Found problems: 3632
2013 AMC 8, 5
Hammie is in the $6^\text{th}$ grade and weighs 106 pounds. His quadruplet sisters are tiny babies and weigh 5, 5, 6, and 8 pounds. Which is greater, the average (mean) weight of these five children or the median weight, and by how many pounds?
$\textbf{(A)}\ \text{median, by 60} \qquad \textbf{(B)}\ \text{median, by 20} \qquad \textbf{(C)}\ \text{average, by 5} \qquad \textbf{(D)}\ \text{average, by 15}$ \\ $\textbf{(E)}\ \text{average, by 20}$
2008 AMC 12/AHSME, 13
Points $ A$ and $ B$ lie on a circle centered at $ O$, and $ \angle AOB\equal{}60^\circ$. A second circle is internally tangent to the first and tangent to both $ \overline{OA}$ and $ \overline{OB}$. What is the ratio of the area of the smaller circle to that of the larger circle?
$ \textbf{(A)}\ \frac{1}{16} \qquad
\textbf{(B)}\ \frac{1}{9} \qquad
\textbf{(C)}\ \frac{1}{8} \qquad
\textbf{(D)}\ \frac{1}{6} \qquad
\textbf{(E)}\ \frac{1}{4}$
2003 AIME Problems, 10
Triangle $ABC$ is isosceles with $AC = BC$ and $\angle ACB = 106^\circ$. Point $M$ is in the interior of the triangle so that $\angle MAC = 7^\circ$ and $\angle MCA = 23^\circ$. Find the number of degrees in $\angle CMB$.
2021 AIME Problems, 13
Find the least positive integer $n$ for which $2^n + 5^n - n$ is a multiple of $1000$.
2023 AMC 8, 1
What is the value of $(8 \times 4 + 2) - (8 + 4 \times 2)?$
$\textbf{(A)}~0\qquad\textbf{(B)}~6\qquad\textbf{(C)}~10\qquad\textbf{(D)}~18\qquad\textbf{(E)}~24$
2021 AMC 12/AHSME Fall, 18
Set $u_0 = \frac{1}{4},$ and for $k \geq 0$ let $u_{k+1}$ be determined by the recurrence $u_{k+1} = 2u_k - 2u_k^2.$ This sequence tends to a limit, call it $L.$ What is the least value of $k$ such that $$|u_k - L| \leq \frac{1}{2^{1000}}?$$
$\textbf{(A)}\ 10 \qquad\textbf{(B)}\ 97 \qquad\textbf{(C)}\ 253 \qquad\textbf{(D)}\
329 \qquad\textbf{(E)}\ 401$
2013 AMC 10, 4
A softball team played ten games, scoring $1,2,3,4,5,6,7,8,9$, and $10$ runs. They lost by one run in exactly five games. In each of the other games, they scored twice as many runs as their opponent. How many total runs did their opponents score?
$ \textbf {(A) } 35 \qquad \textbf {(B) } 40 \qquad \textbf {(C) } 45 \qquad \textbf {(D) } 50 \qquad \textbf {(E) } 55 $
1990 AMC 12/AHSME, 28
A quadrilateral that has consecutive sides of lengths $70, 90, 130$ and $110$ is inscribed in a circle and also has a circle inscribed in it. The point of tangency of the inscribed circle to the side of length $130$ divides that side into segments of lengths $x$ and $y$. Find $|x-y|$.
$ \textbf{(A)}\ 12 \qquad\textbf{(B)}\ 13 \qquad\textbf{(C)}\ 14 \qquad\textbf{(D)}\ 15 \qquad\textbf{(E)}\ 16 $
2022 AIME Problems, 4
Let $w = \frac{\sqrt{3}+i}{2}$ and $z=\frac{-1+i\sqrt{3}}{2}$, where $i=\sqrt{-1}$. Find the number of ordered pairs $(r, s)$ of positive integers not exceeding $100$ that satisfy the equation $i\cdot w^r=z^s$.
2009 AMC 10, 22
Two cubical dice each have removable numbers $ 1$ through $ 6$. The twelve numbers on the two dice are removed, put into a bag, then drawn one at a time and randomly reattached to the faces of the cubes, one number to each face. The dice are then rolled and the numbers on the two top faces are added. What is the probability that the sum is $ 7$?
$ \textbf{(A)}\ \frac{1}{9} \qquad
\textbf{(B)}\ \frac{1}{8} \qquad
\textbf{(C)}\ \frac{1}{6} \qquad
\textbf{(D)}\ \frac{2}{11} \qquad
\textbf{(E)}\ \frac{1}{5}$
2021 USAMO, 3
Let $n \geq 2$ be an integer. An $n \times n$ board is initially empty. Each minute, you may perform one of three moves:
[list]
[*] If there is an L-shaped tromino region of three cells without stones on the board (see figure; rotations not allowed), you may place a stone in each of those cells.
[*] If all cells in a column have a stone, you may remove all stones from that column.
[*] If all cells in a row have a stone, you may remove all stones from that row.
[/list]
[asy]
unitsize(20);
draw((0,0)--(4,0)--(4,4)--(0,4)--(0,0));
fill((0.2,3.8)--(1.8,3.8)--(1.8, 1.8)--(3.8, 1.8)--(3.8, 0.2)--(0.2, 0.2)--cycle, grey);
draw((0.2,3.8)--(1.8,3.8)--(1.8, 1.8)--(3.8, 1.8)--(3.8, 0.2)--(0.2, 0.2)--(0.2, 3.8), linewidth(2));
draw((0,2)--(4,2));
draw((2,4)--(2,0));
[/asy]
For which $n$ is it possible that, after some non-zero number of moves, the board has no stones?
2011 AIME Problems, 1
Jar A contains four liters of a solution that is $45\%$ acid. Jar B contains five liters of a solution that is $48\%$ acid. Jar C contains one liter of a solution that is $k\%$ acid. From jar C, $\tfrac{m}{n}$ liters of the solution is added to jar A, and the remainder of the solution in jar C is added to jar B. At the end, both jar A and jar B contain solutions that are $50\%$ acid. Given that $m$ and $n$ are relatively prime positive integers, find $k+m+n$.
2022 AMC 10, 18
Consider systems of three linear equations with unknowns $x,$ $y,$ and $z,$
\begin{align*}
a_1 x + b_1 y + c_1 z = 0 \\
a_2 x + b_2 y + c_2 z = 0 \\
a_3 x + b_3 y + c_3 z = 0
\end{align*}
where each of the coefficients is either $0$ or $1$ and the system has a solution other than $x = y = z = 0.$ For example, one such system is $\{1x + 1y + 0z = 0, 0x + 1y + 1z = 0, 0x + 0y + 0z = 0\}$ with a nonzero solution of $\{x, y, z\} = \{1, -1, 1\}.$ How many such systems are there? (The equations in a system need not be distinct, and two systems containing the same equations in a different order are considered different.)
$\textbf{(A) } 302 \qquad \textbf{(B) } 338 \qquad \textbf{(C) } 340 \qquad \textbf{(D) } 343 \qquad \textbf{(E) } 344$
2012 AMC 12/AHSME, 1
A bug crawls along a number line, starting at $-2$. It crawls to $-6$, then turns around and crawls to $5$. How many units does the bug crawl altogether?
$ \textbf{(A)}\ 9
\qquad\textbf{(B)}\ 11
\qquad\textbf{(C)}\ 13
\qquad\textbf{(D)}\ 14
\qquad\textbf{(E)}\ 15
$
2025 USAMO, 2
Let $n$ and $k$ be positive integers with $k<n$. Let $P(x)$ be a polynomial of degree $n$ with real coefficients, nonzero constant term, and no repeated roots. Suppose that for any real numbers $a_0,\,a_1,\,\ldots,\,a_k$ such that the polynomial $a_kx^k+\cdots+a_1x+a_0$ divides $P(x)$, the product $a_0a_1\cdots a_k$ is zero. Prove that $P(x)$ has a nonreal root.
2002 AMC 10, 25
In trapezoid $ ABCD$ with bases $ AB$ and $ CD$, we have $ AB\equal{}52$, $ BC\equal{}12$, $ CD\equal{}39$, and $ DA\equal{}5$. The area of $ ABCD$ is
[asy]
pair A,B,C,D;
A=(0,0);
B=(52,0);
C=(38,20);
D=(5,20);
dot(A);
dot(B);
dot(C);
dot(D);
draw(A--B--C--D--cycle);
label("$A$",A,S);
label("$B$",B,S);
label("$C$",C,N);
label("$D$",D,N);
label("52",(A+B)/2,S);
label("39",(C+D)/2,N);
label("12",(B+C)/2,E);
label("5",(D+A)/2,W);[/asy]
$ \text{(A)}\ 182 \qquad
\text{(B)}\ 195 \qquad
\text{(C)}\ 210 \qquad
\text{(D)}\ 234 \qquad
\text{(E)}\ 260$
1970 AMC 12/AHSME, 35
A retiring employee receives and annual pension proportional to the square root of the number of years of his service. Had he served $a$ years more, his pension would have been $p$ dollars greater, whereas, had he served $b$ years more $b\neq a$, his pension would have been $q$ dollars greater than the original annual pension. Find his annual pension in terms of $a,b,p,$ and $q$.
$\textbf{(A) }\dfrac{p^2-q^2}{2(a-b)}\qquad\textbf{(B) }\dfrac{(p-q)^2}{2\sqrt{ab}}\qquad\textbf{(C) }\dfrac{ap^2-bq^2}{2(ap-bq)}\qquad\textbf{(D) }\dfrac{aq^2-bp^2}{2(bp-aq)}\qquad \textbf{(E) }\sqrt{(a-b)(p-q)}$
1967 AMC 12/AHSME, 30
A dealer bought $n$ radios for $d$ dollars, $d$ a positive integer. He contributed two radios to a community bazaar at half their cost. The rest he sold at a profit of $\$8$ on each radio sold. If the overall profit was $\$72$, then the least possible value of $n$ for the given information is:
$\textbf{(A)}\ 18\qquad
\textbf{(B)}\ 16\qquad
\textbf{(C)}\ 15\qquad
\textbf{(D)}\ 12\qquad
\textbf{(E)}\ 11$
2022 AIME Problems, 15
Let $x$, $y$, and $z$ be positive real numbers satisfying the system of equations
\begin{align*}
\sqrt{2x - xy} + \sqrt{2y - xy} & = 1\\
\sqrt{2y - yz} + \hspace{0.1em} \sqrt{2z - yz} & = \sqrt{2}\\
\sqrt{2z - zx\vphantom{y}} + \sqrt{2x - zx\vphantom{y}} & = \sqrt{3}.
\end{align*}Then $\big[ (1-x)(1-y)(1-z) \big] ^2$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
2009 AMC 12/AHSME, 21
Let $ p(x) \equal{} x^3 \plus{} ax^2 \plus{} bx \plus{} c$, where $ a$, $ b$, and $ c$ are complex numbers. Suppose that
\[ p(2009 \plus{} 9002\pi i) \equal{} p(2009) \equal{} p(9002) \equal{} 0
\]What is the number of nonreal zeros of $ x^{12} \plus{} ax^8 \plus{} bx^4 \plus{} c$?
$ \textbf{(A)}\ 4\qquad \textbf{(B)}\ 6\qquad \textbf{(C)}\ 8\qquad \textbf{(D)}\ 10\qquad \textbf{(E)}\ 12$
2016 AMC 12/AHSME, 19
Jerry starts at 0 on the real number line. He tosses a fair coin 8 times. When he gets heads, he moves 1 unit in the positive direction; when he gets tails, he moves 1 unit in the negative direction. The probability that he reaches 4 at some time during this process is $a/b$, where $a$ and $b$ are relatively prime positive integers. What is $a+b$? (For example, he succeeds if his sequence of tosses is $HTHHHHHH$.)
$\textbf{(A)}\ 69\qquad\textbf{(B)}\ 151\qquad\textbf{(C)}\ 257\qquad\textbf{(D)}\ 293\qquad\textbf{(E)}\ 313$
1960 AMC 12/AHSME, 11
For a given value of $k$ the product of the roots of \[ x^2-3kx+2k^2-1=0 \]
is $7$. The roots may be characterized as:
$ \textbf{(A) }\text{integral and positive} \qquad\textbf{(B) }\text{integral and negative} \qquad$
$\textbf{(C) }\text{rational, but not integral} \qquad\textbf{(D) }\text{irrational} \qquad\textbf{(E) } \text{imaginary} $
1966 AMC 12/AHSME, 28
Five points $O,A,B,C,D$ are taken in order on a straight line with distances $OA=a$, $OB=b$, $OC=c$, and $OD=d$. $P$ is a point on the line between $B$ and $C$ and such that $AP:PD=BP:PC$. Then $OP$ equals:
$\text{(A)}\ \dfrac{b^2-bc}{a-b+c-d} \qquad
\text{(B)}\ \dfrac{ac-b}{a-b+c-d} \qquad
\text{(C)}\ -\dfrac{bd+c}{a-b+c-d}\qquad\\
\text{(D)}\ \dfrac{bc+ad}{a+b+c+d}\qquad
\text{(E)}\ \dfrac{ac-bd}{a+b+c+d} \qquad$
2020 AMC 10, 19
As shown in the figure below a regular dodecahedron (the polyhedron consisting of 12 congruent regular pentagonal faces) floats in space with two horizontal faces. Note that there is a ring of five slanted faces adjacent to the top face, and a ring of five slanted faces adjacent to the bottom face. How many ways are there to move from the top face to the bottom face via a sequence of adjacent faces so that each face is visited at most once and moves are not permitted from the bottom ring to the top ring?
[asy]
import graph;
unitsize(4.5cm);
pair A = (0.082, 0.378);
pair B = (0.091, 0.649);
pair C = (0.249, 0.899);
pair D = (0.479, 0.939);
pair E = (0.758, 0.893);
pair F = (0.862, 0.658);
pair G = (0.924, 0.403);
pair H = (0.747, 0.194);
pair I = (0.526, 0.075);
pair J = (0.251, 0.170);
pair K = (0.568, 0.234);
pair L = (0.262, 0.449);
pair M = (0.373, 0.813);
pair N = (0.731, 0.813);
pair O = (0.851, 0.461);
path[] f;
f[0] = A--B--C--M--L--cycle;
f[1] = C--D--E--N--M--cycle;
f[2] = E--F--G--O--N--cycle;
f[3] = G--H--I--K--O--cycle;
f[4] = I--J--A--L--K--cycle;
f[5] = K--L--M--N--O--cycle;
draw(f[0]);
axialshade(f[1], white, M, gray(0.5), (C+2*D)/3);
draw(f[1]);
filldraw(f[2], gray);
filldraw(f[3], gray);
axialshade(f[4], white, L, gray(0.7), J);
draw(f[4]);
draw(f[5]);
[/asy]
$\textbf{(A) } 125 \qquad \textbf{(B) } 250 \qquad \textbf{(C) } 405 \qquad \textbf{(D) } 640 \qquad \textbf{(E) } 810$
1993 AMC 8, 9
Consider the operation $*$ defined by the following table:
\[\begin{tabular}{c|cccc}
* & 1 & 2 & 3 & 4 \\ \hline
1 & 1 & 2 & 3 & 4 \\
2 & 2 & 4 & 1 & 3 \\
3 & 3 & 1 & 4 & 2 \\
4 & 4 & 3 & 2 & 1
\end{tabular}\]
For example, $3*2=1$. Then $(2*4)*(1*3)=$
$\text{(A)}\ 1 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 3 \qquad \text{(D)}\ 4 \qquad \text{(E)}\ 5$