Found problems: 3632
2013 AMC 8, 14
Abe holds 1 green and 1 red jelly bean in his hand. Bea holds 1 green, 1 yellow, and 2 red jelly beans in her hand. Each randomly picks a jelly bean to show the other. What is the probability that the colors match?
$\textbf{(A)}\ \frac14 \qquad \textbf{(B)}\ \frac13 \qquad \textbf{(C)}\ \frac38 \qquad \textbf{(D)}\ \frac12 \qquad \textbf{(E)}\ \frac23$
1991 AMC 12/AHSME, 30
For any set $S$, let $|S|$ denote the number of elements in $S$, and let $n(S)$ be the number of subsets of $S$, including the empty set and the set $S$ itself. If $A$, $B$ and $C$ are sets for which \[n(A) + n(B) + n(C) = n(A \cup B \cup C)\quad\text{and}\quad |A| = |B| = 100,\] then what is the minimum possible value of $|A \cap B \cap C|$?
$ \textbf{(A)}\ 96\qquad\textbf{(B)}\ 97\qquad\textbf{(C)}\ 98\qquad\textbf{(D)}\ 99\qquad\textbf{(E)}\ 100 $
1959 AMC 12/AHSME, 32
The length $l$ of a tangent, drawn from a point $A$ to a circle, is $\frac43$ of the radius $r$. The (shortest) distance from $A$ to the circle is:
$ \textbf{(A)}\ \frac{1}{2}r \qquad\textbf{(B)}\ r\qquad\textbf{(C)}\ \frac{1}{2}l\qquad\textbf{(D)}\ \frac23l \qquad\textbf{(E)}\ \text{a value between r and l.} $
2020 CHMMC Winter (2020-21), 5
Suppose that a professor has $n \ge 4$ students. Let $P$ denote the set of all ordered pairs $(n, k)$ such that the number of ways for the professor to choose one pair of students equals the number of ways for the professor to choose $k > 1$ pairs of students. For each such ordered pair $(n, k) \in P$, consider the sum $n+k=s$. Find the sum of all $s$ over all ordered pairs $(n, k)$ in $P$.
[i]If the same value of $s$ appears in multiple distinct elements $(n, k)$ in $P$, count this value multiple times.[/i]
2022 AIME Problems, 1
Quadratic polynomials $P(x)$ and $Q(x)$ have leading coefficients of $2$ and $-2$, respectively. The graphs of both polynomials pass through the two points $(16,54)$ and $(20,53)$. Find ${P(0) + Q(0)}$.
2022 AMC 12/AHSME, 5
Let the [i]taxicab distance[/i] between points $(x_1,y_1)$ and $(x_2,y_2)$ in the coordinate plane is given by $|x_1-x_2|+|y_1-y_2|$. For how many points $P$ with integer coordinates is the taxicab distance between $P$ and the origin less than or equal to $20$?
$\textbf{(A) }441\qquad\textbf{(B) }761\qquad\textbf{(C) }841\qquad\textbf{(D) }921\qquad\textbf{(E) }924$
2012 AMC 12/AHSME, 10
A triangle has area $30$, one side of length $10$, and the median to that side of length $9$. Let $\theta$ be the acute angle formed by that side and the median. What is $\sin{\theta}$?
$ \textbf{(A)}\ \frac{3}{10}\qquad\textbf{(B)}\ \frac{1}{3}\qquad\textbf{(C)}\ \frac{9}{20}\qquad\textbf{(D)}\ \frac{2}{3}\qquad\textbf{(E)}\ \frac{9}{10} $
2021 AMC 10 Spring, 2
What is the value of $\sqrt{(3-2\sqrt{3})^2}+\sqrt{(3+2\sqrt{3})^2}$?
$\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 4\sqrt{3}-6 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 4\sqrt{3} \qquad \textbf{(E)}\ 4\sqrt{3}+6$
1964 AMC 12/AHSME, 39
The magnitudes of the sides of triangle $ABC$ are $a$, $b$, and $c$, as shown, with $c\le b\le a$. Through interior point $P$ and the vertices $A$, $B$, $C$, lines are drawn meeting the opposite sides in $A'$, $B'$, $C'$, respectively. Let $s=AA'+BB'+CC'$. Then, for all positions of point $P$, $s$ is less than:
$\textbf{(A) }2a+b\qquad\textbf{(B) }2a+c\qquad\textbf{(C) }2b+c\qquad\textbf{(D) }a+2b\qquad \textbf{(E) }$ $a+b+c$
[asy]
import math;
defaultpen(fontsize(11pt));
pair A = (0,0), B = (1,3), C = (5,0), P = (1.5,1);
pair X = extension(B,C,A,P), Y = extension(A,C,B,P), Z = extension(A,B,C,P);
draw(A--B--C--cycle);
draw(A--X);
draw(B--Y);
draw(C--Z);
dot(P);
dot(A);
dot(B);
dot(C);
label("$A$",A,dir(210));
label("$B$",B,dir(90));
label("$C$",C,dir(-30));
label("$A'$",X,dir(-100));
label("$B'$",Y,dir(65));
label("$C'$",Z,dir(20));
label("$P$",P,dir(70));
label("$a$",X,dir(80));
label("$b$",Y,dir(-90));
label("$c$",Z,dir(110));
//Credit to bobthesmartypants for the diagram
[/asy]
2023 AMC 12/AHSME, 11
What is the degree measure of the acute angle formed by lines with slopes $2$ and $\tfrac{1}{3}$?
$\textbf{(A)}~30\qquad\textbf{(B)}~37.5\qquad\textbf{(C)}~45\qquad\textbf{(D)}~52.5\qquad\textbf{(E)}~60$
1978 USAMO, 4
(a) Prove that if the six dihedral (i.e. angles between pairs of faces) of a given tetrahedron are congruent, then the tetrahedron is regular.
(b) Is a tetrahedron necessarily regular if five dihedral angles are congruent?
2014 AMC 8, 22
A $2$-digit number is such that the product of the digits plus the sum of the digits is equal to the number. What is the units digit of the number?
$\textbf{(A) }1\qquad\textbf{(B) }3\qquad\textbf{(C) }5\qquad\textbf{(D) }7\qquad \textbf{(E) }9$
2023 AMC 10, 25
If $A$ and $B$ are vertices of a polyhedron, define the [i]distance[/i] $d(A, B)$ to be the minimum number of edges of the polyhedron one must traverse in order to connect $A$ and $B$. For example, if $\overline{AB}$ is an edge of the polyhedron, then $d(A, B) = 1$, but if $\overline{AC}$ and $\overline{CB}$ are edges and $\overline{AB}$ is not an edge, then $d(A, B) = 2$. Let $Q$, $R$, and $S$ be randomly chosen distinct vertices of a regular icosahedron (regular polyhedron made up of 20 equilateral triangles). What is the probability that $d(Q, R) > d(R, S)$?
$\textbf{(A)}~\frac{7}{22}\qquad\textbf{(B)}~\frac13\qquad\textbf{(C)}~\frac38\qquad\textbf{(D)}~\frac5{12}\qquad\textbf{(E)}~\frac12$
2023 AMC 10, 5
How many digits are in the base-ten representation of $8^5 \cdot 5^{10} \cdot 15^5$?
$\textbf{(A)}~14\qquad\textbf{(B)}~15\qquad\textbf{(C)}~16\qquad\textbf{(D)}~17\qquad\textbf{(E)}~18\qquad$
2006 AMC 10, 2
For real numbers $ x$ and $ y$, define $ x\spadesuit y \equal{} (x \plus{} y)(x \minus{} y)$. What is $ 3\spadesuit(4\spadesuit 5)$?
$ \textbf{(A) } \minus{} 72 \qquad \textbf{(B) } \minus{} 27 \qquad \textbf{(C) } \minus{} 24 \qquad \textbf{(D) } 24 \qquad \textbf{(E) } 72$
2008 AMC 10, 7
An equilateral triangle of side length $ 10$ is completely filled in by non-overlapping equilateral triangles of side length $ 1$. How many small triangles are required?
$ \textbf{(A)}\ 10 \qquad
\textbf{(B)}\ 25 \qquad
\textbf{(C)}\ 100 \qquad
\textbf{(D)}\ 250 \qquad
\textbf{(E)}\ 1000$
1992 AMC 12/AHSME, 13
How many pairs of positive integers $(a,b)$ with $a + b \le 100$ satisfy the equation $\frac{a + b^{-1}}{a^{-1} + b} = 13$?
$ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 13 $
1976 AMC 12/AHSME, 10
If $m,~n,~p,$ and $q$ are real numbers and $f(x)=mx+n$ and $g(x)=px+q$, then the equation $f(g(x))=g(f(x))$ has a solution
$\textbf{(A) }\text{for all choices of }m,~n,~p, \text{ and } q\qquad$
$\textbf{(B) }\text{if and only if }m=p\text{ and }n=q\qquad$
$\textbf{(C) }\text{if and only if }mq-np=0\qquad$
$\textbf{(D) }\text{if and only if }n(1-p)-q(1-m)=0\qquad$
$\textbf{(E) }\text{if and only if }(1-n)(1-p)-(1-q)(1-m)=0$
1992 AMC 12/AHSME, 8
A square floor is tiled with congruent square tiles. The tiles on the two diagonals of the floor are black. The rest of the tiles are white. If there are 101 black tiles, then the total number of tiles is
[asy]
size(200);
defaultpen(linewidth(0.7)+fontsize(10));
draw((0,0)--(0,3)^^(1,0)--(1,3)^^(2,0)--(2,3)^^(3,0)--(3,3)^^(0,0)--(3,0)^^(0,1)--(3,1)^^(0,2)--(3,2)^^(0,3)--(3,3));
draw((0,12)--(0,15)^^(1,12)--(1,15)^^(2,12)--(2,15)^^(3,12)--(3,15)^^(0,12)--(3,12)^^(0,13)--(3,13)^^(0,14)--(3,14)^^(0,15)--(3,15));
draw((12,0)--(12,3)^^(13,0)--(13,3)^^(14,0)--(14,3)^^(15,0)--(15,3)^^(12,0)--(15,0)^^(12,1)--(15,1)^^(12,2)--(15,2)^^(12,3)--(15,3));
draw((12,12)--(12,15)^^(13,12)--(13,15)^^(14,12)--(14,15)^^(15,12)--(15,15)^^(12,12)--(15,12)^^(12,13)--(15,13)^^(12,14)--(15,14)^^(12,15)--(15,15));
draw((5,5)--(5,10)^^(6,5)--(6,10)^^(7,5)--(7,10)^^(8,5)--(8,10)^^(9,5)--(9,10)^^(10,5)--(10,10)^^(5,5)--(10,5)^^(5,6)--(10,6)^^(5,7)--(10,7)^^(5,8)--(10,8)^^(5,9)--(10,9)^^(5,10)--(10,10));
draw((3.5,.2)--(11.5,.2)^^(3.5,1.5)--(11.5,1.5)^^(3.5,13.5)--(11.5,13.5)^^(3.5,14.8)--(11.5,14.8), linetype("1 7"));
draw((.2,3.5)--(.2,11.5)^^(1.5,3.5)--(1.5,11.5)^^(13.5,3.5)--(13.5,11.5)^^(14.8,3.5)--(14.8,11.5), linetype("1 7"));
draw((3.5,3.5)--(4.5,4.5)^^(3.5,11.5)--(4.5,10.5)^^(11.5,3.5)--(10.5,4.5)^^(11.5,11.5)--(10.5,10.5), linetype("1 7"));
fill((0,0)--(1,0)--(1,1)--(0,1)--cycle,black);
fill((1,1)--(2,1)--(2,2)--(1,2)--cycle,black);
fill((2,2)--(3,2)--(3,3)--(2,3)--cycle,black);
fill((0,14)--(1,14)--(1,15)--(0,15)--cycle,black);
fill((1,13)--(2,13)--(2,14)--(1,14)--cycle,black);
fill((2,12)--(3,12)--(3,13)--(2,13)--cycle,black);
fill((14,0)--(15,0)--(15,1)--(14,1)--cycle,black);
fill((13,1)--(14,1)--(14,2)--(13,2)--cycle,black);
fill((12,2)--(13,2)--(13,3)--(12,3)--cycle,black);
fill((14,14)--(15,14)--(15,15)--(14,15)--cycle,black);
fill((13,13)--(14,13)--(14,14)--(13,14)--cycle,black);
fill((12,12)--(13,12)--(13,13)--(12,13)--cycle,black);
fill((5,5)--(6,5)--(6,6)--(5,6)--cycle,black);
fill((6,6)--(7,6)--(7,7)--(6,7)--cycle,black);
fill((7,7)--(8,7)--(8,8)--(7,8)--cycle,black);
fill((8,8)--(9,8)--(9,9)--(8,9)--cycle,black);
fill((9,9)--(10,9)--(10,10)--(9,10)--cycle,black);
fill((5,9)--(6,9)--(6,10)--(5,10)--cycle,black);
fill((6,8)--(7,8)--(7,9)--(6,9)--cycle,black);
fill((8,6)--(9,6)--(9,7)--(8,7)--cycle,black);
fill((9,5)--(10,5)--(10,6)--(9,6)--cycle,black);
[/asy]
$ \textbf{(A)}\ 121\qquad\textbf{(B)}\ 625\qquad\textbf{(C)}\ 676\qquad\textbf{(D)}\ 2500\qquad\textbf{(E)}\ 2601 $
2018 AIME Problems, 11
Find the least positive integer $n$ such that when $3^n$ is written in base $143$, its two right-most digits in base $143$ are $01$.
2017 AMC 12/AHSME, 14
An ice-cream novelty item consists of a cup in the shape of a $4$-inch-tall frustum of a right circular cone, with a $2$-inch-diameter base at the bottom and a $4$-inch-diameter base at the top, packed solid with ice cream, together with a solid cone of ice cream of height $4$ inches, whose base, at the bottom, is the top base of the frustum. What is the total volume of the ice cream, in cubic inches?
$\textbf{(A)}\ 8\pi\qquad\textbf{(B)}\ \frac{28\pi}{3}\qquad\textbf{(C)}\ 12\pi\qquad\textbf{(D)}\ 14\pi\qquad\textbf{(E)}\ \frac{44\pi}{3}$
1961 AMC 12/AHSME, 13
The symbol $|a|$ means $a$ is a positive number or zero, and $-a$ if $a$ is a negative number. For all real values of $t$ the expression $\sqrt{t^4+t^2}$ is equal to:
${{ \textbf{(A)}\ t^3 \qquad\textbf{(B)}\ t^2+t \qquad\textbf{(C)}\ |t^2+t| \qquad\textbf{(D)}\ t\sqrt{t^2+1} }\qquad\textbf{(E)}\ |t|\sqrt{1+t^2} } $
1967 AMC 12/AHSME, 24
The number of solution-pairs in the positive integers of the equation $3x+5y=501$ is:
$\textbf{(A)}\ 33\qquad
\textbf{(B)}\ 34\qquad
\textbf{(C)}\ 35\qquad
\textbf{(D)}\ 100\qquad
\textbf{(E)}\ \text{none of these}$
2014 AMC 12/AHSME, 4
Susie pays for $4$ muffins and $3$ bananas. Calvin spends twice as much paying for $2$ muffins and $16$ bananas. A muffin is how many times as expensive as a banana?
$ \textbf {(A) } \frac{3}{2} \qquad \textbf {(B) } \frac{5}{3} \qquad \textbf {(C) } \frac{7}{4} \qquad \textbf {(D) } 2 \qquad \textbf {(E) } \frac{13}{4}$
2024 USAJMO, 2
Let $m$ and $n$ be positive integers. Let $S$ be the set of integer points $(x,y)$ with $1\leq x\leq 2m$ and $1\leq y\leq 2n$. A configuration of $mn$ rectangles is called [i]happy[/i] if each point in $S$ is a vertex of exactly one rectangle, and all rectangles have sides parallel to the coordinate axes. Prove that the number of happy configurations is odd.
[i]Proposed by Serena An and Claire Zhang[/i]