https://artofproblemsolving.com/wiki/api.php?action=feedcontributions&user=Teal2048&feedformat=atomAoPS Wiki - User contributions [en]2024-03-29T14:04:12ZUser contributionsMediaWiki 1.31.1https://artofproblemsolving.com/wiki/index.php?title=2007_AIME_II_Problems&diff=1047252007 AIME II Problems2019-03-20T17:59:41Z<p>Teal2048: /* Problem 1 */</p>
<hr />
<div>{{AIME Problems|year=2007|n=II}}<br />
<br />
== Problem 1 ==<br />
A mathematical organization is producing a set of commemorative license plates. Each plate contains a sequence of five characters chosen from the four letters in AIME and the four digits in <math>2007</math>. No character may appear in a sequence more times than it appears among the four letters in AIME or the four digits in <math>2007</math>. A set of plates in which each possible sequence appears exactly once contains N license plates. Find N/10.<br />
<br />
[[2007 AIME II Problems/Problem 1|Solution]]<br />
<br />
== Problem 2 ==<br />
Find the number of ordered triples <math>(a,b,c)</math> where <math>a</math>, <math>b</math>, and <math>c</math> are positive [[integer]]s, <math>a</math> is a [[factor]] of <math>b</math>, <math>a</math> is a factor of <math>c</math>, and <math>a+b+c=100</math>.<br />
<br />
[[2007 AIME II Problems/Problem 2|Solution]]<br />
<br />
== Problem 3 ==<br />
[[Square]] <math>ABCD</math> has side length <math>13</math>, and [[point]]s <math>E</math> and <math>F</math> are exterior to the square such that <math>BE=DF=5</math> and <math>AE=CF=12</math>. Find <math>EF^{2}</math>.<br />
<br />
<div style="text-align:center;">[[Image:2007 AIME II-3.png]]</div><br />
<br />
[[2007 AIME II Problems/Problem 3|Solution]]<br />
<br />
== Problem 4 ==<br />
The workers in a factory produce widgets and whoosits. For each product, production time is [[constant]] and identical for all workers, but not necessarily equal for the two products. In one hour, <math>100</math> workers can produce <math>300</math> widgets and <math>200</math> whoosits. In two hours, <math>60</math> workers can produce <math>240</math> widgets and <math>300</math> whoosits. In three hours, <math>50</math> workers can produce <math>150</math> widgets and <math>m</math> whoosits. Find <math>m</math>.<br />
<br />
[[2007 AIME II Problems/Problem 4|Solution]]<br />
<br />
== Problem 5 ==<br />
The [[graph]] of the [[equation]] <math>9x+223y=2007</math> is drawn on graph paper with each [[square]] representing one [[unit square|unit]] in each direction. How many of the <math>1</math> by <math>1</math> graph paper squares have interiors lying entirely below the graph and entirely in the first [[quadrant]]?<br />
<br />
[[2007 AIME II Problems/Problem 5|Solution]]<br />
<br />
== Problem 6 ==<br />
An integer is called ''parity-monotonic'' if its decimal representation <math>a_{1}a_{2}a_{3}\cdots a_{k}</math> satisfies <math>a_{i}<a_{i+1}</math> if <math>a_{i}</math> is [[odd]], and <math>a_{i}>a_{i+1}</math> if <math>a_{i}</math> is [[even]]. How many four-digit parity-monotonic integers are there? <br />
<br />
[[2007 AIME II Problems/Problem 6|Solution]]<br />
<br />
== Problem 7 ==<br />
Given a [[real number]] <math>x,</math> let <math>\lfloor x \rfloor</math> denote the [[floor function|greatest integer]] less than or equal to <math>x.</math> For a certain [[integer]] <math>k,</math> there are exactly <math>70</math> positive integers <math>n_{1}, n_{2}, \ldots, n_{70}</math> such that <math>k=\lfloor\sqrt[3]{n_{1}}\rfloor = \lfloor\sqrt[3]{n_{2}}\rfloor = \cdots = \lfloor\sqrt[3]{n_{70}}\rfloor</math> and <math>k</math> divides <math>n_{i}</math> for all <math>i</math> such that <math>1 \leq i \leq 70.</math><br />
<br />
Find the maximum value of <math>\frac{n_{i}}{k}</math> for <math>1\leq i \leq 70.</math><br />
<br />
[[2007 AIME II Problems/Problem 7|Solution]]<br />
<br />
== Problem 8 ==<br />
A [[rectangle|rectangular]] piece of paper measures 4 units by 5 units. Several [[line]]s are drawn [[parallel]] to the edges of the paper. A rectangle determined by the [[intersection]]s of some of these lines is called ''basic'' if <br />
<br />
:(i) all four sides of the rectangle are segments of drawn line segments, and <br />
:(ii) no [[segment]]s of drawn lines lie inside the rectangle.<br />
<br />
Given that the total length of all lines drawn is exactly 2007 units, let <math>N</math> be the maximum possible number of basic rectangles determined. Find the [[remainder]] when <math>N</math> is divided by 1000.<br />
<br />
[[2007 AIME II Problems/Problem 8|Solution]]<br />
<br />
== Problem 9 ==<br />
[[Rectangle]] <math>ABCD</math> is given with <math>AB=63</math> and <math>BC=448.</math> Points <math>E</math> and <math>F</math> lie on <math>AD</math> and <math>BC</math> respectively, such that <math>AE=CF=84.</math> The [[inscribed circle]] of [[triangle]] <math>BEF</math> is [[tangent]] to <math>EF</math> at point <math>P,</math> and the inscribed circle of triangle <math>DEF</math> is tangent to <math>EF</math> at [[point]] <math>Q.</math> Find <math>PQ.</math><br />
<br />
[[2007 AIME II Problems/Problem 9|Solution]]<br />
<br />
== Problem 10 ==<br />
Let <math>S</math> be a [[set]] with six [[element]]s. Let <math>P</math> be the set of all [[subset]]s of <math>S.</math> Subsets <math>A</math> and <math>B</math> of <math>S</math>, not necessarily distinct, are chosen independently and at random from <math>P</math>. The [[probability]] that <math>B</math> is contained in at least one of <math>A</math> or <math>S-A</math> is <math>\frac{m}{n^{r}},</math> where <math>m</math>, <math>n</math>, and <math>r</math> are [[positive]] [[integer]]s, <math>n</math> is [[prime]], and <math>m</math> and <math>n</math> are [[relatively prime]]. Find <math>m+n+r.</math> (The set <math>S-A</math> is the set of all elements of <math>S</math> which are not in <math>A.</math>)<br />
<br />
[[2007 AIME II Problems/Problem 10|Solution]]<br />
<br />
== Problem 11 ==<br />
Two long [[cylinder|cylindrical]] tubes of the same length but different [[diameter]]s lie [[parallel]] to each other on a [[plane|flat surface]]. The larger tube has [[radius]] <math>72</math> and rolls along the surface toward the smaller tube, which has radius <math>24</math>. It rolls over the smaller tube and continues rolling along the flat surface until it comes to rest on the same point of its [[circumference]] as it started, having made one complete revolution. If the smaller tube never moves, and the rolling occurs with no slipping, the larger tube ends up a [[distance]] <math>x</math> from where it starts. The distance <math>x</math> can be expressed in the form <math>a\pi+b\sqrt{c},</math> where <math>a,</math> <math>b,</math> and <math>c</math> are [[integer]]s and <math>c</math> is not divisible by the [[square]] of any [[prime]]. Find <math>a+b+c.</math><br />
<br />
[[2007 AIME II Problems/Problem 11|Solution]]<br />
<br />
== Problem 12 ==<br />
The increasing [[geometric sequence]] <math>x_{0},x_{1},x_{2},\ldots</math> consists entirely of [[integer|integral]] powers of <math>3.</math> Given that<br />
<br />
<math>\sum_{n=0}^{7}\log_{3}(x_{n}) = 308</math> and <math>56 \leq \log_{3}\left ( \sum_{n=0}^{7}x_{n}\right ) \leq 57,</math><br />
<br />
find <math>\log_{3}(x_{14}).</math><br />
<br />
[[2007 AIME II Problems/Problem 12|Solution]]<br />
<br />
== Problem 13 ==<br />
A [[triangle|triangular]] [[array]] of [[square]]s has one square in the first row, two in the second, and in general, <math>k</math> squares in the <math>k</math>th row for <math>1 \leq k \leq 11.</math> With the exception of the bottom row, each square rests on two squares in the row immediately below (illustrated in given diagram). In each square of the eleventh row, a <math>0</math> or a <math>1</math> is placed. Numbers are then placed into the other squares, with the entry for each square being the sum of the entries in the two squares below it. For how many initial distributions of <math>0</math>'s and <math>1</math>'s in the bottom row is the number in the top square a [[multiple]] of <math>3</math>?<br />
<br />
<asy><br />
for (int i=0; i<12; ++i){<br />
for (int j=0; j<i; ++j){<br />
//dot((-j+i/2,-i));<br />
draw((-j+i/2,-i)--(-j+i/2+1,-i)--(-j+i/2+1,-i+1)--(-j+i/2,-i+1)--cycle);<br />
}<br />
}<br />
</asy><br />
<br />
[[2007 AIME II Problems/Problem 13|Solution]]<br />
<br />
== Problem 14 ==<br />
Let <math>f(x)</math> be a [[polynomial]] with real [[coefficient]]s such that <math>f(0) = 1,</math> <math>f(2)+f(3)=125,</math> and for all <math>x</math>, <math>f(x)f(2x^{2})=f(2x^{3}+x).</math> Find <math>f(5).</math><br />
<br />
[[2007 AIME II Problems/Problem 14|Solution]]<br />
<br />
== Problem 15 ==<br />
Four [[circle]]s <math>\omega,</math> <math>\omega_{A},</math> <math>\omega_{B},</math> and <math>\omega_{C}</math> with the same [[radius]] are drawn in the interior of [[triangle]] <math>ABC</math> such that <math>\omega_{A}</math> is [[tangent]] to sides <math>AB</math> and <math>AC</math>, <math>\omega_{B}</math> to <math>BC</math> and <math>BA</math>, <math>\omega_{C}</math> to <math>CA</math> and <math>CB</math>, and <math>\omega</math> is [[externally tangent]] to <math>\omega_{A},</math> <math>\omega_{B},</math> and <math>\omega_{C}</math>. If the sides of triangle <math>ABC</math> are <math>13,</math> <math>14,</math> and <math>15,</math> the radius of <math>\omega</math> can be represented in the form <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are [[relatively prime]] positive integers. Find <math>m+n.</math><br />
<br />
[[2007 AIME II Problems/Problem 15|Solution]]<br />
<br />
{{AIME box|year=2017|n=I|before=[[2007 AIME I]]|after=[[2008 AIME I]]}}<br />
{{MAA Notice}}</div>Teal2048https://artofproblemsolving.com/wiki/index.php?title=2016_AMC_12A_Problems&diff=997872016 AMC 12A Problems2018-12-27T02:38:11Z<p>Teal2048: /* Problem 2 */</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 />
For what value of <math>x</math> does <math>10^x \cdot 100^{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 />
==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 common 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 />
<asy><br />
real x=.369;<br />
draw((0,0)--(0,1)--(1,1)--(1,0)--cycle);<br />
filldraw((0,0)--(0,x)--(x,x)--(x,0)--cycle, gray);<br />
filldraw((0,1)--(0,1-x)--(x,1-x)--(x,1)--cycle, gray);<br />
filldraw((1,1)--(1,1-x)--(1-x,1-x)--(1-x,1)--cycle, gray);<br />
filldraw((1,0)--(1,x)--(1-x,x)--(1-x,0)--cycle, gray);<br />
filldraw((.5,.5-x*sqrt(2)/2)--(.5+x*sqrt(2)/2,.5)--(.5,.5+x*sqrt(2)/2)--(.5-x*sqrt(2)/2,.5)--cycle, gray);<br />
</asy><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 <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 />
[[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 />
<asy><br />
pair A = (0,0), B=(6,0), C=intersectionpoints(Circle(A,8),Circle(B,7))[0], F=incenter(A,B,C), D=extension(A,F,B,C),E=extension(B,F,A,C);<br />
draw(A--B--C--A--D^^B--E);<br />
label("$A$",A,SW);<br />
label("$B$",B,SE);<br />
label("$C$",C,N);<br />
label("$D$",D,NE);<br />
label("$E$",E,NW);<br />
label("$F$",F,1.5*N);<br />
</asy><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 />
==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 <math>1</math> through <math>8</math>, with each integer being used once, in such a way that the sum of the four numbers on the vertices of a face is the same for each face. Arrangements that can be obtained from each other through rotations of the cube are considered to be the same. How many different arrangements are possible?<br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 12\qquad\textbf{(E)}\ 24</math><br />
<br />
[[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 is <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 />
==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 relatively 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 />
==See also==<br />
<br />
{{AMC12 box|year=2016|ab=A|before=[[2015 AMC 12A Problems]]|after=[[2017 AMC 12A Problems]]}}<br />
<br />
{{MAA Notice}}<br />
<br />
- gorefeebuddie</div>Teal2048https://artofproblemsolving.com/wiki/index.php?title=2016_AMC_12A_Problems&diff=997862016 AMC 12A Problems2018-12-27T02:37:22Z<p>Teal2048: /* Problem 2 */</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 />
For what value of <math>x</math> does <math>10^x \cdot 100^{2x} = 1000^5</math>?<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 common 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 />
<asy><br />
real x=.369;<br />
draw((0,0)--(0,1)--(1,1)--(1,0)--cycle);<br />
filldraw((0,0)--(0,x)--(x,x)--(x,0)--cycle, gray);<br />
filldraw((0,1)--(0,1-x)--(x,1-x)--(x,1)--cycle, gray);<br />
filldraw((1,1)--(1,1-x)--(1-x,1-x)--(1-x,1)--cycle, gray);<br />
filldraw((1,0)--(1,x)--(1-x,x)--(1-x,0)--cycle, gray);<br />
filldraw((.5,.5-x*sqrt(2)/2)--(.5+x*sqrt(2)/2,.5)--(.5,.5+x*sqrt(2)/2)--(.5-x*sqrt(2)/2,.5)--cycle, gray);<br />
</asy><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 <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 />
[[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 />
<asy><br />
pair A = (0,0), B=(6,0), C=intersectionpoints(Circle(A,8),Circle(B,7))[0], F=incenter(A,B,C), D=extension(A,F,B,C),E=extension(B,F,A,C);<br />
draw(A--B--C--A--D^^B--E);<br />
label("$A$",A,SW);<br />
label("$B$",B,SE);<br />
label("$C$",C,N);<br />
label("$D$",D,NE);<br />
label("$E$",E,NW);<br />
label("$F$",F,1.5*N);<br />
</asy><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 />
==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 <math>1</math> through <math>8</math>, with each integer being used once, in such a way that the sum of the four numbers on the vertices of a face is the same for each face. Arrangements that can be obtained from each other through rotations of the cube are considered to be the same. How many different arrangements are possible?<br />
<br />
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 12\qquad\textbf{(E)}\ 24</math><br />
<br />
[[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 is <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 />
==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 relatively 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 />
==See also==<br />
<br />
{{AMC12 box|year=2016|ab=A|before=[[2015 AMC 12A Problems]]|after=[[2017 AMC 12A Problems]]}}<br />
<br />
{{MAA Notice}}<br />
<br />
- gorefeebuddie</div>Teal2048https://artofproblemsolving.com/wiki/index.php?title=2001_AMC_8_Problems/Problem_10&diff=985762001 AMC 8 Problems/Problem 102018-11-10T04:08:00Z<p>Teal2048: /* Problem */</p>
<hr />
<div>==Problem==<br />
<br />
A collector offers to buy state quarters for 2000% of their face value. At that rate how much will Bryden get for his four state quarters?<br />
<br />
<math>\text{(A)}\ 20\text{ dollars} \qquad \text{(B)}\ 50\text{ dollars} \qquad \text{(C)}\ 200\text{ dollars} \qquad \text{(D)}\ 500\text{ dollars} \qquad \text{(E)}\ 2000\text{ dollars}</math><br />
<br />
==Solution==<br />
<br />
<math> 2000\% </math> is equivalent to <math> 20\times100\% </math>. Therefore, <math> 2000\% </math> of a number is the same as <math> 20 </math> times that number. <math> 4 </math> quarters is <math> 1 </math> dollar, so Bryden will get <math> 20\times1={20} </math> dollars, <math> \boxed{\text{A}} </math>.<br />
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==See Also==<br />
{{AMC8 box|year=2001|num-b=9|num-a=11}}<br />
{{MAA Notice}}</div>Teal2048https://artofproblemsolving.com/wiki/index.php?title=1995_AJHSME_Problems/Problem_22&diff=981581995 AJHSME Problems/Problem 222018-10-17T03:51:27Z<p>Teal2048: /* Solution */</p>
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<div>==Problem==<br />
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The number <math>6545</math> can be written as a product of a pair of positive two-digit numbers. What is the sum of this pair of numbers?<br />
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<math>\text{(A)}\ 162 \qquad \text{(B)}\ 172 \qquad \text{(C)}\ 173 \qquad \text{(D)}\ 174 \qquad \text{(E)}\ 222</math><br />
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==Solution==<br />
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The prime factorization of <math>6545</math> is <math>5*7*11*17. 5*7*11=385</math>, which is a three digit number, so every two-digit number pair has to be two number of the form pq. Now we do trial and error: <br />
<cmath>5*7=35 \text{, } 11*17=187 \text{ X}</cmath> <cmath>5*11=55 \text{, } 7*17=119 \text{ X}</cmath> <cmath>5*17=85 \text{, } 7*11=77 \text{ }\surd </cmath> <cmath>85+77= \boxed{\text{(A)}\ 162}</cmath><br />
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==See Also==<br />
{{AJHSME box|year=1995|num-b=21|num-a=23}}</div>Teal2048