <math>60, 100, x, 40, 50, 200, 90</math> are all equal to <math>x</math>. What is the value of <math>x</math>?<br />
<br />
<math>\textbf{(A)}\ 50\qquad\textbf{(B)}\ 60\qquad\textbf{(C)}\ 75\qquad\textbf{(D)}\ 90\qquad\textbf{(E)}\ 100</math><br />
<br />
[[2016 AMC 12A Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
Goldbach's conjecture states that every even integer greater than 2 can be written as the sum of two prime numbers (for example, <math>2016=13+2003</math>). So far, no one has been able to prove that the conjecture is true, and no one has found a counterexample to show that the conjecture is false. What would a counterexample consist of?<br />
<br />
<math> \textbf{(A)}\ \text{an odd integer greater than } 2 \text{ that can be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(B)}\ \text{an odd integer greater than } 2 \text{ that cannot be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(C)}\ \text{an even integer greater than } 2 \text{ that can be written as the sum of two numbers that are not prime}\\<br />
\qquad\textbf{(D)}\ \text{an even integer greater than } 2 \text{ that can be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(E)}\ \text{an even integer greater than } 2 \text{ that cannot be written as the sum of two prime numbers}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
A triangular array of <math>2016</math> coins has <math>1</math> coin in the first row, <math>2</math> coins in the second row, <math>3</math> coins in the third row, and so on up to <math>N</math> coins in the <math>N</math>th row. What is the sum of the digits of <math>N</math> ?<br />
<br />
<math>\textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 10</math><br />
<br />
[[2016 AMC 12A Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
Which of these describes the graph of <math>x^2(x+y+1)=y^2(x+y+1)</math> ?<br />
<br />
<math> \textbf{(A)}\ \text{two parallel lines}\\<br />
\qquad\textbf{(B)}\ \text{two intersecting lines}\\<br />
\qquad\textbf{(C)}\ \text{three lines that all pass through a common point}\\<br />
\qquad\textbf{(D)}\ \text{three lines that do not all pass through a comment point}\\<br />
\qquad\textbf{(E)}\ \text{a line and a parabola}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
What is the area of the shaded reigon of the given <math>8\times 5</math> rectangle?<br />
<br />
<asy><br />
<br />
size(6cm);<br />
defaultpen(fontsize(9pt));<br />
draw((0,0)--(8,0)--(8,5)--(0,5)--cycle);<br />
filldraw((7,0)--(8,0)--(8,1)--(0,4)--(0,5)--(1,5)--cycle,gray(0.8));<br />
<br />
label("$1$",(1/2,5),dir(90));<br />
label("$7$",(9/2,5),dir(90));<br />
<br />
label("$1$",(8,1/2),dir(0));<br />
label("$4$",(8,3),dir(0));<br />
<br />
label("$1$",(15/2,0),dir(270));<br />
label("$7$",(7/2,0),dir(270));<br />
<br />
label("$1$",(0,9/2),dir(180));<br />
label("$4$",(0,2),dir(180));<br />
<br />
</asy><br />
<br />
<math>\textbf{(A)}\ 4\dfrac{3}{4}\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 5\dfrac{1}{4}\qquad\textbf{(D)}\ 6\dfrac{1}{2}\qquad\textbf{(E)}\ 8</math><br />
<br />
[[2016 AMC 12A Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
The five small shaded squares inside this unit square are congruent and have disjoint interiors. The midpoint of each side of the middle square coincides with one of the vertices of the other four small squares as shown. The common side length is <math>\tfrac{a-\sqrt{2}}{b}</math>, where <math>a</math> and <math>b</math> are positive integers. What is <math>a+b</math> ?<br />
<br />
<pre style="color: gray">TODO: Diagram</pre><br />
<br />
<math>\textbf{(A)}\ 7\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 11</math><br />
<br />
[[2016 AMC 12A Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
Five friends sat in a movie theater in a row containing <math>5</math> seats, numbered <math>1</math> to <math>5</math> from left to right. (The directions "left" and "right" are from the point of view of the people as they sit in the seats.) During the movie Ada went to the lobby to get some popcorn. When she returned, she found that Bea had moved two seats to the right, Ceci had moved one seat to the left, and Dee and Edie had switched seats, leaving an end seat for Ada. In which seat had Ada been sitting before she got up?<br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5</math><br />
<br />
[[2016 AMC 12A Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
Each of the 100 students in a certain summer camp can either sing, dance, or act. Some students have more than one talent, but no student has all three talents. There are 42 students who cannot sing, 65 students who cannot dance, and 29 students who cannot act. How many students have two of these talents?<br />
<br />
<math>\textbf{(A)}\ 16\qquad\textbf{(B)}\ 25\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 49\qquad\textbf{(E)}\ 64</math><br />
<br />
[[2016 AMC 12A Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
In <math>\triangle ABC</math>, <math>AB = 6</math>, <math>BC = 7</math>, and <math>CA = 8</math>. Point <math>D</math> lies on <math>\overline{BC}</math>, and <math>\overline{AD}</math> bisects <math>\angle BAC</math>. Point <math>E</math> lies on <math>\overline{AC}</math>, and <math>\overline{BE}</math> bisects <math>\angle ABC</math>. The bisectors intersect at <math>F</math>. What is the ratio <math>AF</math> : <math>FD</math>?<br />
<br />
<pre style="color: gray">TODO: Diagram</pre><br />
<br />
<math>\textbf{(A)}\ 3:2\qquad\textbf{(B)}\ 5:3\qquad\textbf{(C)}\ 2:1\qquad\textbf{(D)}\ 7:3\qquad\textbf{(E)}\ 5:2</math><br />
<br />
[[2016 AMC 12A Problems/Problem 12|Solution]]<br />
<br />
<br />
==Problem 13==<br />
<br />
Let <math>N</math> be a positive multiple of <math>5</math>. One red ball and <math>N</math> green balls are arranged in a line in random order. Let <math>P(N)</math> be the probability that at least <math>\tfrac{3}{5}</math> of the green balls are on the same side of the red ball. Observe that <math>P(5)=1</math> and that <math>P(N)</math> approaches <math>\tfrac{4}{5}</math> as <math>N</math> grows large. What is the sum of the digits of the least value of <math>N</math> such that <math>P(N) < \tfrac{321}{400}</math>?<br />
<br />
<math>\textbf{(A)}\ 12\qquad\textbf{(B)}\ 14\qquad\textbf{(C)}\ 16\qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 20</math><br />
<br />
[[2016 AMC 12A Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
Each vertex of a cube is to be labeled with an integer from 1 through 8, 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 />
[[2016 AMC 12A Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
Circles with centers <math>P, Q</math> and <math>R</math>, having radii <math>1, 2</math> and <math>3</math>, respectively, lie on the same side of line <math>l</math> and are tangent to <math>l</math> at <math>P', Q'</math> and <math>R'</math>, respectively, with <math>Q'</math> between <math>P'</math> and <math>R'</math>. The circle with center <math>Q</math> is externally tangent to each of the other two circles. What is the area of triangle <math>PQR</math>?<br />
<br />
<math>\textbf{(A) } 0\qquad \textbf{(B) } \sqrt{\frac{2}{3}}\qquad\textbf{(C) } 1\qquad\textbf{(D) } \sqrt{6}-\sqrt{2}\qquad\textbf{(E) }\sqrt{\frac{3}{2}}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
The graphs of <math>y=\log_3 x, y=\log_x 3, y=\log_\frac{1}{3} x,</math> and <math>y=\log_x \dfrac{1}{3}</math> are plotted on the same set of axes. How many points in the plane with positive <math>x</math>-coordinates lie on two or more of the graphs? <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 />
[[2016 AMC 12A Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Let <math>ABCD</math> be a square. Let <math>E, F, G</math> and <math>H</math> be the centers, respectively, of equilateral triangles with bases <math>\overline{AB}, \overline{BC}, \overline{CD},</math> and <math>\overline{DA},</math> each exterior to the square. What is rge ration of the area of square <math>EFGH</math> to the area of square <math>ABCD</math>? <br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ \frac{2+\sqrt{3}}{3} \qquad\textbf{(C)}\ \sqrt{2} \qquad\textbf{(D)}\ \frac{\sqrt{2}+\sqrt{3}}{2} \qquad\textbf{(E)}\ \sqrt{3}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
For some positive integer <math>n,</math> the number <math>110n^3</math> has <math>110</math> positive integer divisors, including <math>1</math> and the number <math>110n^3.</math> How many positive integer divisors does the number <math>81n^4</math> have? <br />
<br />
<math>\textbf{(A)}\ 110\qquad\textbf{(B)}\ 191\qquad\textbf{(C)}\ 261\qquad\textbf{(D)}\ 325\qquad\textbf{(E)}\ 425</math><br />
<br />
[[2016 AMC 12A Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
Jerry starts at <math>0</math> on the real number line. He tosses a fair coin <math>8</math> times. When he gets heads, he moves <math>1</math> unit in the positive direction; when he gets tails, he moves <math>1</math> unit in the negative direction. The probability that he reaches <math>4</math> at some time during this process <math>\frac{a}{b},</math> where <math>a</math> and <math>b</math> are relatively prime positive integers. What is <math>a + b?</math> (For example, he succeeds if his sequence of tosses is <math>HTHHHHHH.</math>)<br />
<br />
<math>\textbf{(A)}\ 69\qquad\textbf{(B)}\ 151\qquad\textbf{(C)}\ 257\qquad\textbf{(D)}\ 293\qquad\textbf{(E)}\ 313</math><br />
<br />
[[2016 AMC 12A Problems/Problem 19|Solution]]<br />
<br />
<br />
==Problem 20==<br />
<br />
A binary operation <math>\ \diamondsuit</math> has the properties that <math>a\ \diamondsuit\ (b\ \diamondsuit\ c) = (a\ \diamondsuit\ b)\ \cdot\ c</math> and that <math>a\ \diamondsuit\ a = 1</math> for all nonzero real numbers <math>a, b</math> and <math>c.</math> (Here the dot <math>\ \cdot</math> represents the usual multiplication operation.) The solution to the equation <math>2016\ \diamondsuit\ (6\ \diamondsuit\ x) = 100</math> can be written as <math>\frac{p}{q},</math> where <math>p</math> and <math>q</math> are relativelt prime positive integers. What is <math>p + q?</math> <br />
<br />
<math>\textbf{(A)}\ 109\qquad\textbf{(B)}\ 201\qquad\textbf{(C)}\ 301\qquad\textbf{(D)}\ 3049\qquad\textbf{(E)}\ 33,601</math><br />
<br />
[[2016 AMC 12A Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 21|Solution]]<br />
<br />
<br />
==Problem 22==<br />
<br />
How many ordered triples <math>(x,y,z)</math> of positive integers satisfy <math>\text{lcm}(x,y) = 72, \text{lcm}(x,z) = 600</math> and <math>\text{lcm}(y,z)=900</math>?<br />
<br />
<math>\textbf{(A)}\ 15\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 64</math><br />
<br />
[[2016 AMC 12A Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 23|Solution]]<br />
<br />
<br />
==Problem 24==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 24|Solution]]<br />
<br />
<br />
==Problem 25==<br />
<br />
Let <math>k</math> be a positive integer. Bernardo and Silvia take turns writing and erasing numbers on a blackboard as follows: Bernardo starts by writing the smallest perfect square with <math>k+1</math> digits. Every time Bernardo writes a number, Silvia erases the last <math>k</math> digits of it. Bernardo then writes the next perfect square, Silvia erases the last <math>k</math> digits of it, and this process continues until the last two numbers that remain on the board differ by at least 2. Let <math>f(k)</math> be the smallest positive integer not written on the board. For example, if <math>k = 1</math>, then the numbers that Bernardo writes are <math>16, 25, 36, 49, 64</math>, and the numbers showing on the board after Silvia erases are <math>1, 2, 3, 4,</math> and <math>6</math>, and thus <math>f(1) = 5</math>. What is the sum of the digits of <math>f(2) + f(4)+ f(6) + ... + f(2016)</math>?<br />
<br />
<math>\textbf{(A)}\ 7986\qquad\textbf{(B)}\ 8002\qquad\textbf{(C)}\ 8030\qquad\textbf{(D)}\ 8048\qquad\textbf{(E)}\ 8064</math><br />
<br />
[[2016 AMC 12A Problems/Problem 25|Solution]]<br />
<br />
{{MAA Notice}}</div>Sssuperm1ch4elhttps://artofproblemsolving.com/wiki/index.php?title=2016_AMC_12A_Problems&diff=754042016 AMC 12A Problems2016-02-04T17:11:20Z<p>Sssuperm1ch4el: /* Problem 20 */</p>
<hr />
<div>{{AMC12 Problems|year=2016|ab=A}}<br />
<br />
==Problem 1==<br />
<br />
What is the value of <math>\frac{11!-10!}{9!}</math>?<br />
<br />
<math>\textbf{(A)}\ 99\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 110\qquad\textbf{(D)}\ 121\qquad\textbf{(E)}\ 132</math><br />
<br />
[[2016 AMC 12A Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
<br />
For what value of <math>x</math> does <math>10^x\cdot100^{2x}=1000^5</math>?<br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5</math><br />
<br />
[[2016 AMC 12A Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
The remainder can be defined for all real numbers <math>x</math> and <math>y</math> with <math>y \neq 0</math> by <cmath>\text{rem} (x ,y)=x-y\left \lfloor \frac{x}{y} \right \rfloor</cmath>where <math>\left \lfloor \tfrac{x}{y} \right \rfloor</math> denotes the greatest integer less than or equal to <math>\tfrac{x}{y}</math>. What is the value of <math>\text{rem} (\tfrac{3}{8}, -\tfrac{2}{5} )</math>?<br />
<br />
<math>\textbf{(A) } -\frac{3}{8} \qquad \textbf{(B) } -\frac{1}{40} \qquad \textbf{(C) } 0 \qquad \textbf{(D) } \frac{3}{8} \qquad \textbf{(E) } \frac{31}{40}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
The mean, median, and mode of the <math>7</math> data values <math>60, 100, x, 40, 50, 200, 90</math> are all equal to <math>x</math>. What is the value of <math>x</math>?<br />
<br />
<math>\textbf{(A)}\ 50\qquad\textbf{(B)}\ 60\qquad\textbf{(C)}\ 75\qquad\textbf{(D)}\ 90\qquad\textbf{(E)}\ 100</math><br />
<br />
[[2016 AMC 12A Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
Goldbach's conjecture states that every even integer greater than 2 can be written as the sum of two prime numbers (for example, <math>2016=13+2003</math>). So far, no one has been able to prove that the conjecture is true, and no one has found a counterexample to show that the conjecture is false. What would a counterexample consist of?<br />
<br />
<math> \textbf{(A)}\ \text{an odd integer greater than } 2 \text{ that can be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(B)}\ \text{an odd integer greater than } 2 \text{ that cannot be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(C)}\ \text{an even integer greater than } 2 \text{ that can be written as the sum of two numbers that are not prime}\\<br />
\qquad\textbf{(D)}\ \text{an even integer greater than } 2 \text{ that can be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(E)}\ \text{an even integer greater than } 2 \text{ that cannot be written as the sum of two prime numbers}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
A triangular array of <math>2016</math> coins has <math>1</math> coin in the first row, <math>2</math> coins in the second row, <math>3</math> coins in the third row, and so on up to <math>N</math> coins in the <math>N</math>th row. What is the sum of the digits of <math>N</math> ?<br />
<br />
<math>\textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 10</math><br />
<br />
[[2016 AMC 12A Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
Which of these describes the graph of <math>x^2(x+y+1)=y^2(x+y+1)</math> ?<br />
<br />
<math> \textbf{(A)}\ \text{two parallel lines}\\<br />
\qquad\textbf{(B)}\ \text{two intersecting lines}\\<br />
\qquad\textbf{(C)}\ \text{three lines that all pass through a common point}\\<br />
\qquad\textbf{(D)}\ \text{three lines that do not all pass through a comment point}\\<br />
\qquad\textbf{(E)}\ \text{a line and a parabola}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
What is the area of the shaded reigon of the given <math>8\times 5</math> rectangle?<br />
<br />
<asy><br />
<br />
size(6cm);<br />
defaultpen(fontsize(9pt));<br />
draw((0,0)--(8,0)--(8,5)--(0,5)--cycle);<br />
filldraw((7,0)--(8,0)--(8,1)--(0,4)--(0,5)--(1,5)--cycle,gray(0.8));<br />
<br />
label("$1$",(1/2,5),dir(90));<br />
label("$7$",(9/2,5),dir(90));<br />
<br />
label("$1$",(8,1/2),dir(0));<br />
label("$4$",(8,3),dir(0));<br />
<br />
label("$1$",(15/2,0),dir(270));<br />
label("$7$",(7/2,0),dir(270));<br />
<br />
label("$1$",(0,9/2),dir(180));<br />
label("$4$",(0,2),dir(180));<br />
<br />
</asy><br />
<br />
<math>\textbf{(A)}\ 4\dfrac{3}{4}\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 5\dfrac{1}{4}\qquad\textbf{(D)}\ 6\dfrac{1}{2}\qquad\textbf{(E)}\ 8</math><br />
<br />
[[2016 AMC 12A Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
The five small shaded squares inside this unit square are congruent and have disjoint interiors. The midpoint of each side of the middle square coincides with one of the vertices of the other four small squares as shown. The common side length is <math>\tfrac{a-\sqrt{2}}{b}</math>, where <math>a</math> and <math>b</math> are positive integers. What is <math>a+b</math> ?<br />
<br />
<pre style="color: gray">TODO: Diagram</pre><br />
<br />
<math>\textbf{(A)}\ 7\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 11</math><br />
<br />
[[2016 AMC 12A Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
Five friends sat in a movie theater in a row containing <math>5</math> seats, numbered <math>1</math> to <math>5</math> from left to right. (The directions "left" and "right" are from the point of view of the people as they sit in the seats.) During the movie Ada went to the lobby to get some popcorn. When she returned, she found that Bea had moved two seats to the right, Ceci had moved one seat to the left, and Dee and Edie had switched seats, leaving an end seat for Ada. In which seat had Ada been sitting before she got up?<br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5</math><br />
<br />
[[2016 AMC 12A Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
Each of the 100 students in a certain summer camp can either sing, dance, or act. Some students have more than one talent, but no student has all three talents. There are 42 students who cannot sing, 65 students who cannot dance, and 29 students who cannot act. How many students have two of these talents?<br />
<br />
<math>\textbf{(A)}\ 16\qquad\textbf{(B)}\ 25\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 49\qquad\textbf{(E)}\ 64</math><br />
<br />
[[2016 AMC 12A Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
In <math>\triangle ABC</math>, <math>AB = 6</math>, <math>BC = 7</math>, and <math>CA = 8</math>. Point <math>D</math> lies on <math>\overline{BC}</math>, and <math>\overline{AD}</math> bisects <math>\angle BAC</math>. Point <math>E</math> lies on <math>\overline{AC}</math>, and <math>\overline{BE}</math> bisects <math>\angle ABC</math>. The bisectors intersect at <math>F</math>. What is the ratio <math>AF</math> : <math>FD</math>?<br />
<br />
<pre style="color: gray">TODO: Diagram</pre><br />
<br />
<math>\textbf{(A)}\ 3:2\qquad\textbf{(B)}\ 5:3\qquad\textbf{(C)}\ 2:1\qquad\textbf{(D)}\ 7:3\qquad\textbf{(E)}\ 5:2</math><br />
<br />
[[2016 AMC 12A Problems/Problem 12|Solution]]<br />
<br />
<br />
==Problem 13==<br />
<br />
Let <math>N</math> be a positive multiple of <math>5</math>. One red ball and <math>N</math> green balls are arranged in a line in random order. Let <math>P(N)</math> be the probability that at least <math>\tfrac{3}{5}</math> of the green balls are on the same side of the red ball. Observe that <math>P(5)=1</math> and that <math>P(N)</math> approaches <math>\tfrac{4}{5}</math> as <math>N</math> grows large. What is the sum of the digits of the least value of <math>N</math> such that <math>P(N) < \tfrac{321}{400}</math>?<br />
<br />
<math>\textbf{(A)}\ 12\qquad\textbf{(B)}\ 14\qquad\textbf{(C)}\ 16\qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 20</math><br />
<br />
[[2016 AMC 12A Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
Each vertex of a cube is to be labeled with an integer from 1 through 8, 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 />
[[2016 AMC 12A Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
Circles with centers <math>P, Q</math> and <math>R</math>, having radii <math>1, 2</math> and <math>3</math>, respectively, lie on the same side of line <math>l</math> and are tangent to <math>l</math> at <math>P', Q'</math> and <math>R'</math>, respectively, with <math>Q'</math> between <math>P'</math> and <math>R'</math>. The circle with center <math>Q</math> is externally tangent to each of the other two circles. What is the area of triangle <math>PQR</math>?<br />
<br />
<math>\textbf{(A) } 0\qquad \textbf{(B) } \sqrt{\frac{2}{3}}\qquad\textbf{(C) } 1\qquad\textbf{(D) } \sqrt{6}-\sqrt{2}\qquad\textbf{(E) }\sqrt{\frac{3}{2}}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
The graphs of <math>y=\log_3 x, y=\log_x 3, y=\log_\frac{1}{3} x,</math> and <math>y=\log_x \dfrac{1}{3}</math> are plotted on the same set of axes. How many points in the plane with positive <math>x</math>-coordinates lie on two or more of the graphs? <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 />
[[2016 AMC 12A Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Let <math>ABCD</math> be a square. Let <math>E, F, G</math> and <math>H</math> be the centers, respectively, of equilateral triangles with bases <math>\overline{AB}, \overline{BC}, \overline{CD},</math> and <math>\overline{DA},</math> each exterior to the square. What is rge ration of the area of square <math>EFGH</math> to the area of square <math>ABCD</math>? <br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ \frac{2+\sqrt{3}}{3} \qquad\textbf{(C)}\ \sqrt{2} \qquad\textbf{(D)}\ \frac{\sqrt{2}+\sqrt{3}}{2} \qquad\textbf{(E)}\ \sqrt{3}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
For some positive integer <math>n,</math> the number <math>110n^3</math> has <math>110</math> positive integer divisors, including <math>1</math> and the number <math>110n^3.</math> How many positive integer divisors does the number <math>81n^4</math> have? <br />
<br />
<math>\textbf{(A)}\ 110\qquad\textbf{(B)}\ 191\qquad\textbf{(C)}\ 261\qquad\textbf{(D)}\ 325\qquad\textbf{(E)}\ 425</math><br />
<br />
[[2016 AMC 12A Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
Jerry starts at <math>0</math> on the real number line. He tosses a fair coin <math>8</math> times. When he gets heads, he moves <math>1</math> unit in the positive direction; when he gets tails, he moves <math>1</math> unit in the negative direction. The probability that he reaches <math>4</math> at some time during this process <math>\frac{a}{b},</math> where <math>a</math> and <math>b</math> are relatively prime positive integers. What is <math>a + b?</math> (For example, he succeeds if his sequence of tosses is <math>HTHHHHHH.</math>)<br />
<br />
<math>\textbf{(A)}\ 69\qquad\textbf{(B)}\ 151\qquad\textbf{(C)}\ 257\qquad\textbf{(D)}\ 293\qquad\textbf{(E)}\ 313</math><br />
<br />
[[2016 AMC 12A Problems/Problem 19|Solution]]<br />
<br />
<br />
==Problem 20==<br />
<br />
A binary operation <math>\ \diamondsuit\ </math> has the properties that <math>a\ \diamondsuit\ (b\ \diamondsuit\ c) = (a\ \diamondsuit\ b)\ \cdot\ c</math> and that <math>a\ \diamondsuit\ a = 1</math> for all nonzero real numbers <math>a, b</math> and <math>c.</math> (Here the dot <math>\ \cdot\ </math> represents the usual multiplication operation.) The solution to the equation <math>2016\ \diamondsuit\ (6\ \diamondsuit\ x) = 100</math> can be written as <math>\frac{p}{q},</math> where <math>p</math> and <math>q</math> are relativelt prime positive integers. What is <math>p + q?</math> <br />
<br />
<math>\textbf{(A)}\ 109\qquad\textbf{(B)}\ 201\qquad\textbf{(C)}\ 301\qquad\textbf{(D)}\ 3049\qquad\textbf{(E)}\ 33,601</math><br />
<br />
[[2016 AMC 12A Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 21|Solution]]<br />
<br />
<br />
==Problem 22==<br />
<br />
How many ordered triples <math>(x,y,z)</math> of positive integers satisfy <math>\text{lcm}(x,y) = 72, \text{lcm}(x,z) = 600</math> and <math>\text{lcm}(y,z)=900</math>?<br />
<br />
<math>\textbf{(A)}\ 15\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 64</math><br />
<br />
[[2016 AMC 12A Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 23|Solution]]<br />
<br />
<br />
==Problem 24==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 24|Solution]]<br />
<br />
<br />
==Problem 25==<br />
<br />
Let <math>k</math> be a positive integer. Bernardo and Silvia take turns writing and erasing numbers on a blackboard as follows: Bernardo starts by writing the smallest perfect square with <math>k+1</math> digits. Every time Bernardo writes a number, Silvia erases the last <math>k</math> digits of it. Bernardo then writes the next perfect square, Silvia erases the last <math>k</math> digits of it, and this process continues until the last two numbers that remain on the board differ by at least 2. Let <math>f(k)</math> be the smallest positive integer not written on the board. For example, if <math>k = 1</math>, then the numbers that Bernardo writes are <math>16, 25, 36, 49, 64</math>, and the numbers showing on the board after Silvia erases are <math>1, 2, 3, 4,</math> and <math>6</math>, and thus <math>f(1) = 5</math>. What is the sum of the digits of <math>f(2) + f(4)+ f(6) + ... + f(2016)</math>?<br />
<br />
<math>\textbf{(A)}\ 7986\qquad\textbf{(B)}\ 8002\qquad\textbf{(C)}\ 8030\qquad\textbf{(D)}\ 8048\qquad\textbf{(E)}\ 8064</math><br />
<br />
[[2016 AMC 12A Problems/Problem 25|Solution]]<br />
<br />
{{MAA Notice}}</div>Sssuperm1ch4elhttps://artofproblemsolving.com/wiki/index.php?title=2016_AMC_12A_Problems&diff=754032016 AMC 12A Problems2016-02-04T17:10:16Z<p>Sssuperm1ch4el: /* Problem 20 */</p>
<hr />
<div>{{AMC12 Problems|year=2016|ab=A}}<br />
<br />
==Problem 1==<br />
<br />
What is the value of <math>\frac{11!-10!}{9!}</math>?<br />
<br />
<math>\textbf{(A)}\ 99\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 110\qquad\textbf{(D)}\ 121\qquad\textbf{(E)}\ 132</math><br />
<br />
[[2016 AMC 12A Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
<br />
For what value of <math>x</math> does <math>10^x\cdot100^{2x}=1000^5</math>?<br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5</math><br />
<br />
[[2016 AMC 12A Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
The remainder can be defined for all real numbers <math>x</math> and <math>y</math> with <math>y \neq 0</math> by <cmath>\text{rem} (x ,y)=x-y\left \lfloor \frac{x}{y} \right \rfloor</cmath>where <math>\left \lfloor \tfrac{x}{y} \right \rfloor</math> denotes the greatest integer less than or equal to <math>\tfrac{x}{y}</math>. What is the value of <math>\text{rem} (\tfrac{3}{8}, -\tfrac{2}{5} )</math>?<br />
<br />
<math>\textbf{(A) } -\frac{3}{8} \qquad \textbf{(B) } -\frac{1}{40} \qquad \textbf{(C) } 0 \qquad \textbf{(D) } \frac{3}{8} \qquad \textbf{(E) } \frac{31}{40}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
The mean, median, and mode of the <math>7</math> data values <math>60, 100, x, 40, 50, 200, 90</math> are all equal to <math>x</math>. What is the value of <math>x</math>?<br />
<br />
<math>\textbf{(A)}\ 50\qquad\textbf{(B)}\ 60\qquad\textbf{(C)}\ 75\qquad\textbf{(D)}\ 90\qquad\textbf{(E)}\ 100</math><br />
<br />
[[2016 AMC 12A Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
Goldbach's conjecture states that every even integer greater than 2 can be written as the sum of two prime numbers (for example, <math>2016=13+2003</math>). So far, no one has been able to prove that the conjecture is true, and no one has found a counterexample to show that the conjecture is false. What would a counterexample consist of?<br />
<br />
<math> \textbf{(A)}\ \text{an odd integer greater than } 2 \text{ that can be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(B)}\ \text{an odd integer greater than } 2 \text{ that cannot be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(C)}\ \text{an even integer greater than } 2 \text{ that can be written as the sum of two numbers that are not prime}\\<br />
\qquad\textbf{(D)}\ \text{an even integer greater than } 2 \text{ that can be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(E)}\ \text{an even integer greater than } 2 \text{ that cannot be written as the sum of two prime numbers}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
A triangular array of <math>2016</math> coins has <math>1</math> coin in the first row, <math>2</math> coins in the second row, <math>3</math> coins in the third row, and so on up to <math>N</math> coins in the <math>N</math>th row. What is the sum of the digits of <math>N</math> ?<br />
<br />
<math>\textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 10</math><br />
<br />
[[2016 AMC 12A Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
Which of these describes the graph of <math>x^2(x+y+1)=y^2(x+y+1)</math> ?<br />
<br />
<math> \textbf{(A)}\ \text{two parallel lines}\\<br />
\qquad\textbf{(B)}\ \text{two intersecting lines}\\<br />
\qquad\textbf{(C)}\ \text{three lines that all pass through a common point}\\<br />
\qquad\textbf{(D)}\ \text{three lines that do not all pass through a comment point}\\<br />
\qquad\textbf{(E)}\ \text{a line and a parabola}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
What is the area of the shaded reigon of the given <math>8\times 5</math> rectangle?<br />
<br />
<asy><br />
<br />
size(6cm);<br />
defaultpen(fontsize(9pt));<br />
draw((0,0)--(8,0)--(8,5)--(0,5)--cycle);<br />
filldraw((7,0)--(8,0)--(8,1)--(0,4)--(0,5)--(1,5)--cycle,gray(0.8));<br />
<br />
label("$1$",(1/2,5),dir(90));<br />
label("$7$",(9/2,5),dir(90));<br />
<br />
label("$1$",(8,1/2),dir(0));<br />
label("$4$",(8,3),dir(0));<br />
<br />
label("$1$",(15/2,0),dir(270));<br />
label("$7$",(7/2,0),dir(270));<br />
<br />
label("$1$",(0,9/2),dir(180));<br />
label("$4$",(0,2),dir(180));<br />
<br />
</asy><br />
<br />
<math>\textbf{(A)}\ 4\dfrac{3}{4}\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 5\dfrac{1}{4}\qquad\textbf{(D)}\ 6\dfrac{1}{2}\qquad\textbf{(E)}\ 8</math><br />
<br />
[[2016 AMC 12A Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
The five small shaded squares inside this unit square are congruent and have disjoint interiors. The midpoint of each side of the middle square coincides with one of the vertices of the other four small squares as shown. The common side length is <math>\tfrac{a-\sqrt{2}}{b}</math>, where <math>a</math> and <math>b</math> are positive integers. What is <math>a+b</math> ?<br />
<br />
<pre style="color: gray">TODO: Diagram</pre><br />
<br />
<math>\textbf{(A)}\ 7\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 11</math><br />
<br />
[[2016 AMC 12A Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
Five friends sat in a movie theater in a row containing <math>5</math> seats, numbered <math>1</math> to <math>5</math> from left to right. (The directions "left" and "right" are from the point of view of the people as they sit in the seats.) During the movie Ada went to the lobby to get some popcorn. When she returned, she found that Bea had moved two seats to the right, Ceci had moved one seat to the left, and Dee and Edie had switched seats, leaving an end seat for Ada. In which seat had Ada been sitting before she got up?<br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5</math><br />
<br />
[[2016 AMC 12A Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
Each of the 100 students in a certain summer camp can either sing, dance, or act. Some students have more than one talent, but no student has all three talents. There are 42 students who cannot sing, 65 students who cannot dance, and 29 students who cannot act. How many students have two of these talents?<br />
<br />
<math>\textbf{(A)}\ 16\qquad\textbf{(B)}\ 25\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 49\qquad\textbf{(E)}\ 64</math><br />
<br />
[[2016 AMC 12A Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
In <math>\triangle ABC</math>, <math>AB = 6</math>, <math>BC = 7</math>, and <math>CA = 8</math>. Point <math>D</math> lies on <math>\overline{BC}</math>, and <math>\overline{AD}</math> bisects <math>\angle BAC</math>. Point <math>E</math> lies on <math>\overline{AC}</math>, and <math>\overline{BE}</math> bisects <math>\angle ABC</math>. The bisectors intersect at <math>F</math>. What is the ratio <math>AF</math> : <math>FD</math>?<br />
<br />
<pre style="color: gray">TODO: Diagram</pre><br />
<br />
<math>\textbf{(A)}\ 3:2\qquad\textbf{(B)}\ 5:3\qquad\textbf{(C)}\ 2:1\qquad\textbf{(D)}\ 7:3\qquad\textbf{(E)}\ 5:2</math><br />
<br />
[[2016 AMC 12A Problems/Problem 12|Solution]]<br />
<br />
<br />
==Problem 13==<br />
<br />
Let <math>N</math> be a positive multiple of <math>5</math>. One red ball and <math>N</math> green balls are arranged in a line in random order. Let <math>P(N)</math> be the probability that at least <math>\tfrac{3}{5}</math> of the green balls are on the same side of the red ball. Observe that <math>P(5)=1</math> and that <math>P(N)</math> approaches <math>\tfrac{4}{5}</math> as <math>N</math> grows large. What is the sum of the digits of the least value of <math>N</math> such that <math>P(N) < \tfrac{321}{400}</math>?<br />
<br />
<math>\textbf{(A)}\ 12\qquad\textbf{(B)}\ 14\qquad\textbf{(C)}\ 16\qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 20</math><br />
<br />
[[2016 AMC 12A Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
Each vertex of a cube is to be labeled with an integer from 1 through 8, 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 />
[[2016 AMC 12A Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
Circles with centers <math>P, Q</math> and <math>R</math>, having radii <math>1, 2</math> and <math>3</math>, respectively, lie on the same side of line <math>l</math> and are tangent to <math>l</math> at <math>P', Q'</math> and <math>R'</math>, respectively, with <math>Q'</math> between <math>P'</math> and <math>R'</math>. The circle with center <math>Q</math> is externally tangent to each of the other two circles. What is the area of triangle <math>PQR</math>?<br />
<br />
<math>\textbf{(A) } 0\qquad \textbf{(B) } \sqrt{\frac{2}{3}}\qquad\textbf{(C) } 1\qquad\textbf{(D) } \sqrt{6}-\sqrt{2}\qquad\textbf{(E) }\sqrt{\frac{3}{2}}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
The graphs of <math>y=\log_3 x, y=\log_x 3, y=\log_\frac{1}{3} x,</math> and <math>y=\log_x \dfrac{1}{3}</math> are plotted on the same set of axes. How many points in the plane with positive <math>x</math>-coordinates lie on two or more of the graphs? <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 />
[[2016 AMC 12A Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Let <math>ABCD</math> be a square. Let <math>E, F, G</math> and <math>H</math> be the centers, respectively, of equilateral triangles with bases <math>\overline{AB}, \overline{BC}, \overline{CD},</math> and <math>\overline{DA},</math> each exterior to the square. What is rge ration of the area of square <math>EFGH</math> to the area of square <math>ABCD</math>? <br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ \frac{2+\sqrt{3}}{3} \qquad\textbf{(C)}\ \sqrt{2} \qquad\textbf{(D)}\ \frac{\sqrt{2}+\sqrt{3}}{2} \qquad\textbf{(E)}\ \sqrt{3}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
For some positive integer <math>n,</math> the number <math>110n^3</math> has <math>110</math> positive integer divisors, including <math>1</math> and the number <math>110n^3.</math> How many positive integer divisors does the number <math>81n^4</math> have? <br />
<br />
<math>\textbf{(A)}\ 110\qquad\textbf{(B)}\ 191\qquad\textbf{(C)}\ 261\qquad\textbf{(D)}\ 325\qquad\textbf{(E)}\ 425</math><br />
<br />
[[2016 AMC 12A Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
Jerry starts at <math>0</math> on the real number line. He tosses a fair coin <math>8</math> times. When he gets heads, he moves <math>1</math> unit in the positive direction; when he gets tails, he moves <math>1</math> unit in the negative direction. The probability that he reaches <math>4</math> at some time during this process <math>\frac{a}{b},</math> where <math>a</math> and <math>b</math> are relatively prime positive integers. What is <math>a + b?</math> (For example, he succeeds if his sequence of tosses is <math>HTHHHHHH.</math>)<br />
<br />
<math>\textbf{(A)}\ 69\qquad\textbf{(B)}\ 151\qquad\textbf{(C)}\ 257\qquad\textbf{(D)}\ 293\qquad\textbf{(E)}\ 313</math><br />
<br />
[[2016 AMC 12A Problems/Problem 19|Solution]]<br />
<br />
<br />
==Problem 20==<br />
<br />
A binary operation <math>\diamondsuit</math> has the properties that <math>a\ \diamondsuit\ (b\ \diamondsuit\ c) = (a\ \diamondsuit\ b)\ \cdot\ c</math> and that <math>a\ \diamondsuit\ a = 1</math> for all nonzero real numbers <math>a, b</math> and <math>c.</math> (Here the dot <math>\cdot</math> represents the usual multiplication operation.) The solution to the equation <math>2016\ \diamondsuit\ (6\ \diamondsuit\ x) = 100</math> can be written as <math>\frac{p}{q},</math> where <math>p</math> and <math>q</math> are relativelt prime positive integers. What is <math>p + q?</math> <br />
<br />
<math>\textbf{(A)}\ 109\qquad\textbf{(B)}\ 201\qquad\textbf{(C)}\ 301\qquad\textbf{(D)}\ 3049\qquad\textbf{(E)}\ 33,601</math><br />
<br />
[[2016 AMC 12A Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 21|Solution]]<br />
<br />
<br />
==Problem 22==<br />
<br />
How many ordered triples <math>(x,y,z)</math> of positive integers satisfy <math>\text{lcm}(x,y) = 72, \text{lcm}(x,z) = 600</math> and <math>\text{lcm}(y,z)=900</math>?<br />
<br />
<math>\textbf{(A)}\ 15\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 64</math><br />
<br />
[[2016 AMC 12A Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 23|Solution]]<br />
<br />
<br />
==Problem 24==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 24|Solution]]<br />
<br />
<br />
==Problem 25==<br />
<br />
Let <math>k</math> be a positive integer. Bernardo and Silvia take turns writing and erasing numbers on a blackboard as follows: Bernardo starts by writing the smallest perfect square with <math>k+1</math> digits. Every time Bernardo writes a number, Silvia erases the last <math>k</math> digits of it. Bernardo then writes the next perfect square, Silvia erases the last <math>k</math> digits of it, and this process continues until the last two numbers that remain on the board differ by at least 2. Let <math>f(k)</math> be the smallest positive integer not written on the board. For example, if <math>k = 1</math>, then the numbers that Bernardo writes are <math>16, 25, 36, 49, 64</math>, and the numbers showing on the board after Silvia erases are <math>1, 2, 3, 4,</math> and <math>6</math>, and thus <math>f(1) = 5</math>. What is the sum of the digits of <math>f(2) + f(4)+ f(6) + ... + f(2016)</math>?<br />
<br />
<math>\textbf{(A)}\ 7986\qquad\textbf{(B)}\ 8002\qquad\textbf{(C)}\ 8030\qquad\textbf{(D)}\ 8048\qquad\textbf{(E)}\ 8064</math><br />
<br />
[[2016 AMC 12A Problems/Problem 25|Solution]]<br />
<br />
{{MAA Notice}}</div>Sssuperm1ch4elhttps://artofproblemsolving.com/wiki/index.php?title=2016_AMC_12A_Problems&diff=754022016 AMC 12A Problems2016-02-04T17:09:00Z<p>Sssuperm1ch4el: /* Problem 20 */</p>
<hr />
<div>{{AMC12 Problems|year=2016|ab=A}}<br />
<br />
==Problem 1==<br />
<br />
What is the value of <math>\frac{11!-10!}{9!}</math>?<br />
<br />
<math>\textbf{(A)}\ 99\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 110\qquad\textbf{(D)}\ 121\qquad\textbf{(E)}\ 132</math><br />
<br />
[[2016 AMC 12A Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
<br />
For what value of <math>x</math> does <math>10^x\cdot100^{2x}=1000^5</math>?<br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5</math><br />
<br />
[[2016 AMC 12A Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
The remainder can be defined for all real numbers <math>x</math> and <math>y</math> with <math>y \neq 0</math> by <cmath>\text{rem} (x ,y)=x-y\left \lfloor \frac{x}{y} \right \rfloor</cmath>where <math>\left \lfloor \tfrac{x}{y} \right \rfloor</math> denotes the greatest integer less than or equal to <math>\tfrac{x}{y}</math>. What is the value of <math>\text{rem} (\tfrac{3}{8}, -\tfrac{2}{5} )</math>?<br />
<br />
<math>\textbf{(A) } -\frac{3}{8} \qquad \textbf{(B) } -\frac{1}{40} \qquad \textbf{(C) } 0 \qquad \textbf{(D) } \frac{3}{8} \qquad \textbf{(E) } \frac{31}{40}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
The mean, median, and mode of the <math>7</math> data values <math>60, 100, x, 40, 50, 200, 90</math> are all equal to <math>x</math>. What is the value of <math>x</math>?<br />
<br />
<math>\textbf{(A)}\ 50\qquad\textbf{(B)}\ 60\qquad\textbf{(C)}\ 75\qquad\textbf{(D)}\ 90\qquad\textbf{(E)}\ 100</math><br />
<br />
[[2016 AMC 12A Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
Goldbach's conjecture states that every even integer greater than 2 can be written as the sum of two prime numbers (for example, <math>2016=13+2003</math>). So far, no one has been able to prove that the conjecture is true, and no one has found a counterexample to show that the conjecture is false. What would a counterexample consist of?<br />
<br />
<math> \textbf{(A)}\ \text{an odd integer greater than } 2 \text{ that can be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(B)}\ \text{an odd integer greater than } 2 \text{ that cannot be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(C)}\ \text{an even integer greater than } 2 \text{ that can be written as the sum of two numbers that are not prime}\\<br />
\qquad\textbf{(D)}\ \text{an even integer greater than } 2 \text{ that can be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(E)}\ \text{an even integer greater than } 2 \text{ that cannot be written as the sum of two prime numbers}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
A triangular array of <math>2016</math> coins has <math>1</math> coin in the first row, <math>2</math> coins in the second row, <math>3</math> coins in the third row, and so on up to <math>N</math> coins in the <math>N</math>th row. What is the sum of the digits of <math>N</math> ?<br />
<br />
<math>\textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 10</math><br />
<br />
[[2016 AMC 12A Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
Which of these describes the graph of <math>x^2(x+y+1)=y^2(x+y+1)</math> ?<br />
<br />
<math> \textbf{(A)}\ \text{two parallel lines}\\<br />
\qquad\textbf{(B)}\ \text{two intersecting lines}\\<br />
\qquad\textbf{(C)}\ \text{three lines that all pass through a common point}\\<br />
\qquad\textbf{(D)}\ \text{three lines that do not all pass through a comment point}\\<br />
\qquad\textbf{(E)}\ \text{a line and a parabola}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
What is the area of the shaded reigon of the given <math>8\times 5</math> rectangle?<br />
<br />
<asy><br />
<br />
size(6cm);<br />
defaultpen(fontsize(9pt));<br />
draw((0,0)--(8,0)--(8,5)--(0,5)--cycle);<br />
filldraw((7,0)--(8,0)--(8,1)--(0,4)--(0,5)--(1,5)--cycle,gray(0.8));<br />
<br />
label("$1$",(1/2,5),dir(90));<br />
label("$7$",(9/2,5),dir(90));<br />
<br />
label("$1$",(8,1/2),dir(0));<br />
label("$4$",(8,3),dir(0));<br />
<br />
label("$1$",(15/2,0),dir(270));<br />
label("$7$",(7/2,0),dir(270));<br />
<br />
label("$1$",(0,9/2),dir(180));<br />
label("$4$",(0,2),dir(180));<br />
<br />
</asy><br />
<br />
<math>\textbf{(A)}\ 4\dfrac{3}{4}\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 5\dfrac{1}{4}\qquad\textbf{(D)}\ 6\dfrac{1}{2}\qquad\textbf{(E)}\ 8</math><br />
<br />
[[2016 AMC 12A Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
The five small shaded squares inside this unit square are congruent and have disjoint interiors. The midpoint of each side of the middle square coincides with one of the vertices of the other four small squares as shown. The common side length is <math>\tfrac{a-\sqrt{2}}{b}</math>, where <math>a</math> and <math>b</math> are positive integers. What is <math>a+b</math> ?<br />
<br />
<pre style="color: gray">TODO: Diagram</pre><br />
<br />
<math>\textbf{(A)}\ 7\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 11</math><br />
<br />
[[2016 AMC 12A Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
Five friends sat in a movie theater in a row containing <math>5</math> seats, numbered <math>1</math> to <math>5</math> from left to right. (The directions "left" and "right" are from the point of view of the people as they sit in the seats.) During the movie Ada went to the lobby to get some popcorn. When she returned, she found that Bea had moved two seats to the right, Ceci had moved one seat to the left, and Dee and Edie had switched seats, leaving an end seat for Ada. In which seat had Ada been sitting before she got up?<br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5</math><br />
<br />
[[2016 AMC 12A Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
Each of the 100 students in a certain summer camp can either sing, dance, or act. Some students have more than one talent, but no student has all three talents. There are 42 students who cannot sing, 65 students who cannot dance, and 29 students who cannot act. How many students have two of these talents?<br />
<br />
<math>\textbf{(A)}\ 16\qquad\textbf{(B)}\ 25\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 49\qquad\textbf{(E)}\ 64</math><br />
<br />
[[2016 AMC 12A Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
In <math>\triangle ABC</math>, <math>AB = 6</math>, <math>BC = 7</math>, and <math>CA = 8</math>. Point <math>D</math> lies on <math>\overline{BC}</math>, and <math>\overline{AD}</math> bisects <math>\angle BAC</math>. Point <math>E</math> lies on <math>\overline{AC}</math>, and <math>\overline{BE}</math> bisects <math>\angle ABC</math>. The bisectors intersect at <math>F</math>. What is the ratio <math>AF</math> : <math>FD</math>?<br />
<br />
<pre style="color: gray">TODO: Diagram</pre><br />
<br />
<math>\textbf{(A)}\ 3:2\qquad\textbf{(B)}\ 5:3\qquad\textbf{(C)}\ 2:1\qquad\textbf{(D)}\ 7:3\qquad\textbf{(E)}\ 5:2</math><br />
<br />
[[2016 AMC 12A Problems/Problem 12|Solution]]<br />
<br />
<br />
==Problem 13==<br />
<br />
Let <math>N</math> be a positive multiple of <math>5</math>. One red ball and <math>N</math> green balls are arranged in a line in random order. Let <math>P(N)</math> be the probability that at least <math>\tfrac{3}{5}</math> of the green balls are on the same side of the red ball. Observe that <math>P(5)=1</math> and that <math>P(N)</math> approaches <math>\tfrac{4}{5}</math> as <math>N</math> grows large. What is the sum of the digits of the least value of <math>N</math> such that <math>P(N) < \tfrac{321}{400}</math>?<br />
<br />
<math>\textbf{(A)}\ 12\qquad\textbf{(B)}\ 14\qquad\textbf{(C)}\ 16\qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 20</math><br />
<br />
[[2016 AMC 12A Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
Each vertex of a cube is to be labeled with an integer from 1 through 8, 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 />
[[2016 AMC 12A Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
Circles with centers <math>P, Q</math> and <math>R</math>, having radii <math>1, 2</math> and <math>3</math>, respectively, lie on the same side of line <math>l</math> and are tangent to <math>l</math> at <math>P', Q'</math> and <math>R'</math>, respectively, with <math>Q'</math> between <math>P'</math> and <math>R'</math>. The circle with center <math>Q</math> is externally tangent to each of the other two circles. What is the area of triangle <math>PQR</math>?<br />
<br />
<math>\textbf{(A) } 0\qquad \textbf{(B) } \sqrt{\frac{2}{3}}\qquad\textbf{(C) } 1\qquad\textbf{(D) } \sqrt{6}-\sqrt{2}\qquad\textbf{(E) }\sqrt{\frac{3}{2}}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
The graphs of <math>y=\log_3 x, y=\log_x 3, y=\log_\frac{1}{3} x,</math> and <math>y=\log_x \dfrac{1}{3}</math> are plotted on the same set of axes. How many points in the plane with positive <math>x</math>-coordinates lie on two or more of the graphs? <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 />
[[2016 AMC 12A Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Let <math>ABCD</math> be a square. Let <math>E, F, G</math> and <math>H</math> be the centers, respectively, of equilateral triangles with bases <math>\overline{AB}, \overline{BC}, \overline{CD},</math> and <math>\overline{DA},</math> each exterior to the square. What is rge ration of the area of square <math>EFGH</math> to the area of square <math>ABCD</math>? <br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ \frac{2+\sqrt{3}}{3} \qquad\textbf{(C)}\ \sqrt{2} \qquad\textbf{(D)}\ \frac{\sqrt{2}+\sqrt{3}}{2} \qquad\textbf{(E)}\ \sqrt{3}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
For some positive integer <math>n,</math> the number <math>110n^3</math> has <math>110</math> positive integer divisors, including <math>1</math> and the number <math>110n^3.</math> How many positive integer divisors does the number <math>81n^4</math> have? <br />
<br />
<math>\textbf{(A)}\ 110\qquad\textbf{(B)}\ 191\qquad\textbf{(C)}\ 261\qquad\textbf{(D)}\ 325\qquad\textbf{(E)}\ 425</math><br />
<br />
[[2016 AMC 12A Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
Jerry starts at <math>0</math> on the real number line. He tosses a fair coin <math>8</math> times. When he gets heads, he moves <math>1</math> unit in the positive direction; when he gets tails, he moves <math>1</math> unit in the negative direction. The probability that he reaches <math>4</math> at some time during this process <math>\frac{a}{b},</math> where <math>a</math> and <math>b</math> are relatively prime positive integers. What is <math>a + b?</math> (For example, he succeeds if his sequence of tosses is <math>HTHHHHHH.</math>)<br />
<br />
<math>\textbf{(A)}\ 69\qquad\textbf{(B)}\ 151\qquad\textbf{(C)}\ 257\qquad\textbf{(D)}\ 293\qquad\textbf{(E)}\ 313</math><br />
<br />
[[2016 AMC 12A Problems/Problem 19|Solution]]<br />
<br />
<br />
==Problem 20==<br />
<br />
A binary operation <math>\diamondsuit</math> has the properties that <math>a\ \diamondsuit\ (b\ \diamondsuit\ c) = (a\ \diamondsuit\ b) \cdot\ c</math> and that <math>a\ \diamondsuit\ a = 1</math> for all nonzero real numbers <math>a, b</math> and <math>c.</math> (Here the dot \cdot represents the usual multiplication operation.) The solution to the equation <math>2016 \diamondsuit\ (6 \diamondsuit\ x) = 100</math> can be written as <math>\frac{p}{q},</math> where <math>p</math> and <math>q</math> are relativelt prime positive integers. What is <math>p + q?</math> <br />
<br />
<math>\textbf{(A)}\ 109\qquad\textbf{(B)}\ 201\qquad\textbf{(C)}\ 301\qquad\textbf{(D)}\ 3049\qquad\textbf{(E)}\ 33,601</math><br />
<br />
[[2016 AMC 12A Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 21|Solution]]<br />
<br />
<br />
==Problem 22==<br />
<br />
How many ordered triples <math>(x,y,z)</math> of positive integers satisfy <math>\text{lcm}(x,y) = 72, \text{lcm}(x,z) = 600</math> and <math>\text{lcm}(y,z)=900</math>?<br />
<br />
<math>\textbf{(A)}\ 15\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 64</math><br />
<br />
[[2016 AMC 12A Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 23|Solution]]<br />
<br />
<br />
==Problem 24==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 24|Solution]]<br />
<br />
<br />
==Problem 25==<br />
<br />
Let <math>k</math> be a positive integer. Bernardo and Silvia take turns writing and erasing numbers on a blackboard as follows: Bernardo starts by writing the smallest perfect square with <math>k+1</math> digits. Every time Bernardo writes a number, Silvia erases the last <math>k</math> digits of it. Bernardo then writes the next perfect square, Silvia erases the last <math>k</math> digits of it, and this process continues until the last two numbers that remain on the board differ by at least 2. Let <math>f(k)</math> be the smallest positive integer not written on the board. For example, if <math>k = 1</math>, then the numbers that Bernardo writes are <math>16, 25, 36, 49, 64</math>, and the numbers showing on the board after Silvia erases are <math>1, 2, 3, 4,</math> and <math>6</math>, and thus <math>f(1) = 5</math>. What is the sum of the digits of <math>f(2) + f(4)+ f(6) + ... + f(2016)</math>?<br />
<br />
<math>\textbf{(A)}\ 7986\qquad\textbf{(B)}\ 8002\qquad\textbf{(C)}\ 8030\qquad\textbf{(D)}\ 8048\qquad\textbf{(E)}\ 8064</math><br />
<br />
[[2016 AMC 12A Problems/Problem 25|Solution]]<br />
<br />
{{MAA Notice}}</div>Sssuperm1ch4elhttps://artofproblemsolving.com/wiki/index.php?title=2016_AMC_12A_Problems&diff=753982016 AMC 12A Problems2016-02-04T17:07:04Z<p>Sssuperm1ch4el: /* Problem 20 */</p>
<hr />
<div>{{AMC12 Problems|year=2016|ab=A}}<br />
<br />
==Problem 1==<br />
<br />
What is the value of <math>\frac{11!-10!}{9!}</math>?<br />
<br />
<math>\textbf{(A)}\ 99\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 110\qquad\textbf{(D)}\ 121\qquad\textbf{(E)}\ 132</math><br />
<br />
[[2016 AMC 12A Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
<br />
For what value of <math>x</math> does <math>10^x\cdot100^{2x}=1000^5</math>?<br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5</math><br />
<br />
[[2016 AMC 12A Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
The remainder can be defined for all real numbers <math>x</math> and <math>y</math> with <math>y \neq 0</math> by <cmath>\text{rem} (x ,y)=x-y\left \lfloor \frac{x}{y} \right \rfloor</cmath>where <math>\left \lfloor \tfrac{x}{y} \right \rfloor</math> denotes the greatest integer less than or equal to <math>\tfrac{x}{y}</math>. What is the value of <math>\text{rem} (\tfrac{3}{8}, -\tfrac{2}{5} )</math>?<br />
<br />
<math>\textbf{(A) } -\frac{3}{8} \qquad \textbf{(B) } -\frac{1}{40} \qquad \textbf{(C) } 0 \qquad \textbf{(D) } \frac{3}{8} \qquad \textbf{(E) } \frac{31}{40}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
The mean, median, and mode of the <math>7</math> data values <math>60, 100, x, 40, 50, 200, 90</math> are all equal to <math>x</math>. What is the value of <math>x</math>?<br />
<br />
<math>\textbf{(A)}\ 50\qquad\textbf{(B)}\ 60\qquad\textbf{(C)}\ 75\qquad\textbf{(D)}\ 90\qquad\textbf{(E)}\ 100</math><br />
<br />
[[2016 AMC 12A Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
Goldbach's conjecture states that every even integer greater than 2 can be written as the sum of two prime numbers (for example, <math>2016=13+2003</math>). So far, no one has been able to prove that the conjecture is true, and no one has found a counterexample to show that the conjecture is false. What would a counterexample consist of?<br />
<br />
<math> \textbf{(A)}\ \text{an odd integer greater than } 2 \text{ that can be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(B)}\ \text{an odd integer greater than } 2 \text{ that cannot be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(C)}\ \text{an even integer greater than } 2 \text{ that can be written as the sum of two numbers that are not prime}\\<br />
\qquad\textbf{(D)}\ \text{an even integer greater than } 2 \text{ that can be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(E)}\ \text{an even integer greater than } 2 \text{ that cannot be written as the sum of two prime numbers}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
A triangular array of <math>2016</math> coins has <math>1</math> coin in the first row, <math>2</math> coins in the second row, <math>3</math> coins in the third row, and so on up to <math>N</math> coins in the <math>N</math>th row. What is the sum of the digits of <math>N</math> ?<br />
<br />
<math>\textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 10</math><br />
<br />
[[2016 AMC 12A Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
Which of these describes the graph of <math>x^2(x+y+1)=y^2(x+y+1)</math> ?<br />
<br />
<math> \textbf{(A)}\ \text{two parallel lines}\\<br />
\qquad\textbf{(B)}\ \text{two intersecting lines}\\<br />
\qquad\textbf{(C)}\ \text{three lines that all pass through a common point}\\<br />
\qquad\textbf{(D)}\ \text{three lines that do not all pass through a comment point}\\<br />
\qquad\textbf{(E)}\ \text{a line and a parabola}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
What is the area of the shaded reigon of the given <math>8\times 5</math> rectangle?<br />
<br />
<asy><br />
<br />
size(6cm);<br />
defaultpen(fontsize(9pt));<br />
draw((0,0)--(8,0)--(8,5)--(0,5)--cycle);<br />
filldraw((7,0)--(8,0)--(8,1)--(0,4)--(0,5)--(1,5)--cycle,gray(0.8));<br />
<br />
label("$1$",(1/2,5),dir(90));<br />
label("$7$",(9/2,5),dir(90));<br />
<br />
label("$1$",(8,1/2),dir(0));<br />
label("$4$",(8,3),dir(0));<br />
<br />
label("$1$",(15/2,0),dir(270));<br />
label("$7$",(7/2,0),dir(270));<br />
<br />
label("$1$",(0,9/2),dir(180));<br />
label("$4$",(0,2),dir(180));<br />
<br />
</asy><br />
<br />
<math>\textbf{(A)}\ 4\dfrac{3}{4}\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 5\dfrac{1}{4}\qquad\textbf{(D)}\ 6\dfrac{1}{2}\qquad\textbf{(E)}\ 8</math><br />
<br />
[[2016 AMC 12A Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
The five small shaded squares inside this unit square are congruent and have disjoint interiors. The midpoint of each side of the middle square coincides with one of the vertices of the other four small squares as shown. The common side length is <math>\tfrac{a-\sqrt{2}}{b}</math>, where <math>a</math> and <math>b</math> are positive integers. What is <math>a+b</math> ?<br />
<br />
<pre style="color: gray">TODO: Diagram</pre><br />
<br />
<math>\textbf{(A)}\ 7\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 11</math><br />
<br />
[[2016 AMC 12A Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
Five friends sat in a movie theater in a row containing <math>5</math> seats, numbered <math>1</math> to <math>5</math> from left to right. (The directions "left" and "right" are from the point of view of the people as they sit in the seats.) During the movie Ada went to the lobby to get some popcorn. When she returned, she found that Bea had moved two seats to the right, Ceci had moved one seat to the left, and Dee and Edie had switched seats, leaving an end seat for Ada. In which seat had Ada been sitting before she got up?<br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5</math><br />
<br />
[[2016 AMC 12A Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
Each of the 100 students in a certain summer camp can either sing, dance, or act. Some students have more than one talent, but no student has all three talents. There are 42 students who cannot sing, 65 students who cannot dance, and 29 students who cannot act. How many students have two of these talents?<br />
<br />
<math>\textbf{(A)}\ 16\qquad\textbf{(B)}\ 25\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 49\qquad\textbf{(E)}\ 64</math><br />
<br />
[[2016 AMC 12A Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
In <math>\triangle ABC</math>, <math>AB = 6</math>, <math>BC = 7</math>, and <math>CA = 8</math>. Point <math>D</math> lies on <math>\overline{BC}</math>, and <math>\overline{AD}</math> bisects <math>\angle BAC</math>. Point <math>E</math> lies on <math>\overline{AC}</math>, and <math>\overline{BE}</math> bisects <math>\angle ABC</math>. The bisectors intersect at <math>F</math>. What is the ratio <math>AF</math> : <math>FD</math>?<br />
<br />
<pre style="color: gray">TODO: Diagram</pre><br />
<br />
<math>\textbf{(A)}\ 3:2\qquad\textbf{(B)}\ 5:3\qquad\textbf{(C)}\ 2:1\qquad\textbf{(D)}\ 7:3\qquad\textbf{(E)}\ 5:2</math><br />
<br />
[[2016 AMC 12A Problems/Problem 12|Solution]]<br />
<br />
<br />
==Problem 13==<br />
<br />
Let <math>N</math> be a positive multiple of <math>5</math>. One red ball and <math>N</math> green balls are arranged in a line in random order. Let <math>P(N)</math> be the probability that at least <math>\tfrac{3}{5}</math> of the green balls are on the same side of the red ball. Observe that <math>P(5)=1</math> and that <math>P(N)</math> approaches <math>\tfrac{4}{5}</math> as <math>N</math> grows large. What is the sum of the digits of the least value of <math>N</math> such that <math>P(N) < \tfrac{321}{400}</math>?<br />
<br />
<math>\textbf{(A)}\ 12\qquad\textbf{(B)}\ 14\qquad\textbf{(C)}\ 16\qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 20</math><br />
<br />
[[2016 AMC 12A Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
Each vertex of a cube is to be labeled with an integer from 1 through 8, 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 />
[[2016 AMC 12A Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
Circles with centers <math>P, Q</math> and <math>R</math>, having radii <math>1, 2</math> and <math>3</math>, respectively, lie on the same side of line <math>l</math> and are tangent to <math>l</math> at <math>P', Q'</math> and <math>R'</math>, respectively, with <math>Q'</math> between <math>P'</math> and <math>R'</math>. The circle with center <math>Q</math> is externally tangent to each of the other two circles. What is the area of triangle <math>PQR</math>?<br />
<br />
<math>\textbf{(A) } 0\qquad \textbf{(B) } \sqrt{\frac{2}{3}}\qquad\textbf{(C) } 1\qquad\textbf{(D) } \sqrt{6}-\sqrt{2}\qquad\textbf{(E) }\sqrt{\frac{3}{2}}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
The graphs of <math>y=\log_3 x, y=\log_x 3, y=\log_\frac{1}{3} x,</math> and <math>y=\log_x \dfrac{1}{3}</math> are plotted on the same set of axes. How many points in the plane with positive <math>x</math>-coordinates lie on two or more of the graphs? <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 />
[[2016 AMC 12A Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Let <math>ABCD</math> be a square. Let <math>E, F, G</math> and <math>H</math> be the centers, respectively, of equilateral triangles with bases <math>\overline{AB}, \overline{BC}, \overline{CD},</math> and <math>\overline{DA},</math> each exterior to the square. What is rge ration of the area of square <math>EFGH</math> to the area of square <math>ABCD</math>? <br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ \frac{2+\sqrt{3}}{3} \qquad\textbf{(C)}\ \sqrt{2} \qquad\textbf{(D)}\ \frac{\sqrt{2}+\sqrt{3}}{2} \qquad\textbf{(E)}\ \sqrt{3}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
For some positive integer <math>n,</math> the number <math>110n^3</math> has <math>110</math> positive integer divisors, including <math>1</math> and the number <math>110n^3.</math> How many positive integer divisors does the number <math>81n^4</math> have? <br />
<br />
<math>\textbf{(A)}\ 110\qquad\textbf{(B)}\ 191\qquad\textbf{(C)}\ 261\qquad\textbf{(D)}\ 325\qquad\textbf{(E)}\ 425</math><br />
<br />
[[2016 AMC 12A Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
Jerry starts at <math>0</math> on the real number line. He tosses a fair coin <math>8</math> times. When he gets heads, he moves <math>1</math> unit in the positive direction; when he gets tails, he moves <math>1</math> unit in the negative direction. The probability that he reaches <math>4</math> at some time during this process <math>\frac{a}{b},</math> where <math>a</math> and <math>b</math> are relatively prime positive integers. What is <math>a + b?</math> (For example, he succeeds if his sequence of tosses is <math>HTHHHHHH.</math>)<br />
<br />
<math>\textbf{(A)}\ 69\qquad\textbf{(B)}\ 151\qquad\textbf{(C)}\ 257\qquad\textbf{(D)}\ 293\qquad\textbf{(E)}\ 313</math><br />
<br />
[[2016 AMC 12A Problems/Problem 19|Solution]]<br />
<br />
<br />
==Problem 20==<br />
<br />
A binary operation <math>\diamondsuit</math> has the properties that <math>a\ \diamondsuit (b\ \diamondsuit\ c) = (a\ \diamondsuit b) \cdot\ c</math> and that <math>a\ \diamondsuit\ a = 1</math> for all nonzero real numbers <math>a, b</math> and <math>c.</math> (Here the dot \cdot represents the usual multiplication operation.) The solution to the equation <math>2016 \diamondsuit (6 \diamondsuit x) = 100</math> can be written as <math>\frac{p}{q},</math> where <math>p</math> and <math>q</math> are relativelt prime positive integers. What is <math>p + q?</math> <br />
<br />
<math>\textbf{(A)}\ 109\qquad\textbf{(B)}\ 201\qquad\textbf{(C)}\ 301\qquad\textbf{(D)}\ 3049\qquad\textbf{(E)}\ 33,601</math><br />
<br />
[[2016 AMC 12A Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 21|Solution]]<br />
<br />
<br />
==Problem 22==<br />
<br />
How many ordered triples <math>(x,y,z)</math> of positive integers satisfy <math>\text{lcm}(x,y) = 72, \text{lcm}(x,z) = 600</math> and <math>\text{lcm}(y,z)=900</math>?<br />
<br />
<math>\textbf{(A)}\ 15\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 64</math><br />
<br />
[[2016 AMC 12A Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 23|Solution]]<br />
<br />
<br />
==Problem 24==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 24|Solution]]<br />
<br />
<br />
==Problem 25==<br />
<br />
Let <math>k</math> be a positive integer. Bernardo and Silvia take turns writing and erasing numbers on a blackboard as follows: Bernardo starts by writing the smallest perfect square with <math>k+1</math> digits. Every time Bernardo writes a number, Silvia erases the last <math>k</math> digits of it. Bernardo then writes the next perfect square, Silvia erases the last <math>k</math> digits of it, and this process continues until the last two numbers that remain on the board differ by at least 2. Let <math>f(k)</math> be the smallest positive integer not written on the board. For example, if <math>k = 1</math>, then the numbers that Bernardo writes are <math>16, 25, 36, 49, 64</math>, and the numbers showing on the board after Silvia erases are <math>1, 2, 3, 4,</math> and <math>6</math>, and thus <math>f(1) = 5</math>. What is the sum of the digits of <math>f(2) + f(4)+ f(6) + ... + f(2016)</math>?<br />
<br />
<math>\textbf{(A)}\ 7986\qquad\textbf{(B)}\ 8002\qquad\textbf{(C)}\ 8030\qquad\textbf{(D)}\ 8048\qquad\textbf{(E)}\ 8064</math><br />
<br />
[[2016 AMC 12A Problems/Problem 25|Solution]]<br />
<br />
{{MAA Notice}}</div>Sssuperm1ch4elhttps://artofproblemsolving.com/wiki/index.php?title=2016_AMC_12A_Problems&diff=753952016 AMC 12A Problems2016-02-04T17:05:55Z<p>Sssuperm1ch4el: /* Problem 20 */</p>
<hr />
<div>{{AMC12 Problems|year=2016|ab=A}}<br />
<br />
==Problem 1==<br />
<br />
What is the value of <math>\frac{11!-10!}{9!}</math>?<br />
<br />
<math>\textbf{(A)}\ 99\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 110\qquad\textbf{(D)}\ 121\qquad\textbf{(E)}\ 132</math><br />
<br />
[[2016 AMC 12A Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
<br />
For what value of <math>x</math> does <math>10^x\cdot100^{2x}=1000^5</math>?<br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5</math><br />
<br />
[[2016 AMC 12A Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
The remainder can be defined for all real numbers <math>x</math> and <math>y</math> with <math>y \neq 0</math> by <cmath>\text{rem} (x ,y)=x-y\left \lfloor \frac{x}{y} \right \rfloor</cmath>where <math>\left \lfloor \tfrac{x}{y} \right \rfloor</math> denotes the greatest integer less than or equal to <math>\tfrac{x}{y}</math>. What is the value of <math>\text{rem} (\tfrac{3}{8}, -\tfrac{2}{5} )</math>?<br />
<br />
<math>\textbf{(A) } -\frac{3}{8} \qquad \textbf{(B) } -\frac{1}{40} \qquad \textbf{(C) } 0 \qquad \textbf{(D) } \frac{3}{8} \qquad \textbf{(E) } \frac{31}{40}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
The mean, median, and mode of the <math>7</math> data values <math>60, 100, x, 40, 50, 200, 90</math> are all equal to <math>x</math>. What is the value of <math>x</math>?<br />
<br />
<math>\textbf{(A)}\ 50\qquad\textbf{(B)}\ 60\qquad\textbf{(C)}\ 75\qquad\textbf{(D)}\ 90\qquad\textbf{(E)}\ 100</math><br />
<br />
[[2016 AMC 12A Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
Goldbach's conjecture states that every even integer greater than 2 can be written as the sum of two prime numbers (for example, <math>2016=13+2003</math>). So far, no one has been able to prove that the conjecture is true, and no one has found a counterexample to show that the conjecture is false. What would a counterexample consist of?<br />
<br />
<math> \textbf{(A)}\ \text{an odd integer greater than } 2 \text{ that can be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(B)}\ \text{an odd integer greater than } 2 \text{ that cannot be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(C)}\ \text{an even integer greater than } 2 \text{ that can be written as the sum of two numbers that are not prime}\\<br />
\qquad\textbf{(D)}\ \text{an even integer greater than } 2 \text{ that can be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(E)}\ \text{an even integer greater than } 2 \text{ that cannot be written as the sum of two prime numbers}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
A triangular array of <math>2016</math> coins has <math>1</math> coin in the first row, <math>2</math> coins in the second row, <math>3</math> coins in the third row, and so on up to <math>N</math> coins in the <math>N</math>th row. What is the sum of the digits of <math>N</math> ?<br />
<br />
<math>\textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 10</math><br />
<br />
[[2016 AMC 12A Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
Which of these describes the graph of <math>x^2(x+y+1)=y^2(x+y+1)</math> ?<br />
<br />
<math> \textbf{(A)}\ \text{two parallel lines}\\<br />
\qquad\textbf{(B)}\ \text{two intersecting lines}\\<br />
\qquad\textbf{(C)}\ \text{three lines that all pass through a common point}\\<br />
\qquad\textbf{(D)}\ \text{three lines that do not all pass through a comment point}\\<br />
\qquad\textbf{(E)}\ \text{a line and a parabola}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
What is the area of the shaded reigon of the given <math>8\times 5</math> rectangle?<br />
<br />
<asy><br />
<br />
size(6cm);<br />
defaultpen(fontsize(9pt));<br />
draw((0,0)--(8,0)--(8,5)--(0,5)--cycle);<br />
filldraw((7,0)--(8,0)--(8,1)--(0,4)--(0,5)--(1,5)--cycle,gray(0.8));<br />
<br />
label("$1$",(1/2,5),dir(90));<br />
label("$7$",(9/2,5),dir(90));<br />
<br />
label("$1$",(8,1/2),dir(0));<br />
label("$4$",(8,3),dir(0));<br />
<br />
label("$1$",(15/2,0),dir(270));<br />
label("$7$",(7/2,0),dir(270));<br />
<br />
label("$1$",(0,9/2),dir(180));<br />
label("$4$",(0,2),dir(180));<br />
<br />
</asy><br />
<br />
<math>\textbf{(A)}\ 4\dfrac{3}{4}\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 5\dfrac{1}{4}\qquad\textbf{(D)}\ 6\dfrac{1}{2}\qquad\textbf{(E)}\ 8</math><br />
<br />
[[2016 AMC 12A Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
The five small shaded squares inside this unit square are congruent and have disjoint interiors. The midpoint of each side of the middle square coincides with one of the vertices of the other four small squares as shown. The common side length is <math>\tfrac{a-\sqrt{2}}{b}</math>, where <math>a</math> and <math>b</math> are positive integers. What is <math>a+b</math> ?<br />
<br />
<pre style="color: gray">TODO: Diagram</pre><br />
<br />
<math>\textbf{(A)}\ 7\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 11</math><br />
<br />
[[2016 AMC 12A Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
Five friends sat in a movie theater in a row containing <math>5</math> seats, numbered <math>1</math> to <math>5</math> from left to right. (The directions "left" and "right" are from the point of view of the people as they sit in the seats.) During the movie Ada went to the lobby to get some popcorn. When she returned, she found that Bea had moved two seats to the right, Ceci had moved one seat to the left, and Dee and Edie had switched seats, leaving an end seat for Ada. In which seat had Ada been sitting before she got up?<br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5</math><br />
<br />
[[2016 AMC 12A Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
Each of the 100 students in a certain summer camp can either sing, dance, or act. Some students have more than one talent, but no student has all three talents. There are 42 students who cannot sing, 65 students who cannot dance, and 29 students who cannot act. How many students have two of these talents?<br />
<br />
<math>\textbf{(A)}\ 16\qquad\textbf{(B)}\ 25\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 49\qquad\textbf{(E)}\ 64</math><br />
<br />
[[2016 AMC 12A Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
In <math>\triangle ABC</math>, <math>AB = 6</math>, <math>BC = 7</math>, and <math>CA = 8</math>. Point <math>D</math> lies on <math>\overline{BC}</math>, and <math>\overline{AD}</math> bisects <math>\angle BAC</math>. Point <math>E</math> lies on <math>\overline{AC}</math>, and <math>\overline{BE}</math> bisects <math>\angle ABC</math>. The bisectors intersect at <math>F</math>. What is the ratio <math>AF</math> : <math>FD</math>?<br />
<br />
<pre style="color: gray">TODO: Diagram</pre><br />
<br />
<math>\textbf{(A)}\ 3:2\qquad\textbf{(B)}\ 5:3\qquad\textbf{(C)}\ 2:1\qquad\textbf{(D)}\ 7:3\qquad\textbf{(E)}\ 5:2</math><br />
<br />
[[2016 AMC 12A Problems/Problem 12|Solution]]<br />
<br />
<br />
==Problem 13==<br />
<br />
Let <math>N</math> be a positive multiple of <math>5</math>. One red ball and <math>N</math> green balls are arranged in a line in random order. Let <math>P(N)</math> be the probability that at least <math>\tfrac{3}{5}</math> of the green balls are on the same side of the red ball. Observe that <math>P(5)=1</math> and that <math>P(N)</math> approaches <math>\tfrac{4}{5}</math> as <math>N</math> grows large. What is the sum of the digits of the least value of <math>N</math> such that <math>P(N) < \tfrac{321}{400}</math>?<br />
<br />
<math>\textbf{(A)}\ 12\qquad\textbf{(B)}\ 14\qquad\textbf{(C)}\ 16\qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 20</math><br />
<br />
[[2016 AMC 12A Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
Each vertex of a cube is to be labeled with an integer from 1 through 8, 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 />
[[2016 AMC 12A Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
Circles with centers <math>P, Q</math> and <math>R</math>, having radii <math>1, 2</math> and <math>3</math>, respectively, lie on the same side of line <math>l</math> and are tangent to <math>l</math> at <math>P', Q'</math> and <math>R'</math>, respectively, with <math>Q'</math> between <math>P'</math> and <math>R'</math>. The circle with center <math>Q</math> is externally tangent to each of the other two circles. What is the area of triangle <math>PQR</math>?<br />
<br />
<math>\textbf{(A) } 0\qquad \textbf{(B) } \sqrt{\frac{2}{3}}\qquad\textbf{(C) } 1\qquad\textbf{(D) } \sqrt{6}-\sqrt{2}\qquad\textbf{(E) }\sqrt{\frac{3}{2}}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
The graphs of <math>y=\log_3 x, y=\log_x 3, y=\log_\frac{1}{3} x,</math> and <math>y=\log_x \dfrac{1}{3}</math> are plotted on the same set of axes. How many points in the plane with positive <math>x</math>-coordinates lie on two or more of the graphs? <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 />
[[2016 AMC 12A Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Let <math>ABCD</math> be a square. Let <math>E, F, G</math> and <math>H</math> be the centers, respectively, of equilateral triangles with bases <math>\overline{AB}, \overline{BC}, \overline{CD},</math> and <math>\overline{DA},</math> each exterior to the square. What is rge ration of the area of square <math>EFGH</math> to the area of square <math>ABCD</math>? <br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ \frac{2+\sqrt{3}}{3} \qquad\textbf{(C)}\ \sqrt{2} \qquad\textbf{(D)}\ \frac{\sqrt{2}+\sqrt{3}}{2} \qquad\textbf{(E)}\ \sqrt{3}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
For some positive integer <math>n,</math> the number <math>110n^3</math> has <math>110</math> positive integer divisors, including <math>1</math> and the number <math>110n^3.</math> How many positive integer divisors does the number <math>81n^4</math> have? <br />
<br />
<math>\textbf{(A)}\ 110\qquad\textbf{(B)}\ 191\qquad\textbf{(C)}\ 261\qquad\textbf{(D)}\ 325\qquad\textbf{(E)}\ 425</math><br />
<br />
[[2016 AMC 12A Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
Jerry starts at <math>0</math> on the real number line. He tosses a fair coin <math>8</math> times. When he gets heads, he moves <math>1</math> unit in the positive direction; when he gets tails, he moves <math>1</math> unit in the negative direction. The probability that he reaches <math>4</math> at some time during this process <math>\frac{a}{b},</math> where <math>a</math> and <math>b</math> are relatively prime positive integers. What is <math>a + b?</math> (For example, he succeeds if his sequence of tosses is <math>HTHHHHHH.</math>)<br />
<br />
<math>\textbf{(A)}\ 69\qquad\textbf{(B)}\ 151\qquad\textbf{(C)}\ 257\qquad\textbf{(D)}\ 293\qquad\textbf{(E)}\ 313</math><br />
<br />
[[2016 AMC 12A Problems/Problem 19|Solution]]<br />
<br />
<br />
==Problem 20==<br />
<br />
A binary operation <math>\diamondsuit</math> has the properties that <math>a \diamondsuit (b \diamondsuit c) = (a \diamondsuit b) \cdot c</math> and that <math>a \diamondsuit a = 1</math> for all nonzero real numbers <math>a, b</math> and <math>c.</math> (Here the dot \cdot represents the usual multiplication operation.) The solution to the equation <math>2016 \diamondsuit (6 \diamondsuit x) = 100</math> can be written as <math>\frac{p}{q},</math> where <math>p</math> and <math>q</math> are relativelt prime positive integers. What is <math>p + q?</math> <br />
<br />
<math>\textbf{(A)}\ 109\qquad\textbf{(B)}\ 201\qquad\textbf{(C)}\ 301\qquad\textbf{(D)}\ 3049\qquad\textbf{(E)}\ 33,601</math><br />
<br />
[[2016 AMC 12A Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 21|Solution]]<br />
<br />
<br />
==Problem 22==<br />
<br />
How many ordered triples <math>(x,y,z)</math> of positive integers satisfy <math>\text{lcm}(x,y) = 72, \text{lcm}(x,z) = 600</math> and <math>\text{lcm}(y,z)=900</math>?<br />
<br />
<math>\textbf{(A)}\ 15\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 64</math><br />
<br />
[[2016 AMC 12A Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 23|Solution]]<br />
<br />
<br />
==Problem 24==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 24|Solution]]<br />
<br />
<br />
==Problem 25==<br />
<br />
Let <math>k</math> be a positive integer. Bernardo and Silvia take turns writing and erasing numbers on a blackboard as follows: Bernardo starts by writing the smallest perfect square with <math>k+1</math> digits. Every time Bernardo writes a number, Silvia erases the last <math>k</math> digits of it. Bernardo then writes the next perfect square, Silvia erases the last <math>k</math> digits of it, and this process continues until the last two numbers that remain on the board differ by at least 2. Let <math>f(k)</math> be the smallest positive integer not written on the board. For example, if <math>k = 1</math>, then the numbers that Bernardo writes are <math>16, 25, 36, 49, 64</math>, and the numbers showing on the board after Silvia erases are <math>1, 2, 3, 4,</math> and <math>6</math>, and thus <math>f(1) = 5</math>. What is the sum of the digits of <math>f(2) + f(4)+ f(6) + ... + f(2016)</math>?<br />
<br />
<math>\textbf{(A)}\ 7986\qquad\textbf{(B)}\ 8002\qquad\textbf{(C)}\ 8030\qquad\textbf{(D)}\ 8048\qquad\textbf{(E)}\ 8064</math><br />
<br />
[[2016 AMC 12A Problems/Problem 25|Solution]]<br />
<br />
{{MAA Notice}}</div>Sssuperm1ch4elhttps://artofproblemsolving.com/wiki/index.php?title=2016_AMC_12A_Problems&diff=753812016 AMC 12A Problems2016-02-04T16:56:16Z<p>Sssuperm1ch4el: /* Problem 19 */</p>
<hr />
<div>{{AMC12 Problems|year=2016|ab=A}}<br />
<br />
==Problem 1==<br />
<br />
What is the value of <math>\frac{11!-10!}{9!}</math>?<br />
<br />
<math>\textbf{(A)}\ 99\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 110\qquad\textbf{(D)}\ 121\qquad\textbf{(E)}\ 132</math><br />
<br />
[[2016 AMC 12A Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
<br />
For what value of <math>x</math> does <math>10^x\cdot100^{2x}=1000^5</math>?<br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5</math><br />
<br />
[[2016 AMC 12A Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
The remainder can be defined for all real numbers <math>x</math> and <math>y</math> with <math>y \neq 0</math> by <cmath>\text{rem} (x ,y)=x-y\left \lfloor \frac{x}{y} \right \rfloor</cmath>where <math>\left \lfloor \tfrac{x}{y} \right \rfloor</math> denotes the greatest integer less than or equal to <math>\tfrac{x}{y}</math>. What is the value of <math>\text{rem} (\tfrac{3}{8}, -\tfrac{2}{5} )</math>?<br />
<br />
<math>\textbf{(A) } -\frac{3}{8} \qquad \textbf{(B) } -\frac{1}{40} \qquad \textbf{(C) } 0 \qquad \textbf{(D) } \frac{3}{8} \qquad \textbf{(E) } \frac{31}{40}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
The mean, median, and mode of the <math>7</math> data values <math>60, 100, x, 40, 50, 200, 90</math> are all equal to <math>x</math>. What is the value of <math>x</math>?<br />
<br />
<math>\textbf{(A)}\ 50\qquad\textbf{(B)}\ 60\qquad\textbf{(C)}\ 75\qquad\textbf{(D)}\ 90\qquad\textbf{(E)}\ 100</math><br />
<br />
[[2016 AMC 12A Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
Goldbach's conjecture states that every even integer greater than 2 can be written as the sum of two prime numbers (for example, <math>2016=13+2003</math>). So far, no one has been able to prove that the conjecture is true, and no one has found a counterexample to show that the conjecture is false. What would a counterexample consist of?<br />
<br />
<math> \textbf{(A)}\ \text{an odd integer greater than } 2 \text{ that can be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(B)}\ \text{an odd integer greater than } 2 \text{ that cannot be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(C)}\ \text{an even integer greater than } 2 \text{ that can be written as the sum of two numbers that are not prime}\\<br />
\qquad\textbf{(D)}\ \text{an even integer greater than } 2 \text{ that can be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(E)}\ \text{an even integer greater than } 2 \text{ that cannot be written as the sum of two prime numbers}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
A triangular array of 2016 coins in the first row, 2 coins in the second row, 3 coins in the third row, and so on up to <math>N</math> coins in the <math>N</math>th row. What is the sum of the digits of <math>N</math> ?<br />
<br />
<math>\textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 10</math><br />
<br />
[[2016 AMC 12A Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
Which of these describes the graph of <math>x^2(x+y+1)=y^2(x+y+1)</math> ?<br />
<br />
<math> \textbf{(A)}\ \text{two parallel lines}\\<br />
\qquad\textbf{(B)}\ \text{two intersecting lines}\\<br />
\qquad\textbf{(C)}\ \text{three lines that all pass through a common point}\\<br />
\qquad\textbf{(D)}\ \text{three lines that do not all pass through a comment point}\\<br />
\qquad\textbf{(E)}\ \text{a line and a parabola}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
What is the area of the shaded reigon of the given <math>8\times 5</math> rectangle?<br />
<br />
<asy><br />
<br />
size(6cm);<br />
defaultpen(fontsize(9pt));<br />
draw((0,0)--(8,0)--(8,5)--(0,5)--cycle);<br />
filldraw((7,0)--(8,0)--(8,1)--(0,4)--(0,5)--(1,5)--cycle,gray(0.8));<br />
<br />
label("$1$",(1/2,5),dir(90));<br />
label("$7$",(9/2,5),dir(90));<br />
<br />
label("$1$",(8,1/2),dir(0));<br />
label("$4$",(8,3),dir(0));<br />
<br />
label("$1$",(15/2,0),dir(270));<br />
label("$7$",(7/2,0),dir(270));<br />
<br />
label("$1$",(0,9/2),dir(180));<br />
label("$4$",(0,2),dir(180));<br />
<br />
</asy><br />
<br />
<math>\textbf{(A)}\ 4\dfrac{3}{4}\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 5\dfrac{1}{4}\qquad\textbf{(D)}\ 6\dfrac{1}{2}\qquad\textbf{(E)}\ 8</math><br />
<br />
[[2016 AMC 12A Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
The five small shaded squares inside this unit square are congruent and have disjoint interiors. The midpoint of each side of the middle square coincides with one of the vertices of the other four small squares as shown. The common side length is <math>\tfrac{a-\sqrt{2}}{b}</math>, where <math>a</math> and <math>b</math> are positive integers. What is <math>a+b</math> ?<br />
<br />
<pre style="color: gray">TODO: Diagram</pre><br />
<br />
<math>\textbf{(A)}\ 7\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 11</math><br />
<br />
[[2016 AMC 12A Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
Five friends sat in a movie theater in a row containing <math>5</math> seats, numbered <math>1</math> to <math>5</math> from left to right. (The directions "left" and "right" are from the point of view of the people as they sit in the seats.) During the movie Ada went to the lobby to get some popcorn. When she returned, she found that Bea had moved two seats to the right, Ceci had moved one seat to the left, and Dee and Edie had switched seats, leaving an end seat for Ada. In which seat had Ada been sitting before she got up?<br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5</math><br />
<br />
[[2016 AMC 12A Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
Each of the 100 students in a certain summer camp can either sing, dance, or act. Some students have more than one talent, but no student has all three talents. There are 42 students who cannot sing, 65 students who cannot dance, and 29 students who cannot act. How many students have two of these talents?<br />
<br />
<math>\textbf{(A)}\ 16\qquad\textbf{(B)}\ 25\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 49\qquad\textbf{(E)}\ 64</math><br />
<br />
[[2016 AMC 12A Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
In <math>\triangle ABC</math>, <math>AB = 6</math>, <math>BC = 7</math>, and <math>CA = 8</math>. Point <math>D</math> lies on <math>\overline{BC}</math>, and <math>\overline{AD}</math> bisects <math>\angle BAC</math>. Point <math>E</math> lies on <math>\overline{AC}</math>, and <math>\overline{BE}</math> bisects <math>\angle ABC</math>. The bisectors intersect at <math>F</math>. What is the ratio <math>AF</math> : <math>FD</math>?<br />
<br />
<pre style="color: gray">TODO: Diagram</pre><br />
<br />
<math>\textbf{(A)}\ 3:2\qquad\textbf{(B)}\ 5:3\qquad\textbf{(C)}\ 2:1\qquad\textbf{(D)}\ 7:3\qquad\textbf{(E)}\ 5:2</math><br />
<br />
[[2016 AMC 12A Problems/Problem 12|Solution]]<br />
<br />
<br />
==Problem 13==<br />
<br />
Let <math>N</math> be a positive multiple of <math>5</math>. One red ball and <math>N</math> green balls are arranged in a line in random order. Let <math>P(N)</math> be the probability that at least <math>\tfrac{3}{5}</math> of the green balls are on the same side of the red ball. Observe that <math>P(5)=1</math> and that <math>P(N)</math> approaches <math>\tfrac{4}{5}</math> as <math>N</math> grows large. What is the sum of the digits of the least value of <math>N</math> such that <math>P(N) < \tfrac{321}{400}</math>?<br />
<br />
<math>\textbf{(A)}\ 12\qquad\textbf{(B)}\ 14\qquad\textbf{(C)}\ 16\qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 20</math><br />
<br />
[[2016 AMC 12A Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
Each vertex of a cube is to be labeled with an integer from 1 through 8, 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 />
[[2016 AMC 12A Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
Circles with centers <math>P, Q</math> and <math>R</math>, having radii <math>1, 2</math> and <math>3</math>, respectively, lie on the same side of line <math>l</math> and are tangent to <math>l</math> at <math>P', Q'</math> and <math>R'</math>, respectively, with <math>Q'</math> between <math>P'</math> and <math>R'</math>. The circle with center <math>Q</math> is externally tangent to each of the other two circles. What is the area of triangle <math>PQR</math>?<br />
<br />
<math>\textbf{(A) } 0\qquad \textbf{(B) } \sqrt{\frac{2}{3}}\qquad\textbf{(C) } 1\qquad\textbf{(D) } \sqrt{6}-\sqrt{2}\qquad\textbf{(E) }\sqrt{\frac{3}{2}}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
The graphs of <math>y=\log_3 x, y=\log_x 3, y=\log_\frac{1}{3} x,</math> and <math>y=\log_x \dfrac{1}{3}</math> are plotted on the same set of axes. How many points in the plane with positive <math>x</math>-coordinates lie on two or more of the graphs? <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 />
[[2016 AMC 12A Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Let <math>ABCD</math> be a square. Let <math>E, F, G</math> and <math>H</math> be the centers, respectively, of equilateral triangles with bases <math>\overline{AB}, \overline{BC}, \overline{CD},</math> and <math>\overline{DA},</math> each exterior to the square. What is rge ration of the area of square <math>EFGH</math> to the area of square <math>ABCD</math>? <br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ \frac{2+\sqrt{3}}{3} \qquad\textbf{(C)}\ \sqrt{2} \qquad\textbf{(D)}\ \frac{\sqrt{2}+\sqrt{3}}{2} \qquad\textbf{(E)}\ \sqrt{3}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
For some positive integer <math>n,</math> the number <math>110n^3</math> has <math>110</math> positive integer divisors, including <math>1</math> and the number <math>110n^3.</math> How many positive integer divisors does the number <math>81n^4</math> have? <br />
<br />
<math>\textbf{(A)}\ 110\qquad\textbf{(B)}\ 191\qquad\textbf{(C)}\ 261\qquad\textbf{(D)}\ 325\qquad\textbf{(E)}\ 425</math><br />
<br />
[[2016 AMC 12A Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
Jerry starts at <math>0</math> on the real number line. He tosses a fair coin <math>8</math> times. When he gets heads, he moves <math>1</math> unit in the positive direction; when he gets tails, he moves <math>1</math> unit in the negative direction. The probability that he reaches <math>4</math> at some time during this process <math>\frac{a}{b},</math> where <math>a</math> and <math>b</math> are relatively prime positive integers. What is <math>a + b?</math> (For example, he succeeds if his sequence of tosses is <math>HTHHHHHH.</math>)<br />
<br />
<math>\textbf{(A)}\ 69\qquad\textbf{(B)}\ 151\qquad\textbf{(C)}\ 257\qquad\textbf{(D)}\ 293\qquad\textbf{(E)}\ 313</math><br />
<br />
[[2016 AMC 12A Problems/Problem 19|Solution]]<br />
<br />
<br />
==Problem 20==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 20|Solution]]<br />
<br />
<br />
==Problem 21==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 21|Solution]]<br />
<br />
<br />
==Problem 22==<br />
<br />
How many ordered triples <math>(x,y,z)</math> of positive integers satisfy <math>\text{lcm}(x,y) = 72, \text{lcm}(x,z) = 600</math> and <math>\text{lcm}(y,z)=900</math>?<br />
<br />
<math>\textbf{(A)}\ 15\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 64</math><br />
<br />
[[2016 AMC 12A Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 23|Solution]]<br />
<br />
<br />
==Problem 24==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 24|Solution]]<br />
<br />
<br />
==Problem 25==<br />
<br />
Let <math>k</math> be a positive integer. Bernardo and Silvia take turns writing and erasing numbers on a blackboard as follows: Bernardo starts by writing the smallest perfect square with <math>k+1</math> digits. Every time Bernardo writes a number, Silvia erases the last <math>k</math> digits of it. Bernardo then writes the next perfect square, Silvia erases the last <math>k</math> digits of it, and this process continues until the last two numbers that remain on the board differ by at least 2. Let <math>f(k)</math> be the smallest positive integer not written on the board. For example, if <math>k = 1</math>, then the numbers that Bernardo writes are <math>16, 25, 36, 49, 64</math>, and the numbers showing on the board after Silvia erases are <math>1, 2, 3, 4,</math> and <math>6</math>, and thus <math>f(1) = 5</math>. What is the sum of the digits of <math>f(2) + f(4)+ f(6) + ... + f(2016)</math>?<br />
<br />
<math>\textbf{(A)}\ 7986\qquad\textbf{(B)}\ 8002\qquad\textbf{(C)}\ 8030\qquad\textbf{(D)}\ 8048\qquad\textbf{(E)}\ 8064</math><br />
<br />
[[2016 AMC 12A Problems/Problem 25|Solution]]<br />
<br />
{{MAA Notice}}</div>Sssuperm1ch4elhttps://artofproblemsolving.com/wiki/index.php?title=2016_AMC_12A_Problems&diff=753702016 AMC 12A Problems2016-02-04T16:46:28Z<p>Sssuperm1ch4el: /* Problem 17 */</p>
<hr />
<div>{{AMC12 Problems|year=2016|ab=A}}<br />
<br />
==Problem 1==<br />
<br />
What is the value of <math>\frac{11!-10!}{9!}</math>?<br />
<br />
<math>\textbf{(A)}\ 99\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 110\qquad\textbf{(D)}\ 121\qquad\textbf{(E)}\ 132</math><br />
<br />
[[2016 AMC 12A Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
<br />
For what value of <math>x</math> does <math>10^x\cdot100^{2x}=1000^5</math>?<br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5</math><br />
<br />
[[2016 AMC 12A Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
<br />
The remainder function can be defined for all real numbers <math>x</math> and <math>y</math> with <math>y\neq 0</math> by<br />
<br />
<math>rem(x,y)=x-y\bigg\lfloor \dfrac{x}{y} \bigg\rfloor</math>,<br />
<br />
where <math>\Big\lfloor \tfrac{x}{y} \Big\rfloor</math> denotes the greatest integer less than or equal to <math>\tfrac{x}{y}</math>. What is the value of <math>rem(\tfrac{3}{8} , -\tfrac{2}{5})</math> ?<br />
<br />
<math>\textbf{(A)}\ -\dfrac{3}{8}\qquad\textbf{(B)}\ -\dfrac{1}{40}\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ \dfrac{3}{8}\qquad\textbf{(E)}\ \dfrac{31}{40}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
The mean, median, and mode of the 7 data values <math>60, 100, x, 40, 50, 200, 90</math> are all equal to <math>x</math>. What is the value of <math>x</math>?<br />
<br />
<math>\textbf{(A)}\ 50\qquad\textbf{(B)}\ 60\qquad\textbf{(C)}\ 75\qquad\textbf{(D)}\ 90\qquad\textbf{(E)}\ 100</math><br />
<br />
[[2016 AMC 12A Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
Goldbach's conjecture states that every even integer greater than 2 can be written as the sum of two prime numbers (for example, <math>2016=13+2003</math>). So far, no one has been able to prove that the conjecture is true, and no one has found a counterexample to show that the conjecture is false. What would a counterexample consist of?<br />
<br />
<math> \textbf{(A)}\ \text{an odd integer greater than } 2 \text{ that can be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(B)}\ \text{an odd integer greater than } 2 \text{ that cannot be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(C)}\ \text{an even integer greater than } 2 \text{ that can be written as the sum of two numbers that are not prime}\\<br />
\qquad\textbf{(D)}\ \text{an even integer greater than } 2 \text{ that can be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(E)}\ \text{an even integer greater than } 2 \text{ that cannot be written as the sum of two prime numbers}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
A triangular array of 2016 coins in the first row, 2 coins in the second row, 3 coins in the third row, and so on up to <math>N</math> coins in the <math>N</math>th row. What is the sum of the digits of <math>N</math> ?<br />
<br />
<math>\textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 10</math><br />
<br />
[[2016 AMC 12A Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
Which of these describes the graph of <math>x^2(x+y+1)=y^2(x+y+1)</math> ?<br />
<br />
<math> \textbf{(A)}\ \text{two parallel lines}\\<br />
\qquad\textbf{(B)}\ \text{two intersecting lines}\\<br />
\qquad\textbf{(C)}\ \text{three lines that all pass through a common point}\\<br />
\qquad\textbf{(D)}\ \text{three lines that do not all pass through a comment point}\\<br />
\qquad\textbf{(E)}\ \text{a line and a parabola}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
What is the area of the shaded reigon of the given <math>8\times 5</math> rectangle?<br />
<br />
<asy><br />
<br />
size(6cm);<br />
defaultpen(fontsize(9pt));<br />
draw((0,0)--(8,0)--(8,5)--(0,5)--cycle);<br />
filldraw((7,0)--(8,0)--(8,1)--(0,4)--(0,5)--(1,5)--cycle,gray(0.8));<br />
<br />
label("$1$",(1/2,5),dir(90));<br />
label("$7$",(9/2,5),dir(90));<br />
<br />
label("$1$",(8,1/2),dir(0));<br />
label("$4$",(8,3),dir(0));<br />
<br />
label("$1$",(15/2,0),dir(270));<br />
label("$7$",(7/2,0),dir(270));<br />
<br />
label("$1$",(0,9/2),dir(180));<br />
label("$4$",(0,2),dir(180));<br />
<br />
</asy><br />
<br />
<math>\textbf{(A)}\ 4\dfrac{3}{4}\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 5\dfrac{1}{4}\qquad\textbf{(D)}\ 6\dfrac{1}{2}\qquad\textbf{(E)}\ 8</math><br />
<br />
[[2016 AMC 12A Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
The five small shaded squares inside this unit square are congruent and have disjoint interiors. The midpoint of each side of the middle square coincides with one of the vertices of the other four small squares as shown. The common side length is <math>\tfrac{a-\sqrt{2}}{b}</math>, where <math>a</math> and <math>b</math> are positive integers. What is <math>a+b</math> ?<br />
<br />
<pre style="color: gray">TODO: Diagram</pre><br />
<br />
<math>\textbf{(A)}\ 7\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 11</math><br />
<br />
[[2016 AMC 12A Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
Five friends sat in a movie theater in a row containing <math>5</math> seats, numbered <math>1</math> to <math>5</math> from left to right. (The directions "left" and "right" are from the point of view of the people as they sit in the seats.) During the movie Ada went to the lobby to get some popcorn. When she returned, she found that Bea had moved two seats to the right, Ceci had moved one seat to the left, and Dee and Edie had switched seats, leaving an end seat for Ada. In which seat had Ada been sitting before she got up?<br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5</math><br />
<br />
[[2016 AMC 12A Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
Each of the 100 students in a certain summer camp can either sing, dance, or act. Some students have more than one talent, but no student has all three talents. There are 42 students who cannot sing, 65 students who cannot dance, and 29 students who cannot act. How many students have two of these talents?<br />
<br />
<math>\textbf{(A)}\ 16\qquad\textbf{(B)}\ 25\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 49\qquad\textbf{(E)}\ 64</math><br />
<br />
[[2016 AMC 12A Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
In <math>\triangle ABC</math>, <math>AB = 6</math>, <math>BC = 7</math>, and <math>CA = 8</math>. Point <math>D</math> lies on <math>\overline{BC}</math>, and <math>\overline{AD}</math> bisects <math>\angle BAC</math>. Point <math>E</math> lies on <math>\overline{AC}</math>, and <math>\overline{BE}</math> bisects <math>\angle ABC</math>. The bisectors intersect at <math>F</math>. What is the ratio <math>AF</math> : <math>FD</math>?<br />
<br />
<pre style="color: gray">TODO: Diagram</pre><br />
<br />
<math>\textbf{(A)}\ 3:2\qquad\textbf{(B)}\ 5:3\qquad\textbf{(C)}\ 2:1\qquad\textbf{(D)}\ 7:3\qquad\textbf{(E)}\ 5:2</math><br />
<br />
[[2016 AMC 12A Problems/Problem 12|Solution]]<br />
<br />
<br />
==Problem 13==<br />
<br />
Let <math>N</math> be a positive multiple of <math>5</math>. One red ball and <math>N</math> green balls are arranged in a line in random order. Let <math>P(N)</math> be the probability that at least <math>\tfrac{3}{5}</math> of the green balls are on the same side of the red ball. Observe that <math>P(5)=1</math> and that <math>P(N)</math> approaches <math>\tfrac{4}{5}</math> as <math>N</math> grows large. What is the sum of the digits of the least value of <math>N</math> such that <math>P(N) < \tfrac{321}{400}</math>?<br />
<br />
<math>\textbf{(A)}\ 12\qquad\textbf{(B)}\ 14\qquad\textbf{(C)}\ 16\qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 20</math><br />
<br />
[[2016 AMC 12A Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
Each vertex of a cube is to be labeled with an integer from 1 through 8, 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 />
[[2016 AMC 12A Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
Circles with centers <math>P, Q</math> and <math>R</math>, having radii <math>1, 2</math> and <math>3</math>, respectively, lie on the same side of line <math>l</math> and are tangent to <math>l</math> at <math>P', Q'</math> and <math>R'</math>, respectively, with <math>Q'</math> between <math>P'</math> and <math>R'</math>. The circle with center <math>Q</math> is externally tangent to each of the other two circles. What is the area of triangle <math>PQR</math>?<br />
<br />
<math>\textbf{(A) } 0\qquad \textbf{(B) } \sqrt{\frac{2}{3}}\qquad\textbf{(C) } 1\qquad\textbf{(D) } \sqrt{6}-\sqrt{2}\qquad\textbf{(E) }\sqrt{\frac{3}{2}}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
The graphs of <math>y=\log_3 x, y=\log_x 3, y=\log_\frac{1}{3} x,</math> and <math>y=\log_x \dfrac{1}{3}</math> are plotted on the same set of axes. How many points in the plane with positive <math>x</math>-coordinates lie on two or more of the graphs? <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 />
[[2016 AMC 12A Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Let <math>ABCD</math> be a square. Let <math>E, F, G</math> and <math>H</math> be the centers, respectively, of equilateral triangles with bases <math>\overline{AB}, \overline{BC}, \overline{CD},</math> and <math>\overline{DA},</math> each exterior to the square. What is rge ration of the area of square <math>EFGH</math> to the area of square <math>ABCD</math>? <br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ \frac{2+\sqrt{3}}{3} \qquad\textbf{(C)}\ \sqrt{2} \qquad\textbf{(D)}\ \frac{\sqrt{2}+\sqrt{3}}{2} \qquad\textbf{(E)}\ \sqrt{3}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
For some positive integer <math>n,</math> the number <math>110n^3</math> has <math>110</math> positive integer divisors, including <math>1</math> and the number <math>110n^3.</math> How many positive integer divisors does the number <math>81n^4</math> have? <br />
<br />
<math>\textbf{(A)}\ 110\qquad\textbf{(B)}\ 191\qquad\textbf{(C)}\ 261\qquad\textbf{(D)}\ 325\qquad\textbf{(E)}\ 425</math><br />
<br />
[[2016 AMC 12A Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 19|Solution]]<br />
<br />
<br />
==Problem 20==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 20|Solution]]<br />
<br />
<br />
==Problem 21==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 21|Solution]]<br />
<br />
<br />
==Problem 22==<br />
<br />
How many ordered triples <math>(x,y,z)</math> of positive integers satisfy <math>\text{lcm}(x,y) = 72, \text{lcm}(x,z) = 600</math> and <math>\text{lcm}(y,z)=900</math>?<br />
<br />
<math>\textbf{(A)}\ 15\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 64</math><br />
<br />
[[2016 AMC 12A Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 23|Solution]]<br />
<br />
<br />
==Problem 24==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 24|Solution]]<br />
<br />
<br />
==Problem 25==<br />
<br />
Let <math>k</math> be a positive integer. Bernardo and Silvia take turns writing and erasing numbers on a blackboard as follows: Bernardo starts by writing the smallest perfect square with <math>k+1</math> digits. Every time Bernardo writes a number, Silvia erases the last <math>k</math> digits of it. Bernardo then writes the next perfect square, Silvia erases the last <math>k</math> digits of it, and this process continues until the last two numbers that remain on the board differ by at least 2. Let <math>f(k)</math> be the smallest positive integer not written on the board. For example, if <math>k = 1</math>, then the numbers that Bernardo writes are <math>16, 25, 36, 49, 64</math>, and the numbers showing on the board after Silvia erases are <math>1, 2, 3, 4,</math> and <math>6</math>, and thus <math>f(1) = 5</math>. What is the sum of the digits of <math>f(2) + f(4)+ f(6) + ... + f(2016)</math>?<br />
<br />
<math>\textbf{(A)}\ 7986\qquad\textbf{(B)}\ 8002\qquad\textbf{(C)}\ 8030\qquad\textbf{(D)}\ 8048\qquad\textbf{(E)}\ 8064</math><br />
<br />
[[2016 AMC 12A Problems/Problem 25|Solution]]<br />
<br />
{{MAA Notice}}</div>Sssuperm1ch4elhttps://artofproblemsolving.com/wiki/index.php?title=2016_AMC_12A_Problems&diff=753662016 AMC 12A Problems2016-02-04T16:45:12Z<p>Sssuperm1ch4el: /* Problem 17 */</p>
<hr />
<div>{{AMC12 Problems|year=2016|ab=A}}<br />
<br />
==Problem 1==<br />
<br />
What is the value of <math>\frac{11!-10!}{9!}</math>?<br />
<br />
<math>\textbf{(A)}\ 99\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 110\qquad\textbf{(D)}\ 121\qquad\textbf{(E)}\ 132</math><br />
<br />
[[2016 AMC 12A Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
<br />
For what value of <math>x</math> does <math>10^x\cdot100^{2x}=1000^5</math>?<br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5</math><br />
<br />
[[2016 AMC 12A Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
<br />
The remainder function can be defined for all real numbers <math>x</math> and <math>y</math> with <math>y\neq 0</math> by<br />
<br />
<math>rem(x,y)=x-y\bigg\lfloor \dfrac{x}{y} \bigg\rfloor</math>,<br />
<br />
where <math>\Big\lfloor \tfrac{x}{y} \Big\rfloor</math> denotes the greatest integer less than or equal to <math>\tfrac{x}{y}</math>. What is the value of <math>rem(\tfrac{3}{8} , -\tfrac{2}{5})</math> ?<br />
<br />
<math>\textbf{(A)}\ -\dfrac{3}{8}\qquad\textbf{(B)}\ -\dfrac{1}{40}\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ \dfrac{3}{8}\qquad\textbf{(E)}\ \dfrac{31}{40}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
The mean, median, and mode of the 7 data values <math>60, 100, x, 40, 50, 200, 90</math> are all equal to <math>x</math>. What is the value of <math>x</math>?<br />
<br />
<math>\textbf{(A)}\ 50\qquad\textbf{(B)}\ 60\qquad\textbf{(C)}\ 75\qquad\textbf{(D)}\ 90\qquad\textbf{(E)}\ 100</math><br />
<br />
[[2016 AMC 12A Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
Goldbach's conjecture states that every even integer greater than 2 can be written as the sum of two prime numbers (for example, <math>2016=13+2003</math>). So far, no one has been able to prove that the conjecture is true, and no one has found a counterexample to show that the conjecture is false. What would a counterexample consist of?<br />
<br />
<math> \textbf{(A)}\ \text{an odd integer greater than } 2 \text{ that can be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(B)}\ \text{an odd integer greater than } 2 \text{ that cannot be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(C)}\ \text{an even integer greater than } 2 \text{ that can be written as the sum of two numbers that are not prime}\\<br />
\qquad\textbf{(D)}\ \text{an even integer greater than } 2 \text{ that can be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(E)}\ \text{an even integer greater than } 2 \text{ that cannot be written as the sum of two prime numbers}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
A triangular array of 2016 coins in the first row, 2 coins in the second row, 3 coins in the third row, and so on up to <math>N</math> coins in the <math>N</math>th row. What is the sum of the digits of <math>N</math> ?<br />
<br />
<math>\textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 10</math><br />
<br />
[[2016 AMC 12A Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
Which of these describes the graph of <math>x^2(x+y+1)=y^2(x+y+1)</math> ?<br />
<br />
<math> \textbf{(A)}\ \text{two parallel lines}\\<br />
\qquad\textbf{(B)}\ \text{two intersecting lines}\\<br />
\qquad\textbf{(C)}\ \text{three lines that all pass through a common point}\\<br />
\qquad\textbf{(D)}\ \text{three lines that do not all pass through a comment point}\\<br />
\qquad\textbf{(E)}\ \text{a line and a parabola}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
What is the area of the shaded reigon of the given <math>8\times 5</math> rectangle?<br />
<br />
<asy><br />
<br />
size(6cm);<br />
defaultpen(fontsize(9pt));<br />
draw((0,0)--(8,0)--(8,5)--(0,5)--cycle);<br />
filldraw((7,0)--(8,0)--(8,1)--(0,4)--(0,5)--(1,5)--cycle,gray(0.8));<br />
<br />
label("$1$",(1/2,5),dir(90));<br />
label("$7$",(9/2,5),dir(90));<br />
<br />
label("$1$",(8,1/2),dir(0));<br />
label("$4$",(8,3),dir(0));<br />
<br />
label("$1$",(15/2,0),dir(270));<br />
label("$7$",(7/2,0),dir(270));<br />
<br />
label("$1$",(0,9/2),dir(180));<br />
label("$4$",(0,2),dir(180));<br />
<br />
</asy><br />
<br />
<math>\textbf{(A)}\ 4\dfrac{3}{4}\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 5\dfrac{1}{4}\qquad\textbf{(D)}\ 6\dfrac{1}{2}\qquad\textbf{(E)}\ 8</math><br />
<br />
[[2016 AMC 12A Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
The five small shaded squares inside this unit square are congruent and have disjoint interiors. The midpoint of each side of the middle square coincides with one of the vertices of the other four small squares as shown. The common side length is <math>\tfrac{a-\sqrt{2}}{b}</math>, where <math>a</math> and <math>b</math> are positive integers. What is <math>a+b</math> ?<br />
<br />
<pre style="color: gray">TODO: Diagram</pre><br />
<br />
<math>\textbf{(A)}\ 7\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 11</math><br />
<br />
[[2016 AMC 12A Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
Five friends sat in a movie theater in a row containing <math>5</math> seats, numbered <math>1</math> to <math>5</math> from left to right. (The directions "left" and "right" are from the point of view of the people as they sit in the seats.) During the movie Ada went to the lobby to get some popcorn. When she returned, she found that Bea had moved two seats to the right, Ceci had moved one seat to the left, and Dee and Edie had switched seats, leaving an end seat for Ada. In which seat had Ada been sitting before she got up?<br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5</math><br />
<br />
[[2016 AMC 12A Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
Each of the 100 students in a certain summer camp can either sing, dance, or act. Some students have more than one talent, but no student has all three talents. There are 42 students who cannot sing, 65 students who cannot dance, and 29 students who cannot act. How many students have two of these talents?<br />
<br />
<math>\textbf{(A)}\ 16\qquad\textbf{(B)}\ 25\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 49\qquad\textbf{(E)}\ 64</math><br />
<br />
[[2016 AMC 12A Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
In <math>\triangle ABC</math>, <math>AB = 6</math>, <math>BC = 7</math>, and <math>CA = 8</math>. Point <math>D</math> lies on <math>\overline{BC}</math>, and <math>\overline{AD}</math> bisects <math>\angle BAC</math>. Point <math>E</math> lies on <math>\overline{AC}</math>, and <math>\overline{BE}</math> bisects <math>\angle ABC</math>. The bisectors intersect at <math>F</math>. What is the ratio <math>AF</math> : <math>FD</math>?<br />
<br />
<pre style="color: gray">TODO: Diagram</pre><br />
<br />
<math>\textbf{(A)}\ 3:2\qquad\textbf{(B)}\ 5:3\qquad\textbf{(C)}\ 2:1\qquad\textbf{(D)}\ 7:3\qquad\textbf{(E)}\ 5:2</math><br />
<br />
[[2016 AMC 12A Problems/Problem 12|Solution]]<br />
<br />
<br />
==Problem 13==<br />
<br />
Let <math>N</math> be a positive multiple of <math>5</math>. One red ball and <math>N</math> green balls are arranged in a line in random order. Let <math>P(N)</math> be the probability that at least <math>\tfrac{3}{5}</math> of the green balls are on the same side of the red ball. Observe that <math>P(5)=1</math> and that <math>P(N)</math> approaches <math>\tfrac{4}{5}</math> as <math>N</math> grows large. What is the sum of the digits of the least value of <math>N</math> such that <math>P(N) < \tfrac{321}{400}</math>?<br />
<br />
<math>\textbf{(A)}\ 12\qquad\textbf{(B)}\ 14\qquad\textbf{(C)}\ 16\qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 20</math><br />
<br />
[[2016 AMC 12A Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
Each vertex of a cube is to be labeled with an integer from 1 through 8, 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 />
[[2016 AMC 12A Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
Circles with centers <math>P, Q</math> and <math>R</math>, having radii <math>1, 2</math> and <math>3</math>, respectively, lie on the same side of line <math>l</math> and are tangent to <math>l</math> at <math>P', Q'</math> and <math>R'</math>, respectively, with <math>Q'</math> between <math>P'</math> and <math>R'</math>. The circle with center <math>Q</math> is externally tangent to each of the other two circles. What is the area of triangle <math>PQR</math>?<br />
<br />
<math>\textbf{(A) } 0\qquad \textbf{(B) } \sqrt{\frac{2}{3}}\qquad\textbf{(C) } 1\qquad\textbf{(D) } \sqrt{6}-\sqrt{2}\qquad\textbf{(E) }\sqrt{\frac{3}{2}}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
The graphs of <math>y=\log_3 x, y=\log_x 3, y=\log_\frac{1}{3} x,</math> and <math>y=\log_x \dfrac{1}{3}</math> are plotted on the same set of axes. How many points in the plane with positive <math>x</math>-coordinates lie on two or more of the graphs? <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 />
[[2016 AMC 12A Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Let <math>ABCD</math> be a square. Let <math>E, F, G</math> and <math>H</math> be the centers, respectively, of equilateral triangles with bases <math>\overline{AB}, \overline{BC}, \overline{CD},</math> and <math>\overline{DA},</math> each exterior to the square. What is rge ration of the area of square <math>EFGH</math> to the area of square <math>ABCD</math>? <br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\frac{2+\sqrt{3}}{3} \qquad\textbf{(C)}\sqrt{2} \qquad\textbf{(D)}\frac{\sqrt{2}+\sqrt{3}}{2} \qquad\textbf{(E)}\sqrt{3}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
For some positive integer <math>n,</math> the number <math>110n^3</math> has <math>110</math> positive integer divisors, including <math>1</math> and the number <math>110n^3.</math> How many positive integer divisors does the number <math>81n^4</math> have? <br />
<br />
<math>\textbf{(A)}\ 110\qquad\textbf{(B)}\ 191\qquad\textbf{(C)}\ 261\qquad\textbf{(D)}\ 325\qquad\textbf{(E)}\ 425</math><br />
<br />
[[2016 AMC 12A Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 19|Solution]]<br />
<br />
<br />
==Problem 20==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 20|Solution]]<br />
<br />
<br />
==Problem 21==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 21|Solution]]<br />
<br />
<br />
==Problem 22==<br />
<br />
How many ordered triples <math>(x,y,z)</math> of positive integers satisfy <math>\text{lcm}(x,y) = 72, \text{lcm}(x,z) = 600</math> and <math>\text{lcm}(y,z)=900</math>?<br />
<br />
<math>\textbf{(A)}\ 15\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 64</math><br />
<br />
[[2016 AMC 12A Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 23|Solution]]<br />
<br />
<br />
==Problem 24==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 24|Solution]]<br />
<br />
<br />
==Problem 25==<br />
<br />
Let <math>k</math> be a positive integer. Bernardo and Silvia take turns writing and erasing numbers on a blackboard as follows: Bernardo starts by writing the smallest perfect square with <math>k+1</math> digits. Every time Bernardo writes a number, Silvia erases the last <math>k</math> digits of it. Bernardo then writes the next perfect square, Silvia erases the last <math>k</math> digits of it, and this process continues until the last two numbers that remain on the board differ by at least 2. Let <math>f(k)</math> be the smallest positive integer not written on the board. For example, if <math>k = 1</math>, then the numbers that Bernardo writes are <math>16, 25, 36, 49, 64</math>, and the numbers showing on the board after Silvia erases are <math>1, 2, 3, 4,</math> and <math>6</math>, and thus <math>f(1) = 5</math>. What is the sum of the digits of <math>f(2) + f(4)+ f(6) + ... + f(2016)</math>?<br />
<br />
<math>\textbf{(A)}\ 7986\qquad\textbf{(B)}\ 8002\qquad\textbf{(C)}\ 8030\qquad\textbf{(D)}\ 8048\qquad\textbf{(E)}\ 8064</math><br />
<br />
[[2016 AMC 12A Problems/Problem 25|Solution]]<br />
<br />
{{MAA Notice}}</div>Sssuperm1ch4elhttps://artofproblemsolving.com/wiki/index.php?title=2016_AMC_12A_Problems&diff=753652016 AMC 12A Problems2016-02-04T16:44:39Z<p>Sssuperm1ch4el: /* Problem 17 */</p>
<hr />
<div>{{AMC12 Problems|year=2016|ab=A}}<br />
<br />
==Problem 1==<br />
<br />
What is the value of <math>\frac{11!-10!}{9!}</math>?<br />
<br />
<math>\textbf{(A)}\ 99\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 110\qquad\textbf{(D)}\ 121\qquad\textbf{(E)}\ 132</math><br />
<br />
[[2016 AMC 12A Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
<br />
For what value of <math>x</math> does <math>10^x\cdot100^{2x}=1000^5</math>?<br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5</math><br />
<br />
[[2016 AMC 12A Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
<br />
The remainder function can be defined for all real numbers <math>x</math> and <math>y</math> with <math>y\neq 0</math> by<br />
<br />
<math>rem(x,y)=x-y\bigg\lfloor \dfrac{x}{y} \bigg\rfloor</math>,<br />
<br />
where <math>\Big\lfloor \tfrac{x}{y} \Big\rfloor</math> denotes the greatest integer less than or equal to <math>\tfrac{x}{y}</math>. What is the value of <math>rem(\tfrac{3}{8} , -\tfrac{2}{5})</math> ?<br />
<br />
<math>\textbf{(A)}\ -\dfrac{3}{8}\qquad\textbf{(B)}\ -\dfrac{1}{40}\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ \dfrac{3}{8}\qquad\textbf{(E)}\ \dfrac{31}{40}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
The mean, median, and mode of the 7 data values <math>60, 100, x, 40, 50, 200, 90</math> are all equal to <math>x</math>. What is the value of <math>x</math>?<br />
<br />
<math>\textbf{(A)}\ 50\qquad\textbf{(B)}\ 60\qquad\textbf{(C)}\ 75\qquad\textbf{(D)}\ 90\qquad\textbf{(E)}\ 100</math><br />
<br />
[[2016 AMC 12A Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
Goldbach's conjecture states that every even integer greater than 2 can be written as the sum of two prime numbers (for example, <math>2016=13+2003</math>). So far, no one has been able to prove that the conjecture is true, and no one has found a counterexample to show that the conjecture is false. What would a counterexample consist of?<br />
<br />
<math> \textbf{(A)}\ \text{an odd integer greater than } 2 \text{ that can be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(B)}\ \text{an odd integer greater than } 2 \text{ that cannot be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(C)}\ \text{an even integer greater than } 2 \text{ that can be written as the sum of two numbers that are not prime}\\<br />
\qquad\textbf{(D)}\ \text{an even integer greater than } 2 \text{ that can be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(E)}\ \text{an even integer greater than } 2 \text{ that cannot be written as the sum of two prime numbers}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
A triangular array of 2016 coins in the first row, 2 coins in the second row, 3 coins in the third row, and so on up to <math>N</math> coins in the <math>N</math>th row. What is the sum of the digits of <math>N</math> ?<br />
<br />
<math>\textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 10</math><br />
<br />
[[2016 AMC 12A Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
Which of these describes the graph of <math>x^2(x+y+1)=y^2(x+y+1)</math> ?<br />
<br />
<math> \textbf{(A)}\ \text{two parallel lines}\\<br />
\qquad\textbf{(B)}\ \text{two intersecting lines}\\<br />
\qquad\textbf{(C)}\ \text{three lines that all pass through a common point}\\<br />
\qquad\textbf{(D)}\ \text{three lines that do not all pass through a comment point}\\<br />
\qquad\textbf{(E)}\ \text{a line and a parabola}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
What is the area of the shaded reigon of the given <math>8\times 5</math> rectangle?<br />
<br />
<asy><br />
<br />
size(6cm);<br />
defaultpen(fontsize(9pt));<br />
draw((0,0)--(8,0)--(8,5)--(0,5)--cycle);<br />
filldraw((7,0)--(8,0)--(8,1)--(0,4)--(0,5)--(1,5)--cycle,gray(0.8));<br />
<br />
label("$1$",(1/2,5),dir(90));<br />
label("$7$",(9/2,5),dir(90));<br />
<br />
label("$1$",(8,1/2),dir(0));<br />
label("$4$",(8,3),dir(0));<br />
<br />
label("$1$",(15/2,0),dir(270));<br />
label("$7$",(7/2,0),dir(270));<br />
<br />
label("$1$",(0,9/2),dir(180));<br />
label("$4$",(0,2),dir(180));<br />
<br />
</asy><br />
<br />
<math>\textbf{(A)}\ 4\dfrac{3}{4}\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 5\dfrac{1}{4}\qquad\textbf{(D)}\ 6\dfrac{1}{2}\qquad\textbf{(E)}\ 8</math><br />
<br />
[[2016 AMC 12A Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
The five small shaded squares inside this unit square are congruent and have disjoint interiors. The midpoint of each side of the middle square coincides with one of the vertices of the other four small squares as shown. The common side length is <math>\tfrac{a-\sqrt{2}}{b}</math>, where <math>a</math> and <math>b</math> are positive integers. What is <math>a+b</math> ?<br />
<br />
<pre style="color: gray">TODO: Diagram</pre><br />
<br />
<math>\textbf{(A)}\ 7\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 11</math><br />
<br />
[[2016 AMC 12A Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
Five friends sat in a movie theater in a row containing <math>5</math> seats, numbered <math>1</math> to <math>5</math> from left to right. (The directions "left" and "right" are from the point of view of the people as they sit in the seats.) During the movie Ada went to the lobby to get some popcorn. When she returned, she found that Bea had moved two seats to the right, Ceci had moved one seat to the left, and Dee and Edie had switched seats, leaving an end seat for Ada. In which seat had Ada been sitting before she got up?<br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5</math><br />
<br />
[[2016 AMC 12A Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
Each of the 100 students in a certain summer camp can either sing, dance, or act. Some students have more than one talent, but no student has all three talents. There are 42 students who cannot sing, 65 students who cannot dance, and 29 students who cannot act. How many students have two of these talents?<br />
<br />
<math>\textbf{(A)}\ 16\qquad\textbf{(B)}\ 25\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 49\qquad\textbf{(E)}\ 64</math><br />
<br />
[[2016 AMC 12A Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
In <math>\triangle ABC</math>, <math>AB = 6</math>, <math>BC = 7</math>, and <math>CA = 8</math>. Point <math>D</math> lies on <math>\overline{BC}</math>, and <math>\overline{AD}</math> bisects <math>\angle BAC</math>. Point <math>E</math> lies on <math>\overline{AC}</math>, and <math>\overline{BE}</math> bisects <math>\angle ABC</math>. The bisectors intersect at <math>F</math>. What is the ratio <math>AF</math> : <math>FD</math>?<br />
<br />
<pre style="color: gray">TODO: Diagram</pre><br />
<br />
<math>\textbf{(A)}\ 3:2\qquad\textbf{(B)}\ 5:3\qquad\textbf{(C)}\ 2:1\qquad\textbf{(D)}\ 7:3\qquad\textbf{(E)}\ 5:2</math><br />
<br />
[[2016 AMC 12A Problems/Problem 12|Solution]]<br />
<br />
<br />
==Problem 13==<br />
<br />
Let <math>N</math> be a positive multiple of <math>5</math>. One red ball and <math>N</math> green balls are arranged in a line in random order. Let <math>P(N)</math> be the probability that at least <math>\tfrac{3}{5}</math> of the green balls are on the same side of the red ball. Observe that <math>P(5)=1</math> and that <math>P(N)</math> approaches <math>\tfrac{4}{5}</math> as <math>N</math> grows large. What is the sum of the digits of the least value of <math>N</math> such that <math>P(N) < \tfrac{321}{400}</math>?<br />
<br />
<math>\textbf{(A)}\ 12\qquad\textbf{(B)}\ 14\qquad\textbf{(C)}\ 16\qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 20</math><br />
<br />
[[2016 AMC 12A Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
Each vertex of a cube is to be labeled with an integer from 1 through 8, 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 />
[[2016 AMC 12A Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
Circles with centers <math>P, Q</math> and <math>R</math>, having radii <math>1, 2</math> and <math>3</math>, respectively, lie on the same side of line <math>l</math> and are tangent to <math>l</math> at <math>P', Q'</math> and <math>R'</math>, respectively, with <math>Q'</math> between <math>P'</math> and <math>R'</math>. The circle with center <math>Q</math> is externally tangent to each of the other two circles. What is the area of triangle <math>PQR</math>?<br />
<br />
<math>\textbf{(A) } 0\qquad \textbf{(B) } \sqrt{\frac{2}{3}}\qquad\textbf{(C) } 1\qquad\textbf{(D) } \sqrt{6}-\sqrt{2}\qquad\textbf{(E) }\sqrt{\frac{3}{2}}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
The graphs of <math>y=\log_3 x, y=\log_x 3, y=\log_\frac{1}{3} x,</math> and <math>y=\log_x \dfrac{1}{3}</math> are plotted on the same set of axes. How many points in the plane with positive <math>x</math>-coordinates lie on two or more of the graphs? <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 />
[[2016 AMC 12A Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Let <math>ABCD</math> be a square. Let <math>E, F, G</math> and <math>H</math> be the centers, respectively, of equilateral triangles with bases <math>\overline{AB}, \overline{BC}, \overline{CD},</math> and <math>\overline{DA},</math> each exterior to the square. What is rge ration of the area of square <math>EFGH</math> to the area of square <math>ABCD</math>? <br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\frac{2+\sqrt{3}}{3} \qquad\textbf{(C)}\sqrt{2} \qquad\textbf{(D)}\frac{sqrt{2}+\sqrt{3}}{2} \qquad\textbf{(E)}\ sqrt{3}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
For some positive integer <math>n,</math> the number <math>110n^3</math> has <math>110</math> positive integer divisors, including <math>1</math> and the number <math>110n^3.</math> How many positive integer divisors does the number <math>81n^4</math> have? <br />
<br />
<math>\textbf{(A)}\ 110\qquad\textbf{(B)}\ 191\qquad\textbf{(C)}\ 261\qquad\textbf{(D)}\ 325\qquad\textbf{(E)}\ 425</math><br />
<br />
[[2016 AMC 12A Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 19|Solution]]<br />
<br />
<br />
==Problem 20==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 20|Solution]]<br />
<br />
<br />
==Problem 21==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 21|Solution]]<br />
<br />
<br />
==Problem 22==<br />
<br />
How many ordered triples <math>(x,y,z)</math> of positive integers satisfy <math>\text{lcm}(x,y) = 72, \text{lcm}(x,z) = 600</math> and <math>\text{lcm}(y,z)=900</math>?<br />
<br />
<math>\textbf{(A)}\ 15\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 64</math><br />
<br />
[[2016 AMC 12A Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 23|Solution]]<br />
<br />
<br />
==Problem 24==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 24|Solution]]<br />
<br />
<br />
==Problem 25==<br />
<br />
Let <math>k</math> be a positive integer. Bernardo and Silvia take turns writing and erasing numbers on a blackboard as follows: Bernardo starts by writing the smallest perfect square with <math>k+1</math> digits. Every time Bernardo writes a number, Silvia erases the last <math>k</math> digits of it. Bernardo then writes the next perfect square, Silvia erases the last <math>k</math> digits of it, and this process continues until the last two numbers that remain on the board differ by at least 2. Let <math>f(k)</math> be the smallest positive integer not written on the board. For example, if <math>k = 1</math>, then the numbers that Bernardo writes are <math>16, 25, 36, 49, 64</math>, and the numbers showing on the board after Silvia erases are <math>1, 2, 3, 4,</math> and <math>6</math>, and thus <math>f(1) = 5</math>. What is the sum of the digits of <math>f(2) + f(4)+ f(6) + ... + f(2016)</math>?<br />
<br />
<math>\textbf{(A)}\ 7986\qquad\textbf{(B)}\ 8002\qquad\textbf{(C)}\ 8030\qquad\textbf{(D)}\ 8048\qquad\textbf{(E)}\ 8064</math><br />
<br />
[[2016 AMC 12A Problems/Problem 25|Solution]]<br />
<br />
{{MAA Notice}}</div>Sssuperm1ch4elhttps://artofproblemsolving.com/wiki/index.php?title=2016_AMC_12A_Problems&diff=753602016 AMC 12A Problems2016-02-04T16:35:57Z<p>Sssuperm1ch4el: /* Problem 16 */</p>
<hr />
<div>{{AMC12 Problems|year=2016|ab=A}}<br />
<br />
==Problem 1==<br />
<br />
What is the value of <math>\frac{11!-10!}{9!}</math>?<br />
<br />
<math>\textbf{(A)}\ 99\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 110\qquad\textbf{(D)}\ 121\qquad\textbf{(E)}\ 132</math><br />
<br />
[[2016 AMC 12A Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
<br />
For what value of <math>x</math> does <math>10^x\cdot100^{2x}=1000^5</math>?<br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5</math><br />
<br />
[[2016 AMC 12A Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
<br />
The remainder function can be defined for all real numbers <math>x</math> and <math>y</math> with <math>y\neq 0</math> by<br />
<br />
<math>rem(x,y)=x-y\bigg\lfloor \dfrac{x}{y} \bigg\rfloor</math>,<br />
<br />
where <math>\Big\lfloor \tfrac{x}{y} \Big\rfloor</math> denotes the greatest integer less than or equal to <math>\tfrac{x}{y}</math>. What is the value of <math>rem(\tfrac{3}{8} , -\tfrac{2}{5})</math> ?<br />
<br />
<math>\textbf{(A)}\ -\dfrac{3}{8}\qquad\textbf{(B)}\ -\dfrac{1}{40}\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ \dfrac{3}{8}\qquad\textbf{(E)}\ \dfrac{31}{40}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
The mean, median, and mode of the 7 data values <math>60, 100, x, 40, 50, 200, 90</math> are all equal to <math>x</math>. What is the value of <math>x</math>?<br />
<br />
<math>\textbf{(A)}\ 50\qquad\textbf{(B)}\ 60\qquad\textbf{(C)}\ 75\qquad\textbf{(D)}\ 90\qquad\textbf{(E)}\ 100</math><br />
<br />
[[2016 AMC 12A Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
Goldbach's conjecture states that every even integer greater than 2 can be written as the sum of two prime numbers (for example, <math>2016=13+2003</math>). So far, no one has been able to prove that the conjecture is true, and no one has found a counterexample to show that the conjecture is false. What would a counterexample consist of?<br />
<br />
<math> \textbf{(A)}\ \text{an odd integer greater than } 2 \text{ that can be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(B)}\ \text{an odd integer greater than } 2 \text{ that cannot be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(C)}\ \text{an even integer greater than } 2 \text{ that can be written as the sum of two numbers that are not prime}\\<br />
\qquad\textbf{(D)}\ \text{an even integer greater than } 2 \text{ that can be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(E)}\ \text{an even integer greater than } 2 \text{ that cannot be written as the sum of two prime numbers}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
A triangular array of 2016 coins in the first row, 2 coins in the second row, 3 coins in the third row, and so on up to <math>N</math> coins in the <math>N</math>th row. What is the sum of the digits of <math>N</math> ?<br />
<br />
<math>\textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 10</math><br />
<br />
[[2016 AMC 12A Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
Which of these describes the graph of <math>x^2(x+y+1)=y^2(x+y+1)</math> ?<br />
<br />
<math> \textbf{(A)}\ \text{two parallel lines}\\<br />
\qquad\textbf{(B)}\ \text{two intersecting lines}\\<br />
\qquad\textbf{(C)}\ \text{three lines that all pass through a common point}\\<br />
\qquad\textbf{(D)}\ \text{three lines that do not all pass through a comment point}\\<br />
\qquad\textbf{(E)}\ \text{a line and a parabola}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
What is the area of the shaded reigon of the given <math>8\times 5</math> rectangle?<br />
<br />
<asy><br />
<br />
size(6cm);<br />
defaultpen(fontsize(9pt));<br />
draw((0,0)--(8,0)--(8,5)--(0,5)--cycle);<br />
filldraw((7,0)--(8,0)--(8,1)--(0,4)--(0,5)--(1,5)--cycle,gray(0.8));<br />
<br />
label("$1$",(1/2,5),dir(90));<br />
label("$7$",(9/2,5),dir(90));<br />
<br />
label("$1$",(8,1/2),dir(0));<br />
label("$4$",(8,3),dir(0));<br />
<br />
label("$1$",(15/2,0),dir(270));<br />
label("$7$",(7/2,0),dir(270));<br />
<br />
label("$1$",(0,9/2),dir(180));<br />
label("$4$",(0,2),dir(180));<br />
<br />
</asy><br />
<br />
<math>\textbf{(A)}\ 4\dfrac{3}{4}\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 5\dfrac{1}{4}\qquad\textbf{(D)}\ 6\dfrac{1}{2}\qquad\textbf{(E)}\ 8</math><br />
<br />
[[2016 AMC 12A Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
The five small shaded squares inside this unit square are congruent and have disjoint interiors. The midpoint of each side of the middle square coincides with one of the vertices of the other four small squares as shown. The common side length is <math>\tfrac{a-\sqrt{2}}{b}</math>, where <math>a</math> and <math>b</math> are positive integers. What is <math>a+b</math> ?<br />
<br />
<pre style="color: gray">TODO: Diagram</pre><br />
<br />
<math>\textbf{(A)}\ 7\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 11</math><br />
<br />
[[2016 AMC 12A Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
Five friends sat in a movie theater in a row containing <math>5</math> seats, numbered <math>1</math> to <math>5</math> from left to right. (The directions "left" and "right" are from the point of view of the people as they sit in the seats.) During the movie Ada went to the lobby to get some popcorn. When she returned, she found that Bea had moved two seats to the right, Ceci had moved one seat to the left, and Dee and Edie had switched seats, leaving an end seat for Ada. In which seat had Ada been sitting before she got up?<br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5</math><br />
<br />
[[2016 AMC 12A Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
Each of the 100 students in a certain summer camp can either sing, dance, or act. Some students have more than one talent, but no student has all three talents. There are 42 students who cannot sing, 65 students who cannot dance, and 29 students who cannot act. How many students have two of these talents?<br />
<br />
<math>\textbf{(A)}\ 16\qquad\textbf{(B)}\ 25\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 49\qquad\textbf{(E)}\ 64</math><br />
<br />
[[2016 AMC 12A Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
In <math>\triangle ABC</math>, <math>AB = 6</math>, <math>BC = 7</math>, and <math>CA = 8</math>. Point <math>D</math> lies on <math>\overline{BC}</math>, and <math>\overline{AD}</math> bisects <math>\angle BAC</math>. Point <math>E</math> lies on <math>\overline{AC}</math>, and <math>\overline{BE}</math> bisects <math>\angle ABC</math>. The bisectors intersect at <math>F</math>. What is the ratio <math>AF</math> : <math>FD</math>?<br />
<br />
<pre style="color: gray">TODO: Diagram</pre><br />
<br />
<math>\textbf{(A)}\ 3:2\qquad\textbf{(B)}\ 5:3\qquad\textbf{(C)}\ 2:1\qquad\textbf{(D)}\ 7:3\qquad\textbf{(E)}\ 5:2</math><br />
<br />
[[2016 AMC 12A Problems/Problem 12|Solution]]<br />
<br />
<br />
==Problem 13==<br />
<br />
Let <math>N</math> be a positive multiple of <math>5</math>. One red ball and <math>N</math> green balls are arranged in a line in random order. Let <math>P(N)</math> be the probability that at least <math>\tfrac{3}{5}</math> of the green balls are on the same side of the red ball. Observe that <math>P(5)=1</math> and that <math>P(N)</math> approaches <math>\tfrac{4}{5}</math> as <math>N</math> grows large. What is the sum of the digits of the least value of <math>N</math> such that <math>P(N) < \tfrac{321}{400}</math>?<br />
<br />
<math>\textbf{(A)}\ 12\qquad\textbf{(B)}\ 14\qquad\textbf{(C)}\ 16\qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 20</math><br />
<br />
[[2016 AMC 12A Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
Each vertex of a cube is to be labeled with an integer from 1 through 8, 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 />
[[2016 AMC 12A Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
Circles with centers <math>P, Q</math> and <math>R</math>, having radii <math>1, 2</math> and <math>3</math>, respectively, lie on the same side of line <math>l</math> and are tangent to <math>l</math> at <math>P', Q'</math> and <math>R'</math>, respectively, with <math>Q'</math> between <math>P'</math> and <math>R'</math>. The circle with center <math>Q</math> is externally tangent to each of the other two circles. What is the area of triangle <math>PQR</math>?<br />
<br />
<math>\textbf{(A) } 0\qquad \textbf{(B) } \sqrt{\frac{2}{3}}\qquad\textbf{(C) } 1\qquad\textbf{(D) } \sqrt{6}-\sqrt{2}\qquad\textbf{(E) }\sqrt{\frac{3}{2}}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
The graphs of <math>y=\log_3 x, y=\log_x 3, y=\log_\frac{1}{3} x,</math> and <math>y=\log_x \dfrac{1}{3}</math> are plotted on the same set of axes. How many points in the plane with positive <math>x</math>-coordinates lie on two or more of the graphs? <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 />
[[2016 AMC 12A Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 17|Solution]]<br />
<br />
<br />
==Problem 18==<br />
<br />
For some positive integer <math>n,</math> the number <math>110n^3</math> has <math>110</math> positive integer divisors, including <math>1</math> and the number <math>110n^3.</math> How many positive integer divisors does the number <math>81n^4</math> have? <br />
<br />
<math>\textbf{(A)}\ 110\qquad\textbf{(B)}\ 191\qquad\textbf{(C)}\ 261\qquad\textbf{(D)}\ 325\qquad\textbf{(E)}\ 425</math><br />
<br />
[[2016 AMC 12A Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 19|Solution]]<br />
<br />
<br />
==Problem 20==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 20|Solution]]<br />
<br />
<br />
==Problem 21==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 21|Solution]]<br />
<br />
<br />
==Problem 22==<br />
<br />
How many ordered triples <math>(x,y,z)</math> of positive integers satisfy <math>\text{lcm}(x,y) = 72, \text{lcm}(x,z) = 600</math> and <math>\text{lcm}(y,z)=900</math>?<br />
<br />
<math>\textbf{(A)}\ 15\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 64</math><br />
<br />
[[2016 AMC 12A Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 23|Solution]]<br />
<br />
<br />
==Problem 24==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 24|Solution]]<br />
<br />
<br />
==Problem 25==<br />
<br />
Let <math>k</math> be a positive integer. Bernardo and Silvia take turns writing and erasing numbers on a blackboard as follows: Bernardo starts by writing the smallest perfect square with <math>k+1</math> digits. Every time Bernardo writes a number, Silvia erases the last <math>k</math> digits of it. Bernardo then writes the next perfect square, Silvia erases the last <math>k</math> digits of it, and this process continues until the last two numbers that remain on the board differ by at least 2. Let <math>f(k)</math> be the smallest positive integer not written on the board. For example, if <math>k = 1</math>, then the numbers that Bernardo writes are <math>16, 25, 36, 49, 64</math>, and the numbers showing on the board after Silvia erases are <math>1, 2, 3, 4,</math> and <math>6</math>, and thus <math>f(1) = 5</math>. What is the sum of the digits of <math>f(2) + f(4)+ f(6) + ... + f(2016)</math>?<br />
<br />
<math>\textbf{(A)}\ 7986\qquad\textbf{(B)}\ 8002\qquad\textbf{(C)}\ 8030\qquad\textbf{(D)}\ 8048\qquad\textbf{(E)}\ 8064</math><br />
<br />
[[2016 AMC 12A Problems/Problem 25|Solution]]<br />
<br />
{{MAA Notice}}</div>Sssuperm1ch4elhttps://artofproblemsolving.com/wiki/index.php?title=2016_AMC_12A_Problems&diff=753592016 AMC 12A Problems2016-02-04T16:31:23Z<p>Sssuperm1ch4el: /* Problem 16 */</p>
<hr />
<div>{{AMC12 Problems|year=2016|ab=A}}<br />
<br />
==Problem 1==<br />
<br />
What is the value of <math>\frac{11!-10!}{9!}</math>?<br />
<br />
<math>\textbf{(A)}\ 99\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 110\qquad\textbf{(D)}\ 121\qquad\textbf{(E)}\ 132</math><br />
<br />
[[2016 AMC 12A Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
<br />
For what value of <math>x</math> does <math>10^x\cdot100^{2x}=1000^5</math>?<br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5</math><br />
<br />
[[2016 AMC 12A Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
<br />
The remainder function can be defined for all real numbers <math>x</math> and <math>y</math> with <math>y\neq 0</math> by<br />
<br />
<math>rem(x,y)=x-y\bigg\lfloor \dfrac{x}{y} \bigg\rfloor</math>,<br />
<br />
where <math>\Big\lfloor \tfrac{x}{y} \Big\rfloor</math> denotes the greatest integer less than or equal to <math>\tfrac{x}{y}</math>. What is the value of <math>rem(\tfrac{3}{8} , -\tfrac{2}{5})</math> ?<br />
<br />
<math>\textbf{(A)}\ -\dfrac{3}{8}\qquad\textbf{(B)}\ -\dfrac{1}{40}\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ \dfrac{3}{8}\qquad\textbf{(E)}\ \dfrac{31}{40}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
The mean, median, and mode of the 7 data values <math>60, 100, x, 40, 50, 200, 90</math> are all equal to <math>x</math>. What is the value of <math>x</math>?<br />
<br />
<math>\textbf{(A)}\ 50\qquad\textbf{(B)}\ 60\qquad\textbf{(C)}\ 75\qquad\textbf{(D)}\ 90\qquad\textbf{(E)}\ 100</math><br />
<br />
[[2016 AMC 12A Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
Goldbach's conjecture states that every even integer greater than 2 can be written as the sum of two prime numbers (for example, <math>2016=13+2003</math>). So far, no one has been able to prove that the conjecture is true, and no one has found a counterexample to show that the conjecture is false. What would a counterexample consist of?<br />
<br />
<math> \textbf{(A)}\ \text{an odd integer greater than } 2 \text{ that can be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(B)}\ \text{an odd integer greater than } 2 \text{ that cannot be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(C)}\ \text{an even integer greater than } 2 \text{ that can be written as the sum of two numbers that are not prime}\\<br />
\qquad\textbf{(D)}\ \text{an even integer greater than } 2 \text{ that can be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(E)}\ \text{an even integer greater than } 2 \text{ that cannot be written as the sum of two prime numbers}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
A triangular array of 2016 coins in the first row, 2 coins in the second row, 3 coins in the third row, and so on up to <math>N</math> coins in the <math>N</math>th row. What is the sum of the digits of <math>N</math> ?<br />
<br />
<math>\textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 10</math><br />
<br />
[[2016 AMC 12A Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
Which of these describes the graph of <math>x^2(x+y+1)=y^2(x+y+1)</math> ?<br />
<br />
<math> \textbf{(A)}\ \text{two parallel lines}\\<br />
\qquad\textbf{(B)}\ \text{two intersecting lines}\\<br />
\qquad\textbf{(C)}\ \text{three lines that all pass through a common point}\\<br />
\qquad\textbf{(D)}\ \text{three lines that do not all pass through a comment point}\\<br />
\qquad\textbf{(E)}\ \text{a line and a parabola}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
What is the area of the shaded reigon of the given <math>8\times 5</math> rectangle?<br />
<br />
<asy><br />
<br />
size(6cm);<br />
defaultpen(fontsize(9pt));<br />
draw((0,0)--(8,0)--(8,5)--(0,5)--cycle);<br />
filldraw((7,0)--(8,0)--(8,1)--(0,4)--(0,5)--(1,5)--cycle,gray(0.8));<br />
<br />
label("$1$",(1/2,5),dir(90));<br />
label("$7$",(9/2,5),dir(90));<br />
<br />
label("$1$",(8,1/2),dir(0));<br />
label("$4$",(8,3),dir(0));<br />
<br />
label("$1$",(15/2,0),dir(270));<br />
label("$7$",(7/2,0),dir(270));<br />
<br />
label("$1$",(0,9/2),dir(180));<br />
label("$4$",(0,2),dir(180));<br />
<br />
</asy><br />
<br />
<math>\textbf{(A)}\ 4\dfrac{3}{4}\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 5\dfrac{1}{4}\qquad\textbf{(D)}\ 6\dfrac{1}{2}\qquad\textbf{(E)}\ 8</math><br />
<br />
[[2016 AMC 12A Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
The five small shaded squares inside this unit square are congruent and have disjoint interiors. The midpoint of each side of the middle square coincides with one of the vertices of the other four small squares as shown. The common side length is <math>\tfrac{a-\sqrt{2}}{b}</math>, where <math>a</math> and <math>b</math> are positive integers. What is <math>a+b</math> ?<br />
<br />
<pre style="color: gray">TODO: Diagram</pre><br />
<br />
<math>\textbf{(A)}\ 7\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 11</math><br />
<br />
[[2016 AMC 12A Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
Five friends sat in a movie theater in a row containing <math>5</math> seats, numbered <math>1</math> to <math>5</math> from left to right. (The directions "left" and "right" are from the point of view of the people as they sit in the seats.) During the movie Ada went to the lobby to get some popcorn. When she returned, she found that Bea had moved two seats to the right, Ceci had moved one seat to the left, and Dee and Edie had switched seats, leaving an end seat for Ada. In which seat had Ada been sitting before she got up?<br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5</math><br />
<br />
[[2016 AMC 12A Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
Each of the 100 students in a certain summer camp can either sing, dance, or act. Some students have more than one talent, but no student has all three talents. There are 42 students who cannot sing, 65 students who cannot dance, and 29 students who cannot act. How many students have two of these talents?<br />
<br />
<math>\textbf{(A)}\ 16\qquad\textbf{(B)}\ 25\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 49\qquad\textbf{(E)}\ 64</math><br />
<br />
[[2016 AMC 12A Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
In <math>\triangle ABC</math>, <math>AB = 6</math>, <math>BC = 7</math>, and <math>CA = 8</math>. Point <math>D</math> lies on <math>\overline{BC}</math>, and <math>\overline{AD}</math> bisects <math>\angle BAC</math>. Point <math>E</math> lies on <math>\overline{AC}</math>, and <math>\overline{BE}</math> bisects <math>\angle ABC</math>. The bisectors intersect at <math>F</math>. What is the ratio <math>AF</math> : <math>FD</math>?<br />
<br />
<pre style="color: gray">TODO: Diagram</pre><br />
<br />
<math>\textbf{(A)}\ 3:2\qquad\textbf{(B)}\ 5:3\qquad\textbf{(C)}\ 2:1\qquad\textbf{(D)}\ 7:3\qquad\textbf{(E)}\ 5:2</math><br />
<br />
[[2016 AMC 12A Problems/Problem 12|Solution]]<br />
<br />
<br />
==Problem 13==<br />
<br />
Let <math>N</math> be a positive multiple of <math>5</math>. One red ball and <math>N</math> green balls are arranged in a line in random order. Let <math>P(N)</math> be the probability that at least <math>\tfrac{3}{5}</math> of the green balls are on the same side of the red ball. Observe that <math>P(5)=1</math> and that <math>P(N)</math> approaches <math>\tfrac{4}{5}</math> as <math>N</math> grows large. What is the sum of the digits of the least value of <math>N</math> such that <math>P(N) < \tfrac{321}{400}</math>?<br />
<br />
<math>\textbf{(A)}\ 12\qquad\textbf{(B)}\ 14\qquad\textbf{(C)}\ 16\qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 20</math><br />
<br />
[[2016 AMC 12A Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
Each vertex of a cube is to be labeled with an integer from 1 through 8, 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 />
[[2016 AMC 12A Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
Circles with centers <math>P, Q</math> and <math>R</math>, having radii <math>1, 2</math> and <math>3</math>, respectively, lie on the same side of line <math>l</math> and are tangent to <math>l</math> at <math>P', Q'</math> and <math>R'</math>, respectively, with <math>Q'</math> between <math>P'</math> and <math>R'</math>. The circle with center <math>Q</math> is externally tangent to each of the other two circles. What is the area of triangle <math>PQR</math>?<br />
<br />
<math>\textbf{(A) } 0\qquad \textbf{(B) } \sqrt{\frac{2}{3}}\qquad\textbf{(C) } 1\qquad\textbf{(D) } \sqrt{6}-\sqrt{2}\qquad\textbf{(E) }\sqrt{\frac{3}{2}}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
The graphs of <math>y=log_3 x, y=log_x 3, y=log_\frac{1}{3} x,</math> and <math>y=log_x \frac{1}{3}</math> are plotted on the same set of axes. How many points in the plane with positive <math>x</math>-coordinates lie on two or more of the graphs? <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 />
[[2016 AMC 12A Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 17|Solution]]<br />
<br />
<br />
==Problem 18==<br />
<br />
For some positive integer <math>n,</math> the number <math>110n^3</math> has <math>110</math> positive integer divisors, including <math>1</math> and the number <math>110n^3.</math> How many positive integer divisors does the number <math>81n^4</math> have? <br />
<br />
<math>\textbf{(A)}\ 110\qquad\textbf{(B)}\ 191\qquad\textbf{(C)}\ 261\qquad\textbf{(D)}\ 325\qquad\textbf{(E)}\ 425</math><br />
<br />
[[2016 AMC 12A Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 19|Solution]]<br />
<br />
<br />
==Problem 20==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 20|Solution]]<br />
<br />
<br />
==Problem 21==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 21|Solution]]<br />
<br />
<br />
==Problem 22==<br />
<br />
How many ordered triples <math>(x,y,z)</math> of positive integers satisfy <math>\text{lcm}(x,y) = 72, \text{lcm}(x,z) = 600</math> and <math>\text{lcm}(y,z)=900</math>?<br />
<br />
<math>\textbf{(A)}\ 15\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 64</math><br />
<br />
[[2016 AMC 12A Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 23|Solution]]<br />
<br />
<br />
==Problem 24==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 24|Solution]]<br />
<br />
<br />
==Problem 25==<br />
<br />
Let <math>k</math> be a positive integer. Bernardo and Silvia take turns writing and erasing numbers on a blackboard as follows: Bernardo starts by writing the smallest perfect square with <math>k+1</math> digits. Every time Bernardo writes a number, Silvia erases the last <math>k</math> digits of it. Bernardo then writes the next perfect square, Silvia erases the last <math>k</math> digits of it, and this process continues until the last two numbers that remain on the board differ by at least 2. Let <math>f(k)</math> be the smallest positive integer not written on the board. For example, if <math>k = 1</math>, then the numbers that Bernardo writes are <math>16, 25, 36, 49, 64</math>, and the numbers showing on the board after Silvia erases are <math>1, 2, 3, 4,</math> and <math>6</math>, and thus <math>f(1) = 5</math>. What is the sum of the digits of <math>f(2) + f(4)+ f(6) + ... + f(2016)</math>?<br />
<br />
<math>\textbf{(A)}\ 7986\qquad\textbf{(B)}\ 8002\qquad\textbf{(C)}\ 8030\qquad\textbf{(D)}\ 8048\qquad\textbf{(E)}\ 8064</math><br />
<br />
[[2016 AMC 12A Problems/Problem 25|Solution]]<br />
<br />
{{MAA Notice}}</div>Sssuperm1ch4elhttps://artofproblemsolving.com/wiki/index.php?title=2016_AMC_12A_Problems&diff=753582016 AMC 12A Problems2016-02-04T16:27:36Z<p>Sssuperm1ch4el: /* Problem 18 */</p>
<hr />
<div>{{AMC12 Problems|year=2016|ab=A}}<br />
<br />
==Problem 1==<br />
<br />
What is the value of <math>\frac{11!-10!}{9!}</math>?<br />
<br />
<math>\textbf{(A)}\ 99\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 110\qquad\textbf{(D)}\ 121\qquad\textbf{(E)}\ 132</math><br />
<br />
[[2016 AMC 12A Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
<br />
For what value of <math>x</math> does <math>10^x\cdot100^{2x}=1000^5</math>?<br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5</math><br />
<br />
[[2016 AMC 12A Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
<br />
The remainder function can be defined for all real numbers <math>x</math> and <math>y</math> with <math>y\neq 0</math> by<br />
<br />
<math>rem(x,y)=x-y\bigg\lfloor \dfrac{x}{y} \bigg\rfloor</math>,<br />
<br />
where <math>\Big\lfloor \tfrac{x}{y} \Big\rfloor</math> denotes the greatest integer less than or equal to <math>\tfrac{x}{y}</math>. What is the value of <math>rem(\tfrac{3}{8} , -\tfrac{2}{5})</math> ?<br />
<br />
<math>\textbf{(A)}\ -\dfrac{3}{8}\qquad\textbf{(B)}\ -\dfrac{1}{40}\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ \dfrac{3}{8}\qquad\textbf{(E)}\ \dfrac{31}{40}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
The mean, median, and mode of the 7 data values <math>60, 100, x, 40, 50, 200, 90</math> are all equal to <math>x</math>. What is the value of <math>x</math>?<br />
<br />
<math>\textbf{(A)}\ 50\qquad\textbf{(B)}\ 60\qquad\textbf{(C)}\ 75\qquad\textbf{(D)}\ 90\qquad\textbf{(E)}\ 100</math><br />
<br />
[[2016 AMC 12A Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
Goldbach's conjecture states that every even integer greater than 2 can be written as the sum of two prime numbers (for example, <math>2016=13+2003</math>). So far, no one has been able to prove that the conjecture is true, and no one has found a counterexample to show that the conjecture is false. What would a counterexample consist of?<br />
<br />
<math> \textbf{(A)}\ \text{an odd integer greater than } 2 \text{ that can be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(B)}\ \text{an odd integer greater than } 2 \text{ that cannot be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(C)}\ \text{an even integer greater than } 2 \text{ that can be written as the sum of two numbers that are not prime}\\<br />
\qquad\textbf{(D)}\ \text{an even integer greater than } 2 \text{ that can be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(E)}\ \text{an even integer greater than } 2 \text{ that cannot be written as the sum of two prime numbers}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
A triangular array of 2016 coins in the first row, 2 coins in the second row, 3 coins in the third row, and so on up to <math>N</math> coins in the <math>N</math>th row. What is the sum of the digits of <math>N</math> ?<br />
<br />
<math>\textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 10</math><br />
<br />
[[2016 AMC 12A Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
Which of these describes the graph of <math>x^2(x+y+1)=y^2(x+y+1)</math> ?<br />
<br />
<math> \textbf{(A)}\ \text{two parallel lines}\\<br />
\qquad\textbf{(B)}\ \text{two intersecting lines}\\<br />
\qquad\textbf{(C)}\ \text{three lines that all pass through a common point}\\<br />
\qquad\textbf{(D)}\ \text{three lines that do not all pass through a comment point}\\<br />
\qquad\textbf{(E)}\ \text{a line and a parabola}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
What is the area of the shaded reigon of the given <math>8\times 5</math> rectangle?<br />
<br />
<asy><br />
<br />
size(6cm);<br />
defaultpen(fontsize(9pt));<br />
draw((0,0)--(8,0)--(8,5)--(0,5)--cycle);<br />
filldraw((7,0)--(8,0)--(8,1)--(0,4)--(0,5)--(1,5)--cycle,gray(0.8));<br />
<br />
label("$1$",(1/2,5),dir(90));<br />
label("$7$",(9/2,5),dir(90));<br />
<br />
label("$1$",(8,1/2),dir(0));<br />
label("$4$",(8,3),dir(0));<br />
<br />
label("$1$",(15/2,0),dir(270));<br />
label("$7$",(7/2,0),dir(270));<br />
<br />
label("$1$",(0,9/2),dir(180));<br />
label("$4$",(0,2),dir(180));<br />
<br />
</asy><br />
<br />
<math>\textbf{(A)}\ 4\dfrac{3}{4}\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 5\dfrac{1}{4}\qquad\textbf{(D)}\ 6\dfrac{1}{2}\qquad\textbf{(E)}\ 8</math><br />
<br />
[[2016 AMC 12A Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
The five small shaded squares inside this unit square are congruent and have disjoint interiors. The midpoint of each side of the middle square coincides with one of the vertices of the other four small squares as shown. The common side length is <math>\tfrac{a-\sqrt{2}}{b}</math>, where <math>a</math> and <math>b</math> are positive integers. What is <math>a+b</math> ?<br />
<br />
<pre style="color: gray">TODO: Diagram</pre><br />
<br />
<math>\textbf{(A)}\ 7\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 11</math><br />
<br />
[[2016 AMC 12A Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
Five friends sat in a movie theater in a row containing <math>5</math> seats, numbered <math>1</math> to <math>5</math> from left to right. (The directions "left" and "right" are from the point of view of the people as they sit in the seats.) During the movie Ada went to the lobby to get some popcorn. When she returned, she found that Bea had moved two seats to the right, Ceci had moved one seat to the left, and Dee and Edie had switched seats, leaving an end seat for Ada. In which seat had Ada been sitting before she got up?<br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5</math><br />
<br />
[[2016 AMC 12A Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
Each of the 100 students in a certain summer camp can either sing, dance, or act. Some students have more than one talent, but no student has all three talents. There are 42 students who cannot sing, 65 students who cannot dance, and 29 students who cannot act. How many students have two of these talents?<br />
<br />
<math>\textbf{(A)}\ 16\qquad\textbf{(B)}\ 25\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 49\qquad\textbf{(E)}\ 64</math><br />
<br />
[[2016 AMC 12A Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
In <math>\triangle ABC</math>, <math>AB = 6</math>, <math>BC = 7</math>, and <math>CA = 8</math>. Point <math>D</math> lies on <math>\overline{BC}</math>, and <math>\overline{AD}</math> bisects <math>\angle BAC</math>. Point <math>E</math> lies on <math>\overline{AC}</math>, and <math>\overline{BE}</math> bisects <math>\angle ABC</math>. The bisectors intersect at <math>F</math>. What is the ratio <math>AF</math> : <math>FD</math>?<br />
<br />
<pre style="color: gray">TODO: Diagram</pre><br />
<br />
<math>\textbf{(A)}\ 3:2\qquad\textbf{(B)}\ 5:3\qquad\textbf{(C)}\ 2:1\qquad\textbf{(D)}\ 7:3\qquad\textbf{(E)}\ 5:2</math><br />
<br />
[[2016 AMC 12A Problems/Problem 12|Solution]]<br />
<br />
<br />
==Problem 13==<br />
<br />
Let <math>N</math> be a positive multiple of <math>5</math>. One red ball and <math>N</math> green balls are arranged in a line in random order. Let <math>P(N)</math> be the probability that at least <math>\tfrac{3}{5}</math> of the green balls are on the same side of the red ball. Observe that <math>P(5)=1</math> and that <math>P(N)</math> approaches <math>\tfrac{4}{5}</math> as <math>N</math> grows large. What is the sum of the digits of the least value of <math>N</math> such that <math>P(N) < \tfrac{321}{400}</math>?<br />
<br />
<math>\textbf{(A)}\ 12\qquad\textbf{(B)}\ 14\qquad\textbf{(C)}\ 16\qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 20</math><br />
<br />
[[2016 AMC 12A Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
Each vertex of a cube is to be labeled with an integer from 1 through 8, 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 />
[[2016 AMC 12A Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
Circles with centers <math>P, Q</math> and <math>R</math>, having radii <math>1, 2</math> and <math>3</math>, respectively, lie on the same side of line <math>l</math> and are tangent to <math>l</math> at <math>P', Q'</math> and <math>R'</math>, respectively, with <math>Q'</math> between <math>P'</math> and <math>R'</math>. The circle with center <math>Q</math> is externally tangent to each of the other two circles. What is the area of triangle <math>PQR</math>?<br />
<br />
<math>\textbf{(A) } 0\qquad \textbf{(B) } \sqrt{\frac{2}{3}}\qquad\textbf{(C) } 1\qquad\textbf{(D) } \sqrt{6}-\sqrt{2}\qquad\textbf{(E) }\sqrt{\frac{3}{2}}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
Problem text<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 />
[[2016 AMC 12A Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 17|Solution]]<br />
<br />
<br />
==Problem 18==<br />
<br />
For some positive integer <math>n,</math> the number <math>110n^3</math> has <math>110</math> positive integer divisors, including <math>1</math> and the number <math>110n^3.</math> How many positive integer divisors does the number <math>81n^4</math> have? <br />
<br />
<math>\textbf{(A)}\ 110\qquad\textbf{(B)}\ 191\qquad\textbf{(C)}\ 261\qquad\textbf{(D)}\ 325\qquad\textbf{(E)}\ 425</math><br />
<br />
[[2016 AMC 12A Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 19|Solution]]<br />
<br />
<br />
==Problem 20==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 20|Solution]]<br />
<br />
<br />
==Problem 21==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 21|Solution]]<br />
<br />
<br />
==Problem 22==<br />
<br />
How many ordered triples <math>(x,y,z)</math> of positive integers satisfy <math>\text{lcm}(x,y) = 72, \text{lcm}(x,z) = 600</math> and <math>\text{lcm}(y,z)=900</math>?<br />
<br />
<math>\textbf{(A)}\ 15\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 64</math><br />
<br />
[[2016 AMC 12A Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 23|Solution]]<br />
<br />
<br />
==Problem 24==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 24|Solution]]<br />
<br />
<br />
==Problem 25==<br />
<br />
Let <math>k</math> be a positive integer. Bernardo and Silvia take turns writing and erasing numbers on a blackboard as follows: Bernardo starts by writing the smallest perfect square with <math>k+1</math> digits. Every time Bernardo writes a number, Silvia erases the last <math>k</math> digits of it. Bernardo then writes the next perfect square, Silvia erases the last <math>k</math> digits of it, and this process continues until the last two numbers that remain on the board differ by at least 2. Let <math>f(k)</math> be the smallest positive integer not written on the board. For example, if <math>k = 1</math>, then the numbers that Bernardo writes are <math>16, 25, 36, 49, 64</math>, and the numbers showing on the board after Silvia erases are <math>1, 2, 3, 4,</math> and <math>6</math>, and thus <math>f(1) = 5</math>. What is the sum of the digits of <math>f(2) + f(4)+ f(6) + ... + f(2016)</math>?<br />
<br />
<math>\textbf{(A)}\ 7986\qquad\textbf{(B)}\ 8002\qquad\textbf{(C)}\ 8030\qquad\textbf{(D)}\ 8048\qquad\textbf{(E)}\ 8064</math><br />
<br />
[[2016 AMC 12A Problems/Problem 25|Solution]]<br />
<br />
{{MAA Notice}}</div>Sssuperm1ch4elhttps://artofproblemsolving.com/wiki/index.php?title=2016_AMC_12A_Problems&diff=753552016 AMC 12A Problems2016-02-04T16:26:28Z<p>Sssuperm1ch4el: /* Problem 18 */</p>
<hr />
<div>{{AMC12 Problems|year=2016|ab=A}}<br />
<br />
==Problem 1==<br />
<br />
What is the value of <math>\frac{11!-10!}{9!}</math>?<br />
<br />
<math>\textbf{(A)}\ 99\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 110\qquad\textbf{(D)}\ 121\qquad\textbf{(E)}\ 132</math><br />
<br />
[[2016 AMC 12A Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
<br />
For what value of <math>x</math> does <math>10^x\cdot100^{2x}=1000^5</math>?<br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5</math><br />
<br />
[[2016 AMC 12A Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
<br />
The remainder function can be defined for all real numbers <math>x</math> and <math>y</math> with <math>y\neq 0</math> by<br />
<br />
<math>rem(x,y)=x-y\bigg\lfloor \dfrac{x}{y} \bigg\rfloor</math>,<br />
<br />
where <math>\Big\lfloor \tfrac{x}{y} \Big\rfloor</math> denotes the greatest integer less than or equal to <math>\tfrac{x}{y}</math>. What is the value of <math>rem(\tfrac{3}{8} , -\tfrac{2}{5})</math> ?<br />
<br />
<math>\textbf{(A)}\ -\dfrac{3}{8}\qquad\textbf{(B)}\ -\dfrac{1}{40}\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ \dfrac{3}{8}\qquad\textbf{(E)}\ \dfrac{31}{40}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
The mean, median, and mode of the 7 data values <math>60, 100, x, 40, 50, 200, 90</math> are all equal to <math>x</math>. What is the value of <math>x</math>?<br />
<br />
<math>\textbf{(A)}\ 50\qquad\textbf{(B)}\ 60\qquad\textbf{(C)}\ 75\qquad\textbf{(D)}\ 90\qquad\textbf{(E)}\ 100</math><br />
<br />
[[2016 AMC 12A Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
Goldbach's conjecture states that every even integer greater than 2 can be written as the sum of two prime numbers (for example, <math>2016=13+2003</math>). So far, no one has been able to prove that the conjecture is true, and no one has found a counterexample to show that the conjecture is false. What would a counterexample consist of?<br />
<br />
<math> \textbf{(A)}\ \text{an odd integer greater than } 2 \text{ that can be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(B)}\ \text{an odd integer greater than } 2 \text{ that cannot be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(C)}\ \text{an even integer greater than } 2 \text{ that can be written as the sum of two numbers that are not prime}\\<br />
\qquad\textbf{(D)}\ \text{an even integer greater than } 2 \text{ that can be written as the sum of two prime numbers}\\<br />
\qquad\textbf{(E)}\ \text{an even integer greater than } 2 \text{ that cannot be written as the sum of two prime numbers}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
A triangular array of 2016 coins in the first row, 2 coins in the second row, 3 coins in the third row, and so on up to <math>N</math> coins in the <math>N</math>th row. What is the sum of the digits of <math>N</math> ?<br />
<br />
<math>\textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 10</math><br />
<br />
[[2016 AMC 12A Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
Which of these describes the graph of <math>x^2(x+y+1)=y^2(x+y+1)</math> ?<br />
<br />
<math> \textbf{(A)}\ \text{two parallel lines}\\<br />
\qquad\textbf{(B)}\ \text{two intersecting lines}\\<br />
\qquad\textbf{(C)}\ \text{three lines that all pass through a common point}\\<br />
\qquad\textbf{(D)}\ \text{three lines that do not all pass through a comment point}\\<br />
\qquad\textbf{(E)}\ \text{a line and a parabola}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
What is the area of the shaded reigon of the given <math>8\times 5</math> rectangle?<br />
<br />
<asy><br />
<br />
size(6cm);<br />
defaultpen(fontsize(9pt));<br />
draw((0,0)--(8,0)--(8,5)--(0,5)--cycle);<br />
filldraw((7,0)--(8,0)--(8,1)--(0,4)--(0,5)--(1,5)--cycle,gray(0.8));<br />
<br />
label("$1$",(1/2,5),dir(90));<br />
label("$7$",(9/2,5),dir(90));<br />
<br />
label("$1$",(8,1/2),dir(0));<br />
label("$4$",(8,3),dir(0));<br />
<br />
label("$1$",(15/2,0),dir(270));<br />
label("$7$",(7/2,0),dir(270));<br />
<br />
label("$1$",(0,9/2),dir(180));<br />
label("$4$",(0,2),dir(180));<br />
<br />
</asy><br />
<br />
<math>\textbf{(A)}\ 4\dfrac{3}{4}\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 5\dfrac{1}{4}\qquad\textbf{(D)}\ 6\dfrac{1}{2}\qquad\textbf{(E)}\ 8</math><br />
<br />
[[2016 AMC 12A Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
The five small shaded squares inside this unit square are congruent and have disjoint interiors. The midpoint of each side of the middle square coincides with one of the vertices of the other four small squares as shown. The common side length is <math>\tfrac{a-\sqrt{2}}{b}</math>, where <math>a</math> and <math>b</math> are positive integers. What is <math>a+b</math> ?<br />
<br />
<pre style="color: gray">TODO: Diagram</pre><br />
<br />
<math>\textbf{(A)}\ 7\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 11</math><br />
<br />
[[2016 AMC 12A Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
Five friends sat in a movie theater in a row containing <math>5</math> seats, numbered <math>1</math> to <math>5</math> from left to right. (The directions "left" and "right" are from the point of view of the people as they sit in the seats.) During the movie Ada went to the lobby to get some popcorn. When she returned, she found that Bea had moved two seats to the right, Ceci had moved one seat to the left, and Dee and Edie had switched seats, leaving an end seat for Ada. In which seat had Ada been sitting before she got up?<br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5</math><br />
<br />
[[2016 AMC 12A Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
Each of the 100 students in a certain summer camp can either sing, dance, or act. Some students have more than one talent, but no student has all three talents. There are 42 students who cannot sing, 65 students who cannot dance, and 29 students who cannot act. How many students have two of these talents?<br />
<br />
<math>\textbf{(A)}\ 16\qquad\textbf{(B)}\ 25\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 49\qquad\textbf{(E)}\ 64</math><br />
<br />
[[2016 AMC 12A Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
In <math>\triangle ABC</math>, <math>AB = 6</math>, <math>BC = 7</math>, and <math>CA = 8</math>. Point <math>D</math> lies on <math>\overline{BC}</math>, and <math>\overline{AD}</math> bisects <math>\angle BAC</math>. Point <math>E</math> lies on <math>\overline{AC}</math>, and <math>\overline{BE}</math> bisects <math>\angle ABC</math>. The bisectors intersect at <math>F</math>. What is the ratio <math>AF</math> : <math>FD</math>?<br />
<br />
<pre style="color: gray">TODO: Diagram</pre><br />
<br />
<math>\textbf{(A)}\ 3:2\qquad\textbf{(B)}\ 5:3\qquad\textbf{(C)}\ 2:1\qquad\textbf{(D)}\ 7:3\qquad\textbf{(E)}\ 5:2</math><br />
<br />
[[2016 AMC 12A Problems/Problem 12|Solution]]<br />
<br />
<br />
==Problem 13==<br />
<br />
Let <math>N</math> be a positive multiple of <math>5</math>. One red ball and <math>N</math> green balls are arranged in a line in random order. Let <math>P(N)</math> be the probability that at least <math>\tfrac{3}{5}</math> of the green balls are on the same side of the red ball. Observe that <math>P(5)=1</math> and that <math>P(N)</math> approaches <math>\tfrac{4}{5}</math> as <math>N</math> grows large. What is the sum of the digits of the least value of <math>N</math> such that <math>P(N) < \tfrac{321}{400}</math>?<br />
<br />
<math>\textbf{(A)}\ 12\qquad\textbf{(B)}\ 14\qquad\textbf{(C)}\ 16\qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 20</math><br />
<br />
[[2016 AMC 12A Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
Each vertex of a cube is to be labeled with an integer from 1 through 8, 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 />
[[2016 AMC 12A Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
Circles with centers <math>P, Q</math> and <math>R</math>, having radii <math>1, 2</math> and <math>3</math>, respectively, lie on the same side of line <math>l</math> and are tangent to <math>l</math> at <math>P', Q'</math> and <math>R'</math>, respectively, with <math>Q'</math> between <math>P'</math> and <math>R'</math>. The circle with center <math>Q</math> is externally tangent to each of the other two circles. What is the area of triangle <math>PQR</math>?<br />
<br />
<math>\textbf{(A) } 0\qquad \textbf{(B) } \sqrt{\frac{2}{3}}\qquad\textbf{(C) } 1\qquad\textbf{(D) } \sqrt{6}-\sqrt{2}\qquad\textbf{(E) }\sqrt{\frac{3}{2}}</math><br />
<br />
[[2016 AMC 12A Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
Problem text<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 />
[[2016 AMC 12A Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 17|Solution]]<br />
<br />
<br />
==Problem 18==<br />
<br />
For some positive integer <math>n,</math> the number <math>110n^3</math> has 110 positive integer divisors, including 1 and the number <math>110n^3.</math> How many positive integer divisors does the number <math>81n^4</math> have? <br />
<br />
<math>\textbf{(A)}\ 110\qquad\textbf{(B)}\ 191\qquad\textbf{(C)}\ 261\qquad\textbf{(D)}\ 325\qquad\textbf{(E)}\ 425</math><br />
<br />
[[2016 AMC 12A Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 19|Solution]]<br />
<br />
<br />
==Problem 20==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 20|Solution]]<br />
<br />
<br />
==Problem 21==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 21|Solution]]<br />
<br />
<br />
==Problem 22==<br />
<br />
How many ordered triples <math>(x,y,z)</math> of positive integers satisfy <math>\text{lcm}(x,y) = 72, \text{lcm}(x,z) = 600</math> and <math>\text{lcm}(y,z)=900</math>?<br />
<br />
<math>\textbf{(A)}\ 15\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 64</math><br />
<br />
[[2016 AMC 12A Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 23|Solution]]<br />
<br />
<br />
==Problem 24==<br />
<br />
Problem text<br />
<br />
<math>\textbf{(A)}\ thing\qquad\textbf{(B)}\ thing\qquad\textbf{(C)}\ thng\qquad\textbf{(D)}\ thing\qquad\textbf{(E)}\ thing</math><br />
<br />
[[2016 AMC 12A Problems/Problem 24|Solution]]<br />
<br />
<br />
==Problem 25==<br />
<br />
Let <math>k</math> be a positive integer. Bernardo and Silvia take turns writing and erasing numbers on a blackboard as follows: Bernardo starts by writing the smallest perfect square with <math>k+1</math> digits. Every time Bernardo writes a number, Silvia erases the last <math>k</math> digits of it. Bernardo then writes the next perfect square, Silvia erases the last <math>k</math> digits of it, and this process continues until the last two numbers that remain on the board differ by at least 2. Let <math>f(k)</math> be the smallest positive integer not written on the board. For example, if <math>k = 1</math>, then the numbers that Bernardo writes are <math>16, 25, 36, 49, 64</math>, and the numbers showing on the board after Silvia erases are <math>1, 2, 3, 4,</math> and <math>6</math>, and thus <math>f(1) = 5</math>. What is the sum of the digits of <math>f(2) + f(4)+ f(6) + ... + f(2016)</math>?<br />
<br />
<math>\textbf{(A)}\ 7986\qquad\textbf{(B)}\ 8002\qquad\textbf{(C)}\ 8030\qquad\textbf{(D)}\ 8048\qquad\textbf{(E)}\ 8064</math><br />
<br />
[[2016 AMC 12A Problems/Problem 25|Solution]]<br />
<br />
{{MAA Notice}}</div>Sssuperm1ch4el