Found problems: 3632
1966 AMC 12/AHSME, 37
Three men, Alpha, Beta, and Gamma, working together, do a job in $6$ hours less time than Alpha alone, in $1$ hour less time than Beta alone, and in one-half the time needed by Gamma when working alone. Let $h$ be the number of hours needed by Alpha and Beta, working together to do the job. Then $h$ equals:
$\text{(A)}\ \dfrac{5}{2}\qquad
\text{(B)}\ \frac{3}{2}\qquad
\text{(C)}\ \dfrac{4}{3}\qquad
\text{(D)}\ \dfrac{5}{4}\qquad
\text{(E)}\ \dfrac{3}{4}$
1991 AMC 12/AHSME, 10
Point $P$ is $9$ units from the center of a circle of radius $15$. How many different chords of the circle contain $P$ and have integer lengths?
$ \textbf{(A)}\ 11\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 13\qquad\textbf{(D)}\ 14\qquad\textbf{(E)}\ 29 $
1977 AMC 12/AHSME, 14
How many pairs $(m,n)$ of integers satisfy the equation $m+n=mn$?
$\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad \textbf{(E) }\text{more than }4$
1986 AMC 12/AHSME, 30
The number of real solutions $(x,y,z,w)$ of the simultaneous equations \[2y = x + \frac{17}{x},\quad 2z = y + \frac{17}{y},\quad 2w = z + \frac{17}{z},\quad 2x = w + \frac{17}{w}\] is
$ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 16 $
2010 AIME Problems, 8
For a real number $ a$, let $ \lfloor a \rfloor$ denominate the greatest integer less than or equal to $ a$. Let $ \mathcal{R}$ denote the region in the coordinate plane consisting of points $ (x,y)$ such that \[\lfloor x \rfloor ^2 \plus{} \lfloor y \rfloor ^2 \equal{} 25.\] The region $ \mathcal{R}$ is completely contained in a disk of radius $ r$ (a disk is the union of a circle and its interior). The minimum value of $ r$ can be written as $ \tfrac {\sqrt {m}}{n}$, where $ m$ and $ n$ are integers and $ m$ is not divisible by the square of any prime. Find $ m \plus{} n$.
2006 AMC 10, 1
Sandwiches at Joe's Fast Food cost $ \$3$ each and sodas cost $ \$2$ each. How many dollars will it cost to purchase 5 sandwiches and 8 sodas?
$ \textbf{(A) } 31\qquad \textbf{(B) } 32\qquad \textbf{(C) } 33\qquad \textbf{(D) } 34\qquad \textbf{(E) } 35$
2004 AIME Problems, 8
How many positive integer divisors of $2004^{2004}$ are divisible by exactly $2004$ positive integers?
1967 AMC 12/AHSME, 25
For every odd number $p>1$ we have:
$\textbf{(A)}\ (p-1)^{\frac{1}{2}(p-1)}-1 \; \text{is divisible by} \; p-2\qquad
\textbf{(B)}\ (p-1)^{\frac{1}{2}(p-1)}+1 \; \text{is divisible by} \; p\\
\textbf{(C)}\ (p-1)^{\frac{1}{2}(p-1)} \; \text{is divisible by} \; p\qquad
\textbf{(D)}\ (p-1)^{\frac{1}{2}(p-1)}+1 \; \text{is divisible by} \; p+1\\
\textbf{(E)}\ (p-1)^{\frac{1}{2}(p-1)}-1 \; \text{is divisible by} \; p-1$
1989 AIME Problems, 8
Assume that $x_1,x_2,\ldots,x_7$ are real numbers such that
\[ \begin{array}{r} x_1+4x_2+9x_3+16x_4+25x_5+36x_6+49x_7=1\,\,\,\,\,\,\,\, \\ 4x_1+9x_2+16x_3+25x_4+36x_5+49x_6+64x_7=12\,\,\,\,\, \\ 9x_1+16x_2+25x_3+36x_4+49x_5+64x_6+81x_7=123. \\ \end{array} \] Find the value of \[16x_1+25x_2+36x_3+49x_4+64x_5+81x_6+100x_7.\]
2013 AMC 10, 20
The number $2013$ is expressed in the form \[2013=\frac{a_1!a_2!\cdots a_m!}{b_1!b_2!\cdots b_n!},\] where $a_1\ge a_2\ge\cdots\ge a_m$ and $b_1\ge b_2\ge\cdots\ge b_n$ are positive integers and $a_1+b_1$ is as small as possible. What is $|a_1-b_1|$?
${ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D}}\ 4\qquad\textbf{(E)}\ 5 $
2023 AMC 12/AHSME, 9
A square of area $2$ is inscribed in a square of area $3$, creating four congruent triangles, as shown below. What is the ratio of the shorter leg to the longer leg in the shaded right triangle?
[asy]
size(200);
defaultpen(linewidth(0.6pt)+fontsize(10pt));
real y = sqrt(3);
pair A,B,C,D,E,F,G,H;
A = (0,0);
B = (0,y);
C = (y,y);
D = (y,0);
E = ((y + 1)/2,y);
F = (y, (y - 1)/2);
G = ((y - 1)/2, 0);
H = (0,(y + 1)/2);
fill(H--B--E--cycle, gray);
draw(A--B--C--D--cycle);
draw(E--F--G--H--cycle);
[/asy]
$\textbf{(A) }\frac15\qquad\textbf{(B) }\frac14\qquad\textbf{(C) }2-\sqrt3\qquad\textbf{(D) }\sqrt3-\sqrt2\qquad\textbf{(E) }\sqrt2-1$
2021 AMC 10 Fall, 23
Each of the $5{ }$ sides and the $5{ }$ diagonals of a regular pentagon are randomly and independently colored red or blue with equal probability. What is the probability that there will be a triangle whose vertices are among the vertices of the pentagon such that all of its sides have the same color?
$(\textbf{A})\: \frac23\qquad(\textbf{B}) \: \frac{105}{128}\qquad(\textbf{C}) \: \frac{125}{128}\qquad(\textbf{D}) \: \frac{253}{256}\qquad(\textbf{E}) \: 1$
2005 AMC 12/AHSME, 15
The sum of four two-digit numbers is $ 221$. None of the eight digits is $ 0$ and no two of them are same. Which of the following is [b]not[/b] included among the eight digits?
$ \textbf{(A)}\ 1\qquad
\textbf{(B)}\ 2\qquad
\textbf{(C)}\ 3\qquad
\textbf{(D)}\ 4\qquad
\textbf{(E)}\ 5$
2020 CHMMC Winter (2020-21), 6
Anna and Bob are playing a game on a rectangular board with $i$ rows and $j$ columns. Anna and Bob alternate turns with Anna going first. On each turn, a player places a penny in a square and then all squares in the same row and column of that square are marked. A player cannot place a penny in any marked square. When a player cannot place a penny in any square, they lose and the other player wins. How many ordered pairs of integers $(i, j)$ with $1 \le i \le 2020, 1 \le j \le 2020$ are there such that Anna wins?
1986 AIME Problems, 3
If $\tan x+\tan y=25$ and $\cot x + \cot y=30$, what is $\tan(x+y)$?
2015 AMC 12/AHSME, 9
Larry and Julius are playing a game, taking turns throwing a ball at a bottle sitting on a ledge. Larry throws first. The winner is the first person to knock the bottle off the ledge. At each turn the probability that a player knocks the bottle off the ledge is $\frac{1}{2}$, independently of what has happened before. What is the probability that Larry wins the game?
$\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }\frac{3}{5}\qquad\textbf{(C) }\frac{2}{3}\qquad\textbf{(D) }\frac{3}{4}\qquad\textbf{(E) }\frac{4}{5}$
1974 AMC 12/AHSME, 7
A town's population increased by $1,200$ people, and then this new population decreased by $11 \%$. The town now had $32$ less people than it did before the $1,200$ increase. What is the original population?
$ \textbf{(A)}\ 1,200 \qquad\textbf{(B)}\ 11,200 \qquad\textbf{(C)}\ 9,968 \qquad\textbf{(D)}\ 10,000 \qquad\textbf{(E)}\ \text{none of these} $
2015 AIME Problems, 9
Let $S$ be the set of all ordered triples of integers $(a_1,a_2,a_3)$ with $1 \le a_1,a_2,a_3 \le 10$. Each ordered triple in $S$ generates a sequence according to the rule $a_n=a_{n-1}\cdot | a_{n-2}-a_{n-3} |$ for all $n\ge 4$. Find the number of such sequences for which $a_n=0$ for some $n$.
2020 AMC 10, 8
Points $P$ and $Q$ lie in a plane with $PQ=8$. How many locations for point $R$ in this plane are there such that the triangle with vertices $P,$ $Q,$ and $R$ is a right triangle with area $12$ square units?
$\textbf{(A) } 2 \qquad\textbf{(B) } 4 \qquad\textbf{(C) } 6 \qquad\textbf{(D) }8 \qquad\textbf{(E) } 12$
2014 AMC 10, 8
A truck travels $\frac{b}{6}$ feet every $t$ seconds. There are $3$ feet in a yard. How many yards does the truck travel in $3$ minutes?
$ \textbf {(A) } \frac{b}{1080t} \qquad \textbf {(B) } \frac{30t}{b} \qquad \textbf {(C) } \frac{30b}{t}\qquad \textbf {(D) } \frac{10t}{b} \qquad \textbf {(E) } \frac{10b}{t}$
2020 AMC 10, 4
The acute angles of a right triangle are $a^{\circ}$ and $b^{\circ}$, where $a>b$ and both $a$ and $b$ are prime numbers. What is the least possible value of $b$?
$\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }5\qquad\textbf{(D) }7\qquad\textbf{(E) }11$
1960 AMC 12/AHSME, 2
It takes $5$ seconds for a clock to strike $6$ o'clock beginning at $6:00$ o'clock precisely. If the strikings are uniformly spaced, how long, in seconds, does it take to strike $12$ o'clock?
$ \textbf{(A) }9\frac{1}{5} \qquad\textbf{(B) }10\qquad\textbf{(C) }11\qquad\textbf{(D) }14\frac{2}{5}\qquad\textbf{(E) }\text{none of these} $
2022 AMC 10, 25
Let $R$, $S$, and $T$ be squares that have vertices at lattice points (i.e., points whose coordinates are both integers) in the coordinate plane, together with their interiors. The bottom edge of each square is on the x-axis. The left edge of $R$ and the right edge of $S$ are on the $y$-axis, and $R$ contains $\frac{9}{4}$ as many lattice points as does $S$. The top two vertices of $T$ are in $R \cup S$, and $T$ contains $\frac{1}{4}$ of the lattice points contained in $R \cup S$. See the figure (not drawn to scale).
[asy]
//kaaaaaaaaaante314
size(8cm);
import olympiad;
label(scale(.8)*"$y$", (0,60), N);
label(scale(.8)*"$x$", (60,0), E);
filldraw((0,0)--(55,0)--(55,55)--(0,55)--cycle, yellow+orange+white+white);
label(scale(1.3)*"$R$", (55/2,55/2));
filldraw((0,0)--(0,28)--(-28,28)--(-28,0)--cycle, green+white+white);
label(scale(1.3)*"$S$",(-14,14));
filldraw((-10,0)--(15,0)--(15,25)--(-10,25)--cycle, red+white+white);
label(scale(1.3)*"$T$",(3.5,25/2));
draw((0,-10)--(0,60),EndArrow(TeXHead));
draw((-34,0)--(60,0),EndArrow(TeXHead));[/asy]
The fraction of lattice points in $S$ that are in $S \cap T$ is 27 times the fraction of lattice points in $R$ that are in $R \cap T$. What is the minimum possible value of the edge length of $R$ plus the edge length of $S$ plus the edge length of $T$?
$\textbf{(A) }336\qquad\textbf{(B) }337\qquad\textbf{(C) }338\qquad\textbf{(D) }339\qquad\textbf{(E) }340$
1985 ITAMO, 8
The sum of the following seven numbers is exactly 19:
\[a_1=2.56,\qquad a_2=2.61,\qquad a_3=2.65,\qquad a_4=2.71,\]
\[a_5=2.79,\qquad a_6=2.82,\qquad a_7=2.86.\]
It is desired to replace each $a_i$ by an integer approximation $A_i$, $1 \le i \le 7$, so that the sum of the $A_i$'s is also 19 and so that $M$, the maximum of the "errors" $|A_i - a_i|$, is as small as possible. For this minimum $M$, what is $100M$?
1964 AMC 12/AHSME, 33
$P$ is a point interior to rectangle $ABCD$ and such that $PA=3$ inches, $PD=4$ inches, and $PC=5$ inches. Then $PB$, in inches, equals:
$\textbf{(A) }2\sqrt{3}\qquad\textbf{(B) }3\sqrt{2}\qquad\textbf{(C) }3\sqrt{3}\qquad\textbf{(D) }4\sqrt{2}\qquad \textbf{(E) }2$
[asy]
draw((0,0)--(6.5,0)--(6.5,4.5)--(0,4.5)--cycle);
draw((2.5,1.5)--(0,0));
draw((2.5,1.5)--(0,4.5));
draw((2.5,1.5)--(6.5,4.5));
draw((2.5,1.5)--(6.5,0),linetype("8 8"));
label("$A$",(0,0),dir(-135));
label("$B$",(6.5,0),dir(-45));
label("$C$",(6.5,4.5),dir(45));
label("$D$",(0,4.5),dir(135));
label("$P$",(2.5,1.5),dir(-90));
label("$3$",(1.25,0.75),dir(120));
label("$4$",(1.25,3),dir(35));
label("$5$",(4.5,3),dir(120));
//Credit to bobthesmartypants for the diagram
[/asy]