Found problems: 3632
1992 AMC 12/AHSME, 18
The increasing sequence of positive integers $a_{1},a_{2},a_{3},\ldots$ has the property that $a_{n+2} = a_{n} + a_{n+1}$ for all $n \ge 1$. If $a_{7} = 120$, then $a_{8}$ is
$ \textbf{(A)}\ 128\qquad\textbf{(B)}\ 168\qquad\textbf{(C)}\ 193\qquad\textbf{(D)}\ 194\qquad\textbf{(E)}\ 210 $
1980 USAMO, 5
Prove that for numbers $a,b,c$ in the interval $[0,1]$, \[\frac{a}{b+c+1}+\frac{b}{c+a+1}+\frac{c}{a+b+1}+(1-a)(1-b)(1-c) \le 1.\]
2022 AMC 10, 16
The diagram below shows a rectangle with side lengths $4$ and $8$ and a square with side length $5$. Three vertices of the square lie on three different sides of the rectangle, as shown. What is the area of the region inside both the square and the rectangle?
[asy]
size(5cm);
filldraw((4,0)--(8,3)--(8-3/4,4)--(1,4)--cycle,mediumgray);
draw((0,0)--(8,0)--(8,4)--(0,4)--cycle,linewidth(1.1));
draw((1,0)--(1,4)--(4,0)--(8,3)--(5,7)--(1,4),linewidth(1.1));
label("$4$", (8,2), E);
label("$8$", (4,0), S);
label("$5$", (3,11/2), NW);
draw((1,.35)--(1.35,.35)--(1.35,0),linewidth(.4));
draw((5,7)--(5+21/100,7-28/100)--(5-7/100,7-49/100)--(5-28/100,7-21/100)--cycle,linewidth(.4));
[/asy]
$\textbf{(A) } 15\dfrac{1}{8} \qquad \textbf{(B) } 15\dfrac{3}{8} \qquad \textbf{(C) } 15\dfrac{1}{2} \qquad \textbf{(D) } 15\dfrac{5}{8} \qquad \textbf{(E) } 15\dfrac{7}{8}$
2005 MOP Homework, 6
Given a convex quadrilateral $ABCD$. The points $P$ and $Q$ are the midpoints of the diagonals $AC$ and $BD$ respectively. The line $PQ$ intersects the lines $AB$ and $CD$ at $N$ and $M$ respectively. Prove that the circumcircles of triangles $NAP$, $NBQ$, $MQD$, and $MPC$ have a common point.
2010 AMC 12/AHSME, 15
For how many ordered triples $ (x,y,z)$ of nonnegative integers less than $ 20$ are there exactly two distinct elements in the set $ \{i^x,(1 \plus{} i)^y,z\}$, where $ i \equal{} \sqrt { \minus{} 1}$?
$ \textbf{(A)}\ 149 \qquad
\textbf{(B)}\ 205 \qquad
\textbf{(C)}\ 215 \qquad
\textbf{(D)}\ 225 \qquad
\textbf{(E)}\ 235$
2018 AIME Problems, 4
In \(\triangle ABC, AB = AC = 10\) and \(BC = 12\). Point \(D\) lies strictly between \(A\) and \(B\) on \(\overline{AB}\) and point \(E\) lies strictly between \(A\) and \(C\) on \(\overline{AC}\) so that \(AD = DE = EC\). Then \(AD\) can be expressed in the form \(\tfrac{p}{q}\), where \(p\) and \(q\) are relatively prime positive integers. Find \(p + q\).
1960 AMC 12/AHSME, 12
The locus of the centers of all circles of given radius $a$, in the same plane, passing through a fixed point, is:
$ \textbf{(A) }\text{a point}\qquad\textbf{(B) }\text{ a straight line} \qquad\textbf{(C) }\text{two straight lines}\qquad\textbf{(D) }\text{a circle}\qquad$
$\textbf{(E) }\text{two circles} $
2015 AMC 12/AHSME, 15
What is the minimum number of digits to the right of the decimal point needed to express the fraction $\dfrac{123\,456\,789}{2^{26}\cdot 5^4}$ as a decimal?
$\textbf{(A) }4\qquad\textbf{(B) }22\qquad\textbf{(C) }26\qquad\textbf{(D) }30\qquad\textbf{(E) }104$
1990 AIME Problems, 12
A regular 12-gon is inscribed in a circle of radius 12. The sum of the lengths of all sides and diagonals of the 12-gon can be written in the form
\[ a + b \sqrt{2} + c \sqrt{3} + d \sqrt{6}, \]
where $a$, $b$, $c$, and $d$ are positive integers. Find $a + b + c + d$.
1970 AMC 12/AHSME, 1
The fourth power of $\sqrt{1+\sqrt{1+\sqrt{1}}}$ is:
${\textbf{(A) }\sqrt{2}+\sqrt{3}\qquad\textbf{(B) }\frac{1}{2}(7+3\sqrt{5}})\qquad\textbf{(C) }1+2\sqrt{3}\qquad\textbf{(D) }3\qquad \textbf{(E) }3+2\sqrt{2}$
2004 AIME Problems, 15
For all positive integers $ x$, let
\[ f(x) \equal{} \begin{cases}1 & \text{if }x \equal{} 1 \\
\frac x{10} & \text{if }x\text{ is divisible by 10} \\
x \plus{} 1 & \text{otherwise}\end{cases}\]and define a sequence as follows: $ x_1 \equal{} x$ and $ x_{n \plus{} 1} \equal{} f(x_n)$ for all positive integers $ n$. Let $ d(x)$ be the smallest $ n$ such that $ x_n \equal{} 1$. (For example, $ d(100) \equal{} 3$ and $ d(87) \equal{} 7$.) Let $ m$ be the number of positive integers $ x$ such that $ d(x) \equal{} 20$. Find the sum of the distinct prime factors of $ m$.
2014 AMC 12/AHSME, 8
In the addition shown below $A$, $B$, $C$, and $D$ are distinct digits. How many different values are possible for $D$?
\[\begin{array}{lr}
&ABBCB \\
+& BCADA \\
\hline
& DBDDD
\end{array}\]
$\textbf{(A) }2\qquad\textbf{(B) }4\qquad\textbf{(C) }7\qquad\textbf{(D) }8\qquad\textbf{(E) }9$
2014 AMC 10, 8
Which of the following numbers is a perfect square?
$ \textbf{(A)}\ \dfrac{14!15!}2\qquad\textbf{(B)}\ \dfrac{15!16!}2\qquad\textbf{(C)}\ \dfrac{16!17!}2\qquad\textbf{(D)}\ \dfrac{17!18!}2\qquad\textbf{(E)}\ \dfrac{18!19!}2 $
2025 AIME, 9
There are $n$ values of $x$ in the interval $0<x<2\pi$ where $f(x)=\sin(7\pi\cdot\sin(5x))=0$. For $t$ of these $n$ values of $x$, the graph of $y=f(x)$ is tangent to the $x$-axis. Find $n+t$.
2008 AMC 12/AHSME, 4
On circle $ O$, points $ C$ and $ D$ are on the same side of diameter $ \overline{AB}$, $ \angle AOC \equal{} 30^\circ$, and $ \angle DOB \equal{} 45^\circ$. What is the ratio of the area of the smaller sector $ COD$ to the area of the circle?
[asy]unitsize(6mm);
defaultpen(linewidth(0.7)+fontsize(8pt));
pair C = 3*dir (30);
pair D = 3*dir (135);
pair A = 3*dir (0);
pair B = 3*dir(180);
pair O = (0,0);
draw (Circle ((0, 0), 3));
label ("$C$", C, NE);
label ("$D$", D, NW);
label ("$B$", B, W);
label ("$A$", A, E);
label ("$O$", O, S);
label ("$45^\circ$", (-0.3,0.1), WNW);
label ("$30^\circ$", (0.5,0.1), ENE);
draw (A--B);
draw (O--D);
draw (O--C);[/asy]$ \textbf{(A)}\ \frac {2}{9} \qquad \textbf{(B)}\ \frac {1}{4} \qquad \textbf{(C)}\ \frac {5}{18} \qquad \textbf{(D)}\ \frac {7}{24} \qquad \textbf{(E)}\ \frac {3}{10}$
1960 AMC 12/AHSME, 28
The equation $x-\frac{7}{x-3}=3-\frac{7}{x-3}$ has:
$ \textbf{(A)}\ \text{infinitely many integral roots} \qquad\textbf{(B)}\ \text{no root} \qquad\textbf{(C)}\ \text{one integral root}\qquad$
$\textbf{(D)}\ \text{two equal integral roots} \qquad\textbf{(E)}\ \text{two equal non-integral roots} $
1963 AMC 12/AHSME, 8
The smallest positive integer $x$ for which $1260x=N^3$, where $N$ is an integer, is:
$\textbf{(A)}\ 1050 \qquad
\textbf{(B)}\ 1260 \qquad
\textbf{(C)}\ 1260^2 \qquad
\textbf{(D)}\ 7350 \qquad
\textbf{(E)}\ 44100$
2006 AMC 12/AHSME, 15
Circles with centers $ O$ and $ P$ have radii 2 and 4, respectively, and are externally tangent. Points $ A$ and $ B$ are on the circle centered at $ O$, and points $ C$ and $ D$ are on the circle centered at $ P$, such that $ \overline{AD}$ and $ \overline{BC}$ are common external tangents to the circles. What is the area of hexagon $ AOBCPD$?
[asy]
unitsize(0.4 cm); defaultpen(linewidth(0.7) + fontsize(11));
pair A, B, C, D;
pair[] O;
O[1] = (6,0);
O[2] = (12,0);
A = (32/6,8*sqrt(2)/6);
B = (32/6,-8*sqrt(2)/6);
C = 2*B;
D = 2*A;
draw(Circle(O[1],2));
draw(Circle(O[2],4));
draw((0.7*A)--(1.2*D));
draw((0.7*B)--(1.2*C));
draw(O[1]--O[2]);
draw(A--O[1]);
draw(B--O[1]);
draw(C--O[2]);
draw(D--O[2]);
label("$A$", A, NW);
label("$B$", B, SW);
label("$C$", C, SW);
label("$D$", D, NW);
dot("$O$", O[1], SE);
dot("$P$", O[2], SE);
label("$2$", (A + O[1])/2, E);
label("$4$", (D + O[2])/2, E);[/asy]
$ \textbf{(A) } 18\sqrt {3} \qquad \textbf{(B) } 24\sqrt {2} \qquad \textbf{(C) } 36 \qquad \textbf{(D) } 24\sqrt {3} \qquad \textbf{(E) } 32\sqrt {2}$
1986 AMC 12/AHSME, 16
In $\triangle ABC$, $AB = 8$, $BC = 7$, $CA = 6$ and side $BC$ is extended, as shown in the figure, to a point $P$ so that $\triangle PAB$ is similar to $\triangle PCA$. The length of $PC$ is
[asy]
size(200);
defaultpen(linewidth(0.7)+fontsize(10));
pair A=origin, P=(1.5,5), B=(8,0), C=P+2.5*dir(P--B);
draw(A--P--C--A--B--C);
label("$A$", A, W);
label("$B$", B, E);
label("$C$", C, NE);
label("$P$", P, NW);
label("$6$", 3*dir(A--C), SE);
label("$7$", B+3*dir(B--C), NE);
label("$8$", (4,0), S);[/asy]
$ \textbf{(A)}\ 7\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 11$
2017 AMC 10, 8
At a gathering of $30$ people, there are $20$ people who all know each other and $10$ people who know no one. People who know each other hug, and people who do not know each other shake hands. How many handshakes occur?
$\textbf{(A)}\ 240\qquad\textbf{(B)}\ 245\qquad\textbf{(C)}\ 290\qquad\textbf{(D)}\ 480\qquad\textbf{(E)}\ 490$
2006 AMC 10, 14
A number of linked rings, each 1 cm thick, are hanging on a peg. The top ring has an outside diameter of 20 cm. The outside diameter of each of the outer rings is 1 cm less than that of the ring above it. The bottom ring has an outside diameter of 3 cm. What is the distance, in cm, from the top of the top ring to the bottom of the bottom ring?
[asy]
size(200);
defaultpen(linewidth(3));
real[] inrad = {40,34,28,21};
real[] outrad = {55,49,37,30};
real[] center;
path[][] quad = new path[4][4];
center[0] = 0;
for(int i=0;i<=3;i=i+1) {
if(i != 0) {
center[i] = center[i-1] - inrad[i-1] - inrad[i]+3.5;
}
quad[0][i] = arc((0,center[i]),inrad[i],0,90)--arc((0,center[i]),outrad[i],90,0)--cycle;
quad[1][i] = arc((0,center[i]),inrad[i],90,180)--arc((0,center[i]),outrad[i],180,90)--cycle;
quad[2][i] = arc((0,center[i]),inrad[i],180,270)--arc((0,center[i]),outrad[i],270,180)--cycle;
quad[3][i] = arc((0,center[i]),inrad[i],270,360)--arc((0,center[i]),outrad[i],360,270)--cycle;
draw(circle((0,center[i]),inrad[i])^^circle((0,center[i]),outrad[i]));
}
void fillring(int i,int j) {
if ((j % 2) == 0) {
fill(quad[i][j],white);
} else {
filldraw(quad[i][j],black);
} }
for(int i=0;i<=3;i=i+1) {
for(int j=0;j<=3;j=j+1) {
fillring(((2-i) % 4),j);
} }
for(int k=0;k<=2;k=k+1) {
filldraw(circle((0,-228 - 25 * k),3),black);
}
real r = 130, s = -90;
draw((0,57)--(r,57)^^(0,-57)--(r,-57),linewidth(0.7));
draw((2*r/3,56)--(2*r/3,-56),linewidth(0.7),Arrows(size=3));
label("$20$",(2*r/3,-10),E);
draw((0,39)--(s,39)^^(0,-39)--(s,-39),linewidth(0.7));
draw((9*s/10,38)--(9*s/10,-38),linewidth(0.7),Arrows(size=3));
label("$18$",(9*s/10,0),W);
[/asy]
$ \textbf{(A) } 171\qquad \textbf{(B) } 173\qquad \textbf{(C) } 182\qquad \textbf{(D) } 188\qquad \textbf{(E) } 210$
1959 AMC 12/AHSME, 13
The arithmetic mean (average) of a set of $50$ numbers is $38$. If two numbers, namely, $45$ and $55$, are discarded, the mean of the remaining set of numbers is:
$ \textbf{(A)}\ 36.5 \qquad\textbf{(B)}\ 37\qquad\textbf{(C)}\ 37.2\qquad\textbf{(D)}\ 37.5\qquad\textbf{(E)}\ 37.52 $
2008 AMC 10, 2
A square is drawn inside a rectangle. The ratio of the width of the rectangle to a side of the square is $ 2: 1$. The ratio of the rectangle's length to its width is $ 2: 1$. What percent of the rectangle's area is inside the square?
$ \textbf{(A)}\ 12.5 \qquad \textbf{(B)}\ 25 \qquad \textbf{(C)}\ 50 \qquad \textbf{(D)}\ 75 \qquad \textbf{(E)}\ 87.5$
2011 AIME Problems, 11
Let $M_n$ be the $n\times n$ matrix with entries as follows: for $1\leq i \leq n$, $m_{i,i}=10$; for $1\leq i \leq n-1, m_{i+1,i}=m_{i,i+1}=3$; all other entries in $M_n$ are zero. Let $D_n$ be the determinant of matrix $M_n$. Then $\displaystyle \sum_{n=1}^{\infty} \dfrac{1}{8D_n+1}$ can be represented as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
Note: The determinant of the $1\times 1$ matrix $[a]$ is $a$, and the determinant of the $2\times 2$ matrix $\left[ \begin{array}{cc} a & b \\ c & d \end{array} \right]=ad-bc$; for $n\geq 2$, the determinant of an $n\times n$ matrix with first row or first column $a_1\ a_2\ a_3 \dots\ a_n$ is equal to $a_1C_1 - a_2C_2 + a_3C_3 - \dots + (-1)^{n+1} a_nC_n$, where $C_i$ is the determinant of the $(n-1)\times (n-1)$ matrix found by eliminating the row and column containing $a_i$.
2008 Harvard-MIT Mathematics Tournament, 28
Let $ P$ be a polyhedron where every face is a regular polygon, and every edge has length $ 1$. Each vertex of $ P$ is incident to two regular hexagons and one square. Choose a vertex $ V$ of the polyhedron. Find the volume of the set of all points contained in $ P$ that are closer to $ V$ than to any other vertex.