Found problems: 3632
2019 AIME Problems, 2
Lily pads $1,2,3,\ldots$ lie in a row on a pond. A frog makes a sequence of jumps starting on pad $1$. From any pad $k$ the frog jumps to either pad $k+1$ or pad $k+2$ chosen randomly and independently with probability $\tfrac12$. The probability that the frog visits pad $7$ is $\tfrac pq$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
2000 USAMO, 2
Let $S$ be the set of all triangles $ABC$ for which \[ 5 \left( \dfrac{1}{AP} + \dfrac{1}{BQ} + \dfrac{1}{CR} \right) - \dfrac{3}{\min\{ AP, BQ, CR \}} = \dfrac{6}{r}, \] where $r$ is the inradius and $P, Q, R$ are the points of tangency of the incircle with sides $AB, BC, CA,$ respectively. Prove that all triangles in $S$ are isosceles and similar to one another.
1997 AMC 8, 24
Diameter $ACE$ is divided at $C$ in the ratio $2:3$. The two semicircles, $ABC$ and $CDE$, divide the circular region into an upper (shaded) region and a lower region. The ratio of the area of the upper region to that of the lower region is
[asy]pair A,B,C,D,EE;
A = (0,0); B = (2,2); C = (4,0); D = (7,-3); EE = (10,0);
fill(arc((2,0),A,C,CW)--arc((7,0),C,EE,CCW)--arc((5,0),EE,A,CCW)--cycle,gray);
draw(arc((2,0),A,C,CW)--arc((7,0),C,EE,CCW));
draw(circle((5,0),5));
dot(A); dot(B); dot(C); dot(D); dot(EE);
label("$A$",A,W);
label("$B$",B,N);
label("$C$",C,E);
label("$D$",D,N);
label("$E$",EE,W);
[/asy]
$\textbf{(A)}\ 2:3 \qquad \textbf{(B)}\ 1:1 \qquad \textbf{(C)}\ 3:2 \qquad \textbf{(D)}\ 9:4 \qquad \textbf{(E)}\ 5:2$
2008 AMC 10, 6
A triathlete competes in a triathlon in which the swimming, biking, and running segments are all of the same length. The triathlete swims at a rate of $ 3$ kilometers per hour, bikes at a rate of $ 20$ kilometers per hour, and runs at a rate of $ 10$ kilometers per hour. Which of the following is closest to the triathlete's average speed, in kilometers per hour, for the entire race?
$ \textbf{(A)}\ 3 \qquad
\textbf{(B)}\ 4 \qquad
\textbf{(C)}\ 5 \qquad
\textbf{(D)}\ 6 \qquad
\textbf{(E)}\ 7$
2019 AIME Problems, 4
A soccer team has 22 available players. A fixed set of 11 players starts the game, while the other 11 are available as substitutes. During the game, the coach may make as many as 3 substitutions, where any one of the 11 players in the game is replaced by one of the substitutes. No player removed from the game may reenter the game, although a substitute entering the game may be replaced later. No two substitutions can happen at the same time. The players involved and the order of the substitutions matter. Let $n$ be the number of ways the coach can make substitutions during the game (including the possibility of making no substitutions). Find the remainder when $n$ is divided by 1000.
1993 AMC 8, 13
The word "'''HELP'''" in block letters is painted in black with strokes $1$ unit wide on a $5$ by $15$ rectangular white sign with dimensions as shown. The area of the white portion of the sign, in square units, is
[asy]
unitsize(12);
fill((0,0)--(0,5)--(1,5)--(1,3)--(2,3)--(2,5)--(3,5)--(3,0)--(2,0)--(2,2)--(1,2)--(1,0)--cycle,black);
fill((4,0)--(4,5)--(7,5)--(7,4)--(5,4)--(5,3)--(7,3)--(7,2)--(5,2)--(5,1)--(7,1)--(7,0)--cycle,black);
fill((8,0)--(8,5)--(9,5)--(9,1)--(11,1)--(11,0)--cycle,black);
fill((12,0)--(12,5)--(15,5)--(15,2)--(13,2)--(13,0)--cycle,black);
fill((13,3)--(14,3)--(14,4)--(13,4)--cycle,white);
draw((0,0)--(15,0)--(15,5)--(0,5)--cycle);
label("$5\left\{ \begin{tabular}{c} \\ \\ \\ \\ \end{tabular}\right.$",(1,2.5),W);
label(rotate(90)*"$\{$",(0.5,0.1),S);
label("$1$",(0.5,-0.6),S);
label(rotate(90)*"$\{$",(3.5,0.1),S);
label("$1$",(3.5,-0.6),S);
label(rotate(90)*"$\{$",(7.5,0.1),S);
label("$1$",(7.5,-0.6),S);
label(rotate(90)*"$\{$",(11.5,0.1),S);
label("$1$",(11.5,-0.6),S);
label(rotate(270)*"$\left\{ \begin{tabular}{c} \\ \\ \end{tabular}\right.$",(1.5,4),N);
label("$3$",(1.5,5.8),N);
label(rotate(270)*"$\left\{ \begin{tabular}{c} \\ \\ \end{tabular}\right.$",(5.5,4),N);
label("$3$",(5.5,5.8),N);
label(rotate(270)*"$\left\{ \begin{tabular}{c} \\ \\ \end{tabular}\right.$",(9.5,4),N);
label("$3$",(9.5,5.8),N);
label(rotate(270)*"$\left\{ \begin{tabular}{c} \\ \\ \end{tabular}\right.$",(13.5,4),N);
label("$3$",(13.5,5.8),N);
label("$\left. \begin{tabular}{c} \\ \end{tabular}\right\} 2$",(14,1),E);
[/asy]
$\text{(A)}\ 30 \qquad \text{(B)}\ 32 \qquad \text{(C)}\ 34 \qquad \text{(D)}\ 36 \qquad \text{(E)}\ 38$
2015 AMC 12/AHSME, 14
A circle of radius $2$ is centered at $A$. An equilateral triangle with side $4$ has a vertex at $A$. What is the difference between the area of the region that lies inside the circle but outside the triangle and the area of the region that lies inside the triangle but outside the circle?
$ \textbf {(A) } 8-\pi \qquad \textbf {(B) } \pi + 2 \qquad \textbf {(C) } 2\pi - \frac {\sqrt{2}}{2} \qquad \textbf {(D) } 4(\pi - \sqrt{3}) \qquad \textbf {(E) } 2\pi + \frac {\sqrt{3}}{2} $
2013 AMC 8, 17
The sum of six consecutive positive integers is 2013. What is the largest of these six integers?
$\textbf{(A)}\ 335 \qquad \textbf{(B)}\ 338 \qquad \textbf{(C)}\ 340 \qquad \textbf{(D)}\ 345 \qquad \textbf{(E)}\ 350$
2022 AMC 12/AHSME, 22
Let $c$ be a real number, and let $z_1, z_2$ be the two complex numbers satisfying the quadratic $z^2 - cz + 10 = 0$. Points $z_1, z_2, \frac{1}{z_1}$, and $\frac{1}{z_2}$ are the vertices of a (convex) quadrilateral $Q$ in the complex plane. When the area of $Q$ obtains its maximum value, $c$ is the closest to which of the following?
$\textbf{(A)}~4.5\qquad\textbf{(B)}~5\qquad\textbf{(C)}~5.5\qquad\textbf{(D)}~6\qquad\textbf{(E)}~6.5$
1996 AMC 8, 17
Figure $OPQR$ is a square. Point $O$ is the origin, and point $Q$ has coordinates $(2,2)$. What are the coordinates for $T$ so that the area of triangle $PQT$ equals the area of square $OPQR$?
[asy]
pair O,P,Q,R,T;
O = (0,0); P = (2,0); Q = (2,2); R = (0,2); T = (-4,0);
draw((-5,0)--(3,0)); draw((0,-1)--(0,3));
draw(P--Q--R);
draw((-0.2,-0.8)--(0,-1)--(0.2,-0.8));
draw((-0.2,2.8)--(0,3)--(0.2,2.8));
draw((-4.8,-0.2)--(-5,0)--(-4.8,0.2));
draw((2.8,-0.2)--(3,0)--(2.8,0.2));
draw(Q--T);
label("$O$",O,SW);
label("$P$",P,S);
label("$Q$",Q,NE);
label("$R$",R,W);
label("$T$",T,S);
[/asy]
NOT TO SCALE
$\text{(A)}\ (-6,0) \qquad \text{(B)}\ (-4,0) \qquad \text{(C)}\ (-2,0) \qquad \text{(D)}\ (2,0) \qquad \text{(E)}\ (4,0)$
2020 AMC 12/AHSME, 4
How many $4$-digit positive integers (that is, integers between $1000$ and $9999$, inclusive) having only even digits are divisible by $5?$
$\textbf{(A) } 80 \qquad \textbf{(B) } 100 \qquad \textbf{(C) } 125 \qquad \textbf{(D) } 200 \qquad \textbf{(E) } 500$
1994 AMC 12/AHSME, 2
A large rectangle is partitioned into four rectangles by two segments parallel to its sides. The areas of three of the resulting rectangles are shown. What is the area of the fourth rectangle?
[asy]
draw((0,0)--(10,0)--(10,7)--(0,7)--cycle);
draw((0,5)--(10,5));
draw((3,0)--(3,7));
label("6", (1.5,6));
label("?", (1.5,2.5));
label("14", (6.5,6));
label("35", (6.5,2.5));
[/asy]
$ \textbf{(A)}\ 10 \qquad\textbf{(B)}\ 15 \qquad\textbf{(C)}\ 20 \qquad\textbf{(D)}\ 21 \qquad\textbf{(E)}\ 25 $
2020 AMC 12/AHSME, 22
Let $(a_n)$ and $(b_n)$ be the sequences of real numbers such that
\[
(2 + i)^n = a_n + b_ni
\]
for all integers $n\geq 0$, where $i = \sqrt{-1}$. What is \[\sum_{n=0}^\infty\frac{a_nb_n}{7^n}\,?\]
$\textbf{(A) }\frac 38\qquad\textbf{(B) }\frac7{16}\qquad\textbf{(C) }\frac12\qquad\textbf{(D) }\frac9{16}\qquad\textbf{(E) }\frac47$
2018 AMC 10, 13
How many of the first $2018$ numbers in the sequence $101, 1001, 10001, 100001, \dots$ are divisible by $101$?
$
\textbf{(A) }253 \qquad
\textbf{(B) }504 \qquad
\textbf{(C) }505 \qquad
\textbf{(D) }506 \qquad
\textbf{(E) }1009 \qquad
$
2019 AMC 10, 1
What is the value of $$2^{\left(0^{\left(1^9\right)}\right)}+\left(\left(2^0\right)^1\right)^9\,?$$
$\textbf{(A) } 0 \qquad\textbf{(B) } 1 \qquad\textbf{(C) } 2 \qquad\textbf{(D) } 3 \qquad\textbf{(E) } 4$
1989 AIME Problems, 14
Given a positive integer $n$, it can be shown that every complex number of the form $r+si$, where $r$ and $s$ are integers, can be uniquely expressed in the base $-n+i$ using the integers $1,2,\ldots,n^2$ as digits. That is, the equation\[ r+si=a_m(-n+i)^m+a_{m-1}(-n+i)^{m-1}+\cdots +a_1(-n+i)+a_0 \]is true for a unique choice of non-negative integer $m$ and digits $a_0,a_1,\ldots,a_m$ chosen from the set $\{0,1,2,\ldots,n^2\}$, with $a_m\ne 0$. We write \[ r+si=(a_ma_{m-1}\ldots a_1a_0)_{-n+i} \]to denote the base $-n+i$ expansion of $r+si$. There are only finitely many integers $k+0i$ that have four-digit expansions \[ k=(a_3a_2a_1a_0)_{-3+i}~~~~a_3\ne 0. \]Find the sum of all such $k$.
1970 AMC 12/AHSME, 27
In a triangle, the area is numerically equal to the perimeter. What is the radius of the inscribed circle?
$\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad \textbf{(E) }6$
2020 CHMMC Winter (2020-21), 6
Let $P_0P_5Q_5Q_0$ be a rectangular chocolate bar, one half dark chocolate and one half white chocolate, as shown in the diagram below. We randomly select $4$ points on the segment $P_0P_5$, and immediately after selecting those points, we label those $4$ selected points $P_1, P_2, P_3, P_4$ from left to right. Similarly, we randomly select $4$ points on the segment $Q_0Q_5$, and immediately after selecting those points, we label those $4$ points $Q_1, Q_2, Q_3, Q_4$ from left to right. The segments $P_1Q_1, P_2Q_2, P_3Q_3, P_4Q_4$ divide the rectangular chocolate bar into $5$ smaller trapezoidal pieces of chocolate. The probability that exactly $3$ pieces of chocolate contain both dark and white chocolate can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
[Diagram in the individuals file for this exam on the Chmmc website]
2012 AMC 12/AHSME, 17
Let $S$ be a subset of $\{1,2,3,\dots,30\}$ with the property that no pair of distinct elements in $S$ has a sum divisible by $5$. What is the largest possible size of $S$?
$ \textbf{(A)}\ 10\qquad\textbf{(B)}\ 13\qquad\textbf{(C)}\ 15\qquad\textbf{(D)}\ 16\qquad\textbf{(E)}\ 18 $
2002 AMC 10, -1
This test and the matching AMC 12P were developed for the use of a group of Taiwan schools, in early January of 2002. When Taiwan had taken the contests, the AMC released the questions here as a set of practice questions for the 2002 AMC 10 and AMC 12 contests.
2023 AIME, 2
If $\sqrt{\log_bn}=\log_b\sqrt n$ and $b\log_bn=\log_bbn,$ then the value of $n$ is equal to $\frac jk,$ where $j$ and $k$ are relatively prime. What is $j+k$?
2024 AMC 12/AHSME, 14
How many different remainders can result when the $100$th power of an integer is divided by $125$?
$
\textbf{(A) }1 \qquad
\textbf{(B) }2 \qquad
\textbf{(C) }5 \qquad
\textbf{(D) }25 \qquad
\textbf{(E) }125 \qquad
$
2020 AMC 12/AHSME, 9
How many solutions does the equation $\tan{(2x)} = \cos{(\tfrac{x}{2})}$ have on the interval $[0, 2\pi]?$
$\textbf{(A) } 1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 3 \qquad \textbf{(D) } 4 \qquad \textbf{(E) } 5$
2012 AMC 12/AHSME, 7
Small lights are hung on a string 6 inches apart in the order red, red, green, green, green, red, red, green, green, green, and so on continuing this pattern of 2 red lights followed by 3 green lights. How many feet separate the third red light and the 21st red light?
[b]Note:[/b] 1 foot is equal to 12 inches.
$\textbf{(A)}\ 18 \qquad\textbf{(B)}\ 18.5 \qquad\textbf{(C)}\ 20 \qquad\textbf{(D)}\ 20.5 \qquad\textbf{(E)}\ 22.5 $
2011 AMC 10, 23
Seven students count from $1$ to $1000$ as follows:
[list]
[*]Alice says all the numbers, except she skips the middle number in each consecutive group of three numbers. That is, Alice says $1, 3, 4, 6, 7, 9, \cdots, 997, 999, 1000.$
[*]Barbara says all of the numbers that Alice doesn't say, except she also skips the middle number in each consecutive group of three numbers.
[*]Candice says all of the numbers that neither Alice nor Barbara says, except she also skips the middle number in each consecutive group of three numbers.
[*]Debbie, Eliza, and Fatima say all of the numbers that none of the students with the first names beginning before theirs in the alphabet say, except each also skips the middle number in each of her consecutive groups of three numbers.
[*]Finally, George says the only number that no one else says.
[/list]
What number does George say?
$ \textbf{(A)}\ 37\qquad\textbf{(B)}\ 242\qquad\textbf{(C)}\ 365\qquad\textbf{(D)}\ 728\qquad\textbf{(E)}\ 998 $