Found problems: 3632
2011 AMC 10, 12
Keiko walks once around a track at exactly the same constant speed every day. The sides of the track are straight, and the ends are semicircles. The track has width 6 meters, and it takes her 36 seconds longer to walk around the outside edge of the track than around the inside edge. What is Keiko's speed in meters per second?
$ \textbf{(A)}\ \frac{\pi}{3} \qquad
\textbf{(B)}\ \frac{2\pi}{3} \qquad
\textbf{(C)}\ \pi \qquad
\textbf{(D)}\ \frac{4\pi}{3} \qquad
\textbf{(E)}\ \frac{5\pi}{3} $
2006 AMC 10, 1
What is $ ( \minus{} 1)^1 \plus{} ( \minus{} 1)^2 \plus{} \cdots \plus{} ( \minus{} 1)^{2006}$?
$ \textbf{(A) } \minus{} 2006 \qquad \textbf{(B) } \minus{} 1 \qquad \textbf{(C) } 0 \qquad \textbf{(D) } 1 \qquad \textbf{(E) } 2006$
2024 AMC 10, 23
Integers $a$, $b$, and $c$ satisfy $ab + c = 100$, $bc + a = 87$, and $ca + b = 60$. What is $ab + bc + ca$?
$
\textbf{(A) }212 \qquad
\textbf{(B) }247 \qquad
\textbf{(C) }258 \qquad
\textbf{(D) }276 \qquad
\textbf{(E) }284 \qquad
$
1992 AMC 12/AHSME, 7
The ratio of $w$ to $x$ is $4:3$, of $y$ to $z$ is $3:2$ and of $z$ to $x$ is $1:6$. What is the ratio of $w$ to $y$?
$ \textbf{(A)}\ 1:3\qquad\textbf{(B)}\ 16:3\qquad\textbf{(C)}\ 20:3\qquad\textbf{(D)}\ 27:4\qquad\textbf{(E)}\ 12:1 $
1964 AMC 12/AHSME, 15
A line through the point $(-a,0)$ cuts from the second quadrant a triangular region with area $T$. The equation of the line is:
$ \textbf{(A)}\ 2Tx+a^2y+2aT=0 \qquad\textbf{(B)}\ 2Tx-a^2y+2aT=0 \qquad$
${{\textbf{(C)}\ 2Tx+a^2y-2aT=0 \qquad\textbf{(D)}\ 2Tx-a^2y-2aT=0 }\qquad\textbf{(E)}\ \text{none of these} } $
1967 AMC 12/AHSME, 33
[asy]
fill(circle((4,0),4),grey);
fill((0,0)--(8,0)--(8,-4)--(0,-4)--cycle,white);
fill(circle((7,0),1),white);
fill(circle((3,0),3),white);
draw((0,0)--(8,0),black+linewidth(1));
draw((6,0)--(6,sqrt(12)),black+linewidth(1));
MP("A", (0,0), W); MP("B", (8,0), E); MP("C", (6,0), S); MP("D",(6,sqrt(12)), N);
[/asy]
In this diagram semi-circles are constructed on diameters $\overline{AB}$, $\overline{AC}$, and $\overline{CB}$, so that they are mutually tangent. If $\overline{CD} \bot \overline{AB}$, then the ratio of the shaded area to the area of a circle with $\overline{CD}$ as radius is:
$\textbf{(A)}\ 1:2\qquad
\textbf{(B)}\ 1:3\qquad
\textbf{(C)}\ \sqrt{3}:7\qquad
\textbf{(D)}\ 1:4\qquad
\textbf{(E)}\ \sqrt{2}:6$
2010 AMC 12/AHSME, 9
A solid cube has side length $ 3$ inches. A $ 2$-inch by $ 2$-inch square hole is cut into the center of each face. The edges of each cut are parallel to the edges of the cube, and each hole goes all the way through the cube. What is the volume, in cubic inches, of the remaining solid?
$ \textbf{(A)}\ 7\qquad \textbf{(B)}\ 8\qquad \textbf{(C)}\ 10\qquad \textbf{(D)}\ 12\qquad \textbf{(E)}\ 15$
2014 AMC 10, 17
What is the greatest power of 2 that is a factor of $10^{1002}-4^{501}$?
$ \textbf{(A) }2^{1002}\qquad\textbf{(B) }2^{1003}\qquad\textbf{(C) }2^{1004}\qquad\textbf{(D) }2^{1005} \qquad\textbf{(E) }2^{1006} \qquad $
1981 USAMO, 1
The measure of a given angle is $\frac{180^{\circ}}{n}$ where $n$ is a positive integer not divisible by $3$. Prove that the angle can be trisected by Euclidean means (straightedge and compasses).
2011 AMC 12/AHSME, 17
Let $f\left(x\right)=10^{10x}, g\left(x\right)=\log_{10}\left(\frac{x}{10}\right), h_1\left(x\right)=g\left(f\left(x\right)\right),$ and $h_n\left(x\right)=h_1\left(h_{n-1}\left(x\right)\right)$ for integers $n \ge 2$. What is the sum of the digits of $h_{2011}\left(1\right)$?
$ \textbf{(A)}\ 16,081 \qquad
\textbf{(B)}\ 16,089 \qquad
\textbf{(C)}\ 18,089 \qquad
\textbf{(D)}\ 18,098 \qquad
\textbf{(E)}\ 18,099 $
2014 AIME Problems, 5
Let the set $S = \{P_1, P_2, \cdots, P_{12}\}$ consist of the twelve vertices of a regular $12$-gon. A subset $Q$ of $S$ is called communal if there is a circle such that all points of $Q$ are inside the circle, and all points of $S$ not in $Q$ are outside of the circle. How many communal subsets are there? (Note that the empty set is a communal subset.)
2021 AMC 10 Spring, 2
Portia’s high school has $3$ times as many students as Lara’s high school. The two high schools have a total of
$2600$ students. How many students does Portia’s high school have?
$\textbf{(A) }600 \qquad \textbf{(B) }650 \qquad \textbf{(C) }1950 \qquad \textbf{(D) }2000 \qquad \textbf{(E) }2050$
2018 AIME Problems, 1
Let $S$ be the number of ordered pairs of integers $(a,b)$ with $1 \leq a \leq 100$ and $b \geq 0$ such that the polynomial $x^2+ax+b$ can be factored into the product of two (not necessarily distinct) linear factors with integer coefficients. Find the remainder when $S$ is divided by $1000$.
2015 AMC 12/AHSME, 3
Mr. Patrick teaches math to $15$ students. He was grading tests and found that when he graded everyone's test except Payton's, the average grade for the class was $80$. After he graded Payton's test, the class average became $81$. What was Payton's score on the test?
$\textbf{(A) }81\qquad\textbf{(B) }85\qquad\textbf{(C) }91\qquad\textbf{(D) }94\qquad\textbf{(E) }95$
2004 AMC 10, 1
You and five friends need to raise $ \$1500$ in donations for a charity, dividing the fundraising equally. How many dollars will each of you need to raise?
$ \textbf{(A)}\ 250\qquad
\textbf{(B)}\ 300\qquad
\textbf{(C)}\ 1500\qquad
\textbf{(D)}\ 7500\qquad
\textbf{(E)}\ 9000$
2002 USAMO, 2
Let $ABC$ be a triangle such that
\[ \left( \cot \dfrac{A}{2} \right)^2 + \left( 2\cot \dfrac{B}{2} \right)^2 + \left( 3\cot \dfrac{C}{2} \right)^2 = \left( \dfrac{6s}{7r} \right)^2, \]
where $s$ and $r$ denote its semiperimeter and its inradius, respectively. Prove that triangle $ABC$ is similar to a triangle $T$ whose side lengths are all positive integers with no common divisors and determine these integers.
2012 AMC 12/AHSME, 3
For a science project, Sammy observed a chipmunk and a squirrel stashing acorns in holes. The chipmunk hid $3$ acorns in each of the holes it dug. The squirrel hid $4$ acorns in each of the holes it dug. They each hid the same
number of acorns, although the squirrel needed $4$ fewer holes. How many acorns did the chipmunk hide?
${{ \textbf{(A)}\ 30\qquad\textbf{(B)}\ 36\qquad\textbf{(C)}\ 42\qquad\textbf{(D)}\ 48}\qquad\textbf{(E)}\ 54} $
1999 AIME Problems, 3
Find the sum of all positive integers $n$ for which $n^2-19n+99$ is a perfect square.
1970 AMC 12/AHSME, 23
The number $10!$ $(10$ is written in base $10)$, when written in the base $12$ system, ends in exactly $k$ zeroes. The value of $k$ is
$\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad \textbf{(E) } 5$
2013 AMC 10, 5
Tom, Dorothy, and Sammy went on a vacation and agreed to split the costs evenly. During their trip Tom paid $\$105$, Dorothy paid $\$125$, and Sammy paid $\$175$. In order to share the costs equally, Tom gave Sammy $t$ dollars, and Dorothy gave Sammy $d$ dollars. What is $t-d$?
$ \textbf{(A)}\ 15\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 25\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 35 $
2006 AMC 10, 11
Which of the following describes the graph of the equation $ (x \plus{} y)^2 \equal{} x^2 \plus{} y^2$?
$ \textbf{(A)}\text{ the empty set}\qquad \textbf{(B)}\text{ one point}\qquad \textbf{(C)}\text{ two lines}$
$\textbf{(D)}\text{ a circle}\qquad \textbf{(E)}\text{ the entire plane}$
2015 AMC 10, 21
Cozy the Cat and Dash the Dog are going up a staircase with a certain number of steps. However, instead of walking up the steps one at a time, both Cozy and Dash jump. Cozy goes two steps up with each jump (though if necessary, he will just jump the last step). Dash goes five steps up with each jump (though if necessary, he will just jump the last steps if there are fewer than 5 steps left). Suppose the Dash takes 19 fewer jumps than Cozy to reach the top of the staircase. Let $s$ denote the sum of all possible numbers of steps this staircase can have. What is the sum of the digits of $s$?
$\textbf{(A) } 9
\qquad\textbf{(B) } 11
\qquad\textbf{(C) } 12
\qquad\textbf{(D) } 13
\qquad\textbf{(E) } 15
$
2023 AIME, 12
In $\triangle ABC$ with side lengths $AB=13$, $BC=14$, and $CA=15$, let $M$ be the midpoint of $\overline{BC}$. Let $P$ be the point on the circumcircle of $\triangle ABC$ such that $M$ is on $\overline{AP}$. There exists a unique point $Q$ on segment $\overline{AM}$ such that $\angle PBQ = \angle PCQ$. Then $AQ$ can be written as $\frac{m}{\sqrt{n}}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
2023 AMC 10, 8
What is the units digit of $2022^{2023} + 2023^{2022}$?
$\textbf{(A)}~7\qquad\textbf{(B)}~1\qquad\textbf{(C)}~3\qquad\textbf{(D)}~5\qquad\textbf{(E)}~9$
2022 AMC 12/AHSME, 7
Camila writes down five positive integers. The unique mode of these integers is $2$ greater than their median, and the median is $2$ greater than their arithmetic mean. What is the least possible value for the mode?
$\textbf{(A) }5\qquad\textbf{(B) }7\qquad\textbf{(C) }9\qquad\textbf{(D) }11\qquad\textbf{(E) }13$