Found problems: 3632
2011 AMC 12/AHSME, 21
The arithmetic mean of two distinct positive integers $x$ and $y$ is a two-digit integer. The geometric mean of $x$ and $y$ is obtained by reversing the digits of the arithmetic mean. What is $|x-y|$?
$ \textbf{(A)}\ 24 \qquad
\textbf{(B)}\ 48 \qquad
\textbf{(C)}\ 54 \qquad
\textbf{(D)}\ 66 \qquad
\textbf{(E)}\ 70 $
2006 AMC 10, 16
Leap Day, February 29, 2004, occurred on a Sunday. On what day of the week will Leap Day, February 29, 2020, occur?
$ \textbf{(A) } \text{Tuesday} \qquad \textbf{(B) } \text{Wednesday} \qquad \textbf{(C) } \text{Thursday} \qquad \textbf{(D) } \text{Friday} \qquad \textbf{(E) } \text{Saturday}$
2025 AIME, 1
Find the sum of all integer bases $b>9$ for which $17_b$ is a divisor of $97_b.$
1983 AMC 12/AHSME, 30
Distinct points $A$ and $B$ are on a semicircle with diameter $MN$ and center $C$. The point $P$ is on $CN$ and $\angle CAP = \angle CBP = 10^{\circ}$. If $\stackrel{\frown}{MA} = 40^{\circ}$, then $\stackrel{\frown}{BN}$ equals
[asy]
size(200);
defaultpen(linewidth(0.7)+fontsize(10));
pair C=origin, N=dir(0), B=dir(20), A=dir(135), M=dir(180), P=(3/7)*dir(C--N);
draw(M--N^^C--A--P--B--C^^Arc(origin,1,0,180));
markscalefactor=0.03;
draw(anglemark(C,A,P));
draw(anglemark(C,B,P));
pair point=C;
label("$A$", A, dir(point--A));
label("$B$", B, dir(point--B));
label("$C$", C, S);
label("$M$", M, dir(point--M));
label("$N$", N, dir(point--N));
label("$P$", P, S);
label("$40^\circ$", C+(-0.15,0), NW);
label("$10^\circ$", B+(0,0.05), W);
label("$10^\circ$", A+(0.05,0.02), E);[/asy]
$ \textbf{(A)}\ 10^{\circ}\qquad\textbf{(B)}\ 15^{\circ}\qquad\textbf{(C)}\ 20^{\circ}\qquad\textbf{(D)}\ 25^{\circ}\qquad\textbf{(E)}\ 30^{\circ}$
2017 AMC 12/AHSME, 16
In the figure below, semicircles with centers at $A$ and $B$ and with radii $2$ and $1$, respectively, are drawn in the interior of, and sharing bases with, a semicircle with diameter $\overline{JK}$. The two smaller semicircles are externally tangent to each other and internally tangent to the largest semicircle. A circle centered at $P$ is drawn externally tangent to the two smaller semicircles and internally tangent to the largest semicircle. What is the radius of the circle centered at $P$?
[asy]
size(8cm);
draw(arc((0,0),3,0,180));
draw(arc((2,0),1,0,180));
draw(arc((-1,0),2,0,180));
draw((-3,0)--(3,0));
pair P = (-1,0)+(2+6/7)*dir(36.86989);
draw(circle(P,6/7));
dot((-1,0)); dot((2,0)); dot((-3,0)); dot((3,0)); dot(P);
label("$J$",(-3,0),W);
label("$A$",(-1,0),NW);
label("$B$",(2,0),NE);
label("$K$",(3,0),E);
label("$P$",P,NW);
[/asy]
$ \textbf{(A)}\ \frac{3}{4}
\qquad \textbf{(B)}\ \frac{6}{7}
\qquad\textbf{(C)}\ \frac{1}{2}\sqrt{3}
\qquad\textbf{(D)}\ \frac{5}{8}\sqrt{2}
\qquad\textbf{(E)}\ \frac{11}{12} $
2008 AMC 10, 21
Ten chairs are evenly spaced around a round table and numbered clockwise from $ 1$ through $ 10$. Five married couples are to sit in the chairs with men and women alternating, and no one is to sit either next to or directly across from his or her spouse. How many seating arrangements are possible?
$ \textbf{(A)}\ 240\qquad
\textbf{(B)}\ 360\qquad
\textbf{(C)}\ 480\qquad
\textbf{(D)}\ 540\qquad
\textbf{(E)}\ 720$
2019 AIME Problems, 8
Let $x$ be a real number such that $\sin^{10}x+\cos^{10} x = \tfrac{11}{36}$. Then $\sin^{12}x+\cos^{12} x = \tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
2016 AMC 10, 21
What is the area of the region enclosed by the graph of the equation $x^2+y^2=|x|+|y|?$
$\textbf{(A)}\ \pi+\sqrt{2} \qquad\textbf{(B)}\ \pi+2 \qquad\textbf{(C)}\ \pi+2\sqrt{2} \qquad\textbf{(D)}\ 2\pi+\sqrt{2} \qquad\textbf{(E)}\ 2\pi+2\sqrt{2}$
1961 AMC 12/AHSME, 7
When simplified, the third term in the expansion of $\left(\frac{a}{\sqrt{x}}-\frac{\sqrt{x}}{a^2}\right)^6$ is:
${{ \textbf{(A)}\ \frac{15}{x}\qquad\textbf{(B)}\ -\frac{15}{x}\qquad\textbf{(C)}\ -\frac{6x^2}{a^9} \qquad\textbf{(D)}\ \frac{20}{a^3} }\qquad\textbf{(E)}\ -\frac{20}{a^3} } $
2002 AMC 10, 9
Using the letters $ A$, $ M$, $ O$, $ S$, and $ U$, we can form $ 120$ five-letter "words". If these "words" are arranged in alphabetical order, then the "word" $ USAMO$ occupies position
$ \textbf{(A)}\ 112 \qquad
\textbf{(B)}\ 113 \qquad
\textbf{(C)}\ 114 \qquad
\textbf{(D)}\ 115 \qquad
\textbf{(E)}\ 116$
1963 AMC 12/AHSME, 18
Chord $EF$ is the perpendicular bisector of chord $BC$, intersecting it in $M$. Between $B$ and $M$ point $U$ is taken, and $EU$ extended meets the circle in $A$. Then, for any selection of $U$, as described, triangle $EUM$ is similar to triangle:
[asy]
pair B = (-0.866, -0.5);
pair C = (0.866, -0.5);
pair E = (0, -1);
pair F = (0, 1);
pair M = midpoint(B--C);
pair A = (-0.99, -0.141);
pair U = intersectionpoints(A--E, B--C)[0];
draw(B--C);
draw(F--E--A);
draw(unitcircle);
label("$B$", B, SW);
label("$C$", C, SE);
label("$A$", A, W);
label("$E$", E, S);
label("$U$", U, NE);
label("$M$", M, NE);
label("$F$", F, N);
//Credit to MSTang for the asymptote
[/asy]
$\textbf{(A)}\ EFA \qquad
\textbf{(B)}\ EFC \qquad
\textbf{(C)}\ ABM \qquad
\textbf{(D)}\ ABU \qquad
\textbf{(E)}\ FMC$
2008 AMC 10, 17
A poll shows that $ 70\%$ of all voters approve of the mayor's work. On three separate occasions a pollster selects a voter at random. What is the probability that on exactly one of these three occasions the voter approves of the mayor's work?
$ \textbf{(A)}\ 0.063 \qquad
\textbf{(B)}\ 0.189 \qquad
\textbf{(C)}\ 0.233 \qquad
\textbf{(D)}\ 0.333 \qquad
\textbf{(E)}\ 0.441$
2024 AMC 12/AHSME, 22
The figure below shows a dotted grid $8$ cells wide and $3$ cells tall consisting of $1''\times1''$ squares. Carl places $1$-inch toothpicks along some of the sides of the squares to create a closed loop that does not intersect itself. The numbers in the cells indicate the number of sides of that square that are to be covered by toothpicks, and any number of toothpicks are allowed if no number is written. In how many ways can Carl place the toothpicks? [asy]
size(6cm);
for (int i=0; i<9; ++i) {
draw((i,0)--(i,3),dotted);
}
for (int i=0; i<4; ++i){
draw((0,i)--(8,i),dotted);
}
for (int i=0; i<8; ++i) {
for (int j=0; j<3; ++j) {
if (j==1) {
label("1",(i+0.5,1.5));
}}}
[/asy] $\textbf{(A) }130\qquad\textbf{(B) }144\qquad\textbf{(C) }146\qquad\textbf{(D) }162\qquad\textbf{(E) }196$
2015 AMC 10, 25
A rectangular box measures $a \times b \times c$, where $a,$ $b,$ and $c$ are integers and $1 \leq a \leq b \leq c$. The volume and surface area of the box are numerically equal. How many ordered triples $(a,b,c)$ are possible?
$ \textbf{(A) }4\qquad\textbf{(B) }10\qquad\textbf{(C) }12\qquad\textbf{(D) }21\qquad\textbf{(E) }26 $
2021 AMC 12/AHSME Fall, 25
For $n$ a positive integer, let $R(n)$ be the sum of the remainders when $n$ is divided by $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, and $10$. For example, $R(15) = 1+0+3+0+3+1+7+6+5=26$. How many two-digit positive integers $n$ satisfy $R(n) = R(n+1)\,?$
$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }4$
2024 AMC 10, 14
A dartboard is the region B in the coordinate plane consisting of points $(x, y)$ such that $|x| + |y| \le 8$. A target T is the region where $(x^2 + y^2 - 25)^2 \le 49$. A dart is thrown at a random point in B. The probability that the dart lands in T can be expressed as $\frac{m}{n} \pi$, where $m$ and $n$ are relatively prime positive integers. What is $m + n$?
$
\textbf{(A) }39 \qquad
\textbf{(B) }71 \qquad
\textbf{(C) }73 \qquad
\textbf{(D) }75 \qquad
\textbf{(E) }135 \qquad
$
2002 AIME Problems, 11
Two distinct, real, infinite geometric series each have a sum of $1$ and have the same second term. The third term of one of the series is $1/8,$ and the second term of both series can be written in the form $\frac{\sqrt{m}-n}{p},$ where $m,$ $n,$ and $p$ are positive integers and $m$ is not divisible by the square of any prime. Find $100m+10n+p.$
2012 AMC 8, 1
Rachelle uses 3 pounds of meat to make 8 hamburgers for her family. How many pounds of meat does she need to make 24 hamburgers for a neighborhood picnic?
$\textbf{(A)}\hspace{.05in}6 \qquad \textbf{(B)}\hspace{.05in}6\dfrac23 \qquad \textbf{(C)}\hspace{.05in}7\dfrac12 \qquad \textbf{(D)}\hspace{.05in}8 \qquad \textbf{(E)}\hspace{.05in}9 $
1986 USAMO, 5
By a partition $\pi$ of an integer $n\ge 1$, we mean here a representation of $n$ as a sum of one or more positive integers where the summands must be put in nondecreasing order. (E.g., if $n=4$, then the partitions $\pi$ are $1+1+1+1$, $1+1+2$, $1+3, 2+2$, and $4$).
For any partition $\pi$, define $A(\pi)$ to be the number of $1$'s which appear in $\pi$, and define $B(\pi)$ to be the number of distinct integers which appear in $\pi$. (E.g., if $n=13$ and $\pi$ is the partition $1+1+2+2+2+5$, then $A(\pi)=2$ and $B(\pi) = 3$).
Prove that, for any fixed $n$, the sum of $A(\pi)$ over all partitions of $\pi$ of $n$ is equal to the sum of $B(\pi)$ over all partitions of $\pi$ of $n$.
2016 USAJMO, 4
Find, with proof, the least integer $N$ such that if any $2016$ elements are removed from the set ${1, 2,...,N}$, one can still find $2016$ distinct numbers among the remaining elements with sum $N$.
1987 AMC 12/AHSME, 6
In the $\triangle ABC$ shown, $D$ is some interior point, and $x, y, z, w$ are the measures of angles in degrees. Solve for $x$ in terms of $y, z$ and $w$.
[asy]
draw((0,0)--(10,0)--(2,7)--cycle);
draw((0,0)--(4,3)--(10,0));
label("A", (0,0), SW);
label("B", (10,0), SE);
label("C", (2,7), W);
label("D", (4,3), N);
label("x", (2.25,6));
label("y", (1.5,2), SW);
label("$z$", (7.88,1.5));
label("w", (4,2.85), S);
[/asy]
$ \textbf{(A)}\ w-y-z \qquad\textbf{(B)}\ w-2y-2z \qquad\textbf{(C)}\ 180-w-y-z \\
\qquad\textbf{(D)}\ 2w-y-z \qquad\textbf{(E)}\ 180-w+y+z $
2021 AMC 12/AHSME Fall, 4
Let $n = 8^{2022}$. Which of the following is equal to $\frac{n}{4}$?
$\textbf{(A) }4^{1010}\qquad\textbf{(B) }2^{2022}\qquad\textbf{(C) }8^{2018}\qquad\textbf{(D) }4^{3031}\qquad\textbf{(E) }4^{3032}$
2006 AMC 12/AHSME, 23
Given a finite sequence $ S \equal{} (a_1,a_2,\ldots,a_n)$ of $ n$ real numbers, let $ A(S)$ be the sequence
\[ \left(\frac {a_1 \plus{} a_2}2,\frac {a_2 \plus{} a_3}2,\ldots,\frac {a_{n \minus{} 1} \plus{} a_n}2\right)
\]of $ n \minus{} 1$ real numbers. Define $ A^1(S) \equal{} A(S)$ and, for each integer $ m$, $ 2\le m\le n \minus{} 1$, define $ A^m(S) \equal{} A(A^{m \minus{} 1}(S)).$ Suppose $ x > 0$, and let $ S \equal{} (1,x,x^2,\ldots,x^{100})$. If $ A^{100}(S) \equal{} (1/2^{50})$, then what is $ x$?
$ \textbf{(A) } 1 \minus{} \frac {\sqrt {2}}2\qquad \textbf{(B) } \sqrt {2} \minus{} 1\qquad \textbf{(C) } \frac 12\qquad \textbf{(D) } 2 \minus{} \sqrt {2}\qquad \textbf{(E) } \frac {\sqrt {2}}2$
2020 AMC 10, 1
What value of $x$ satisfies
$$x- \frac{3}{4} = \frac{5}{12} - \frac{1}{3}?$$
$\textbf{(A)}\ -\frac{2}{3}\qquad\textbf{(B)}\ \frac{7}{36}\qquad\textbf{(C)}\ \frac{7}{12}\qquad\textbf{(D)}\ \frac{2}{3}\qquad\textbf{(E)}\ \frac{5}{6}$
2018 AMC 12/AHSME, 8
All of the triangles in the diagram below are similar to iscoceles triangle $ABC$, in which $AB=AC$. Each of the 7 smallest triangles has area 1, and $\triangle ABC$ has area 40. What is the area of trapezoid $DBCE$?
[asy]
unitsize(5);
dot((0,0));
dot((60,0));
dot((50,10));
dot((10,10));
dot((30,30));
draw((0,0)--(60,0)--(50,10)--(30,30)--(10,10)--(0,0));
draw((10,10)--(50,10));
label("$B$",(0,0),SW);
label("$C$",(60,0),SE);
label("$E$",(50,10),E);
label("$D$",(10,10),W);
label("$A$",(30,30),N);
draw((10,10)--(15,15)--(20,10)--(25,15)--(30,10)--(35,15)--(40,10)--(45,15)--(50,10));
draw((15,15)--(45,15));
[/asy]
$\textbf{(A) } 16 \qquad \textbf{(B) } 18 \qquad \textbf{(C) } 20 \qquad \textbf{(D) } 22 \qquad \textbf{(E) } 24 $