Found problems: 3632
2006 AMC 12/AHSME, 25
How many non-empty subsets $ S$ of $ \{1, 2, 3, \ldots, 15\}$ have the following two properties?
(1) No two consecutive integers belong to $ S$.
(2) If $ S$ contains $ k$ elements, then $ S$ contains no number less than $ k$.
$ \textbf{(A) } 277\qquad \textbf{(B) } 311\qquad \textbf{(C) } 376\qquad \textbf{(D) } 377\qquad \textbf{(E) } 405$
2023 AMC 10, 16
In a table tennis tournament every participant played every other participant exactly once. Although there were twice as many right-handed players as left-handed players, the number of games won by left-handed players was $40\%$ more than the number of games won by right-handed players. (There were no ties and no ambidextrous players.) What is the total number of games played?
$\textbf{(A) }15\qquad\textbf{(B) }36\qquad\textbf{(C) }45\qquad\textbf{(D) }48\qquad\textbf{(E) }66$
2008 AMC 12/AHSME, 9
Points $ A$ and $ B$ are on a circle of radius $ 5$ and $ AB\equal{}6$. Point $ C$ is the midpoint of the minor arc $ AB$. What is the length of the line segment $ AC$?
$ \textbf{(A)}\ \sqrt{10} \qquad
\textbf{(B)}\ \frac{7}{2} \qquad
\textbf{(C)}\ \sqrt{14} \qquad
\textbf{(D)}\ \sqrt{15} \qquad
\textbf{(E)}\ 4$
2011 AMC 10, 4
Let $X$ and $Y$ be the following sums of arithmetic sequences: \begin{eqnarray*} X &=& 10 + 12 + 14 + \cdots + 100, \\ Y &=& 12 + 14 + 16 + \cdots + 102. \end{eqnarray*} What is the value of $Y - X$?
$ \textbf{(A)}\ 92\qquad\textbf{(B)}\ 98\qquad\textbf{(C)}\ 100\qquad\textbf{(D)}\ 102\qquad\textbf{(E)}\ 112 $
1997 AMC 8, 12
$\angle 1 + \angle 2 = 180^\circ $
$\angle 3 = \angle 4$
Find $\angle 4.$
[asy]pair H,I,J,K,L;
H = (0,0); I = 10*dir(70); J = I + 10*dir(290); K = J + 5*dir(110); L = J + 5*dir(0);
draw(H--I--J--cycle);
draw(K--L--J);
draw(arc((0,0),dir(70),(1,0),CW)); label("$70^\circ$",dir(35),NE);
draw(arc(I,I+dir(250),I+dir(290),CCW)); label("$40^\circ$",I+1.25*dir(270),S);
label("$1$",J+0.25*dir(162.5),NW); label("$2$",J+0.25*dir(17.5),NE);
label("$3$",L+dir(162.5),WNW); label("$4$",K+dir(-52.5),SE);
[/asy]
$\textbf{(A)}\ 20^\circ \qquad \textbf{(B)}\ 25^\circ \qquad \textbf{(C)}\ 30^\circ \qquad \textbf{(D)}\ 35^\circ \qquad \textbf{(E)}\ 40^\circ$
1986 AMC 12/AHSME, 15
A student attempted to compute the average $A$ of $x$, $y$ and $z$ by computing the average of $x$ and $y$, and then computing the average of the result and $z$. Whenever $x < y < z$, the student's final result is
$\textbf{(A)}\ \text{correct}$
$\textbf{(B)}\ \text{always less than A}$
$\textbf{(C)}\ \text{always greater than A}$
$\textbf{(D)}\ \text{sometimes less than A and sometimes equal to A}$
$\textbf{(E)}\ \text{sometimes greater than A and sometimes equal to A}$
2006 AMC 12/AHSME, 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$
2018 AMC 10, 3
A unit of blood expires after $10!=10\cdot 9 \cdot 8 \cdots 1$ seconds. Yasin donates a unit of blood at noon of January 1. On what day does his unit of blood expire?
$\textbf{(A) }\text{January 2}\qquad\textbf{(B) }\text{January 12}\qquad\textbf{(C) }\text{January 22}\qquad\textbf{(D) }\text{February 11}\qquad\textbf{(E) }\text{February 12}$
2020 AIME Problems, 14
Let $P(x)$ be a quadratic polynomial with complex coefficients whose $x^2$ coefficient is $1$. Suppose the equation $P(P(x))=0$ has four distinct solutions, $x=3,4,a,b$. Find the sum of all possible values of $(a+b)^2$.
2009 AIME Problems, 8
Dave rolls a fair six-sided die until a six appears for the first time. Independently, Linda rolls a fair six-sided die until a six appears for the first time. Let $ m$ and $ n$ be relatively prime positive integers such that $ \frac{m}{n}$ is the probability that the number of times Dave rolls his die is equal to or within one of the number of times Linda rolls her die. Find $ m\plus{}n$.
2016 AMC 10, 18
In how many ways can $345$ be written as the sum of an increasing sequence of two or more consecutive positive integers?
$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 7$
2019 AMC 12/AHSME, 18
Square pyramid $ABCDE$ has base $ABCD,$ which measures $3$ cm on a side, and altitude $\overline{AE}$ perpendicular to the base$,$ which measures $6$ cm. Point $P$ lies on $\overline{BE},$ one third of the way from $B$ to $E;$ point $Q$ lies on $\overline{DE},$ one third of the way from $D$ to $E;$ and point $R$ lies on $\overline{CE},$ two thirds of the way from $C$ to $E.$ What is the area, in square centimeters, of $\triangle PQR?$
$\textbf{(A) } \frac{3\sqrt2}{2} \qquad\textbf{(B) } \frac{3\sqrt3}{2} \qquad\textbf{(C) } 2\sqrt2 \qquad\textbf{(D) } 2\sqrt3 \qquad\textbf{(E) } 3\sqrt2$
2017 AMC 10, 4
Suppose that $x$ and $y$ are nonzero real numbers such that \[\frac{3x+y}{x-3y}= -2.\] What is the value of \[\frac{x+3y}{3x-y}?\]
$\textbf{(A) } {-3} \qquad \textbf{(B) } {-1} \qquad \textbf{(C) } 1 \qquad \textbf{(D) }2 \qquad \textbf{(E) } 3$
2013 AMC 10, 17
Daphne is visited periodically by her three best friends: Alice, Beatrix, and Claire. Alice visits every third day, Beatrix visits every fourth day, and Claire visits every fifth day. All three friends visited Daphne yesterday. How many days of the next $365$-day period will exactly two friends visit her?
$\textbf{(A) }48\qquad
\textbf{(B) }54\qquad
\textbf{(C) }60\qquad
\textbf{(D) }66\qquad
\textbf{(E) }72\qquad$
1968 AMC 12/AHSME, 15
Let $P$ be the product of any three consecutive positive odd integers. The largest integer dividing all such $P$ is:
$\textbf{(A)}\ 15 \qquad
\textbf{(B)}\ 6 \qquad
\textbf{(C)}\ 5 \qquad
\textbf{(D)}\ 3 \qquad
\textbf{(E)}\ 1 $
2021 AMC 12/AHSME Fall, 14
Suppose that $P(z), Q(z)$, and $R(z)$ are polynomials with real coefficients, having degrees $2$, $3$, and $6$, respectively, and constant terms $1$, $2$, and $3$, respectively. Let $N$ be the number of distinct complex numbers $z$ that satisfy the equation $P(z) \cdot Q(z)=R(z)$. What is the minimum possible value of $N$?
$\textbf{(A)}\: 0\qquad\textbf{(B)} \: 1\qquad\textbf{(C)} \: 2\qquad\textbf{(D)} \: 3\qquad\textbf{(E)} \: 5$
2019 AMC 10, 18
For some positive integer $k$, the repeating base-$k$ representation of the (base-ten) fraction $\frac{7}{51}$ is $0.\overline{23}_k = 0.232323..._k$. What is $k$?
$\textbf{(A) } 13 \qquad\textbf{(B) } 14 \qquad\textbf{(C) } 15 \qquad\textbf{(D) } 16 \qquad\textbf{(E) } 17$
2007 AMC 10, 21
A sphere is inscribed in a cube that has a surface area of $ 24$ square meters. A second cube is then inscribed within the sphere. What is the surface area in square meters of the inner cube?
$ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ 12$
2003 AMC 8, 22
The following figures are composed of squares and circles. Which figure has a shaded region with largest area?
[asy]/* AMC8 2003 #22 Problem */
size(3inch, 2inch);
unitsize(1cm);
pen outline = black+linewidth(1);
filldraw((0,0)--(2,0)--(2,2)--(0,2)--cycle, mediumgrey, outline);
filldraw(shift(3,0)*((0,0)--(2,0)--(2,2)--(0,2)--cycle), mediumgrey, outline);
filldraw(Circle((7,1), 1), mediumgrey, black+linewidth(1));
filldraw(Circle((1,1), 1), white, outline);
filldraw(Circle((3.5,.5), .5), white, outline);
filldraw(Circle((4.5,.5), .5), white, outline);
filldraw(Circle((3.5,1.5), .5), white, outline);
filldraw(Circle((4.5,1.5), .5), white, outline);
filldraw((6.3,.3)--(7.7,.3)--(7.7,1.7)--(6.3,1.7)--cycle, white, outline);
label("A", (1, 2), N);
label("B", (4, 2), N);
label("C", (7, 2), N);
draw((0,-.5)--(.5,-.5), BeginArrow);
draw((1.5, -.5)--(2, -.5), EndArrow);
label("2 cm", (1, -.5));
draw((3,-.5)--(3.5,-.5), BeginArrow);
draw((4.5, -.5)--(5, -.5), EndArrow);
label("2 cm", (4, -.5));
draw((6,-.5)--(6.5,-.5), BeginArrow);
draw((7.5, -.5)--(8, -.5), EndArrow);
label("2 cm", (7, -.5));
draw((6,1)--(6,-.5), linetype("4 4"));
draw((8,1)--(8,-.5), linetype("4 4"));[/asy]
$ \textbf{(A)}\ \text{A only}\qquad\textbf{(B)}\ \text{B only}\qquad\textbf{(C)}\ \text{C only}\qquad\textbf{(D)}\ \text{both A and B}\qquad\textbf{(E)}\ \text{all are equal}$
2024 AMC 12/AHSME, 21
The measures of the smallest angles of three different right triangles sum to $90^\circ$. All three triangles have side lengths that are primitive Pythagorean triples. Two of them are $3-4-5$ and $5-12-13$. What is the perimeter of the third triangle?
$
\textbf{(A) }40 \qquad
\textbf{(B) }126 \qquad
\textbf{(C) }154 \qquad
\textbf{(D) }176 \qquad
\textbf{(E) }208 \qquad
$
1993 AMC 12/AHSME, 23
Points $A, B, C$ and $D$ are on a circle of diameter $1$, and $X$ is on diameter $\overline{AD}$. If $BX=CX$ and $3 \angle BAC=\angle BXC=36^{\circ}$, then $AX=$
[asy]
draw(Circle((0,0),10));
draw((-10,0)--(8,6)--(2,0)--(8,-6)--cycle);
draw((-10,0)--(10,0));
dot((-10,0));
dot((2,0));
dot((10,0));
dot((8,6));
dot((8,-6));
label("A", (-10,0), W);
label("B", (8,6), NE);
label("C", (8,-6), SE);
label("D", (10,0), E);
label("X", (2,0), NW);
[/asy]
$ \textbf{(A)}\ \cos 6^{\circ}\cos 12^{\circ} \sec 18^{\circ} \qquad\textbf{(B)}\ \cos 6^{\circ}\sin 12^{\circ} \csc 18^{\circ} \qquad\textbf{(C)}\ \cos 6^{\circ}\sin 12^{\circ} \sec 18^{\circ} \\ \qquad\textbf{(D)}\ \sin 6^{\circ}\sin 12^{\circ} \csc 18^{\circ} \qquad\textbf{(E)}\ \sin 6^{\circ} \sin 12^{\circ} \sec 18^{\circ} $
2002 AIME Problems, 10
In the diagram below, angle $ABC$ is a right angle. Point $D$ is on $\overline{BC}$, and $\overline{AD}$ bisects angle $CAB$. Points $E$ and $F$ are on $\overline{AB}$ and $\overline{AC}$, respectively, so that $AE=3$ and $AF=10.$ Given that $EB=9$ and $FC=27$, find the integer closest to the area of quadrilateral $DCFG.$
[asy]
size(250);
pair A=(0,12), E=(0,8), B=origin, C=(24*sqrt(2),0), D=(6*sqrt(2),0), F=A+10*dir(A--C), G=intersectionpoint(E--F, A--D);
draw(A--B--C--A--D^^E--F);
pair point=G+1*dir(250);
label("$A$", A, dir(point--A));
label("$B$", B, dir(point--B));
label("$C$", C, dir(point--C));
label("$D$", D, dir(point--D));
label("$E$", E, dir(point--E));
label("$F$", F, dir(point--F));
label("$G$", G, dir(point--G));
markscalefactor=0.1;
draw(rightanglemark(A,B,C));
label("10", A--F, dir(90)*dir(A--F));
label("27", F--C, dir(90)*dir(F--C));
label("3", (0,10), W);
label("9", (0,4), W);[/asy]
1967 AMC 12/AHSME, 34
Points $D$, $E$, $F$ are taken respectively on sides $AB$, $BC$, and $CA$ of triangle $ABC$ so that $AD:DB=BE:CE=CF:FA=1:n$. The ratio of the area of triangle $DEF$ to that of triangle $ABC$ is:
$\textbf{(A)}\ \frac{n^2-n+1}{(n+1)^2}\qquad
\textbf{(B)}\ \frac{1}{(n+1)^2}\qquad
\textbf{(C)}\ \frac{2n^2}{(n+1)^2}\qquad
\textbf{(D)}\ \frac{n^2}{(n+1)^2}\qquad
\textbf{(E)}\ \frac{n(n-1)}{n+1}$
2011 USAJMO, 6
Consider the assertion that for each positive integer $n\geq2$, the remainder upon dividing $2^{2^n}$ by $2^n-1$ is a power of $4$. Either prove the assertion or find (with proof) a counterexample.
2022 AMC 10, 24
How many strings of length $5$ formed from the digits $0$,$1$,$2$,$3$,$4$ are there such that for each $j\in\{1,2,3,4\}$, at least $j$ of the digits are less than $j$? (For example, $02214$ satisfies the condition because it contains at least $1$ digit less than $1$, at least $2$ digits less than $2$, at least $3$ digits less than $3$, and at least $4$ digits less than $4$. The string $23404$ does not satisfy the condition because it does not contain at least $2$ digits less than $2$.)
$\textbf{(A) }500\qquad\textbf{(B) }625\qquad\textbf{(C) }1089\qquad\textbf{(D) }1199\qquad\textbf{(E) }1296$