Found problems: 3632
2001 AIME Problems, 1
Find the sum of all positive two-digit integers that are divisible by each of their digits.
2011 USAMO, 4
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.
2011 AIME Problems, 2
In rectangle $ABCD$, $AB=12$ and $BC=10$. Points $E$ and $F$ lie inside rectangle $ABCD$ so that $BE=9$, $DF=8$, $\overline{BE} \parallel \overline{DF}$, $\overline{EF} \parallel \overline{AB}$, and line $BE$ intersects segment $\overline{AD}$. The length $EF$ can be expressed in the form $m\sqrt{n}-p$, where $m,n,$ and $p$ are positive integers and $n$ is not divisible by the square of any prime. Find $m+n+p$.
2014 NIMO Problems, 3
In triangle $ABC$, we have $AB=AC=20$ and $BC=14$. Consider points $M$ on $\overline{AB}$ and $N$ on $\overline{AC}$. If the minimum value of the sum $BN + MN + MC$ is $x$, compute $100x$.
[i]Proposed by Lewis Chen[/i]
1961 AMC 12/AHSME, 17
In the base ten number system the number $526$ means $5 \cdot 10^2+2 \cdot 10 + 6$. In the Land of Mathesis, however, numbers are written in the base $r$. Jones purchases an automobile there for $440$ monetary units (abbreviated m.u). He gives the salesman a $1000$ m.u bill, and receives, in change, $340$ m.u. The base $r$ is:
${{ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 8}\qquad\textbf{(E)}\ 12} $
1991 AIME Problems, 10
Two three-letter strings, $aaa$ and $bbb$, are transmitted electronically. Each string is sent letter by letter. Due to faulty equipment, each of the six letters has a 1/3 chance of being received incorrectly, as an $a$ when it should have been a $b$, or as a $b$ when it should be an $a$. However, whether a given letter is received correctly or incorrectly is independent of the reception of any other letter. Let $S_a$ be the three-letter string received when $aaa$ is transmitted and let $S_b$ be the three-letter string received when $bbb$ is transmitted. Let $p$ be the probability that $S_a$ comes before $S_b$ in alphabetical order. When $p$ is written as a fraction in lowest terms, what is its numerator?
1969 AMC 12/AHSME, 24
When the natural numbers $P$ and $P'$, with $P>P'$, are divided by the natural number $D$, the remainders are $R$ and $R'$, respectively. When $PP'$ and $RR'$ are divided by $D$, the remainders are $r$ and $r'$, respectively. Then:
$\textbf{(A) }r>r'\text{ always}\qquad
\textbf{(B) }r<r'\text{ always}\qquad$
$\textbf{(C) }r>r'\text{ sometimes, and }r<r'\text{ sometimes}$
$\textbf{(D) }r>r'\text{ sometimes, and }r=r'\text{ sometimes}$
$\textbf{(E) }r=r'\text{ always}$
2021 AMC 12/AHSME Spring, 11
A laser is placed at the point (3,5). The laser bean travels in a straight line. Larry wants the beam to hit and bounce off the $y$-axis, then hit and bounce off the $x$-axis, then hit the point $(7,5)$. What is the total distance the beam will travel along this path?
$\textbf{(A) }2\sqrt{10} \qquad \textbf{(B) }5\sqrt2 \qquad \textbf{(C) }10\sqrt2 \qquad \textbf{(D) }15\sqrt2 \qquad \textbf{(E) }10\sqrt5$
1990 AMC 12/AHSME, 9
Each edge of a cube is colored either red or black. Every face of the cube has at least one black edge. The smallest possible number of black edges is
$\textbf{(A) }2\qquad
\textbf{(B) }3\qquad
\textbf{(C) }4\qquad
\textbf{(D) }5\qquad
\textbf{(E) }6\qquad$
1992 AMC 12/AHSME, 27
A circle of radius $r$ has chords $\overline{AB}$ of length $10$ and $\overline{CD}$ of length $7$. When $\overline{AB}$ and $\overline{CD}$ are extended through $B$ and $C$, respectively, they intersect at $P$, which is outside the circle. If $\angle APD = 60^{\circ}$ and $BP = 8$, then $r^{2} =$
$ \textbf{(A)}\ 70\qquad\textbf{(B)}\ 71\qquad\textbf{(C)}\ 72\qquad\textbf{(D)}\ 73\qquad\textbf{(E)}\ 74 $
1983 AMC 12/AHSME, 26
The probability that event $A$ occurs is $\frac{3}{4}$; the probability that event $B$ occurs is $\frac{2}{3}$. Let $p$ be the probability that both $A$ and $B$ occur. The smallest interval necessarily containing $p$ is the interval
$ \textbf{(A)}\ \Big[\frac{1}{12},\frac{1}{2}\Big]\qquad\textbf{(B)}\ \Big[\frac{5}{12},\frac{1}{2}\Big]\qquad\textbf{(C)}\ \Big[\frac{1}{2},\frac{2}{3}\Big]\qquad\textbf{(D)}\ \Big[\frac{5}{12},\frac{2}{3}\Big]\qquad\textbf{(E)}\ \Big[\frac{1}{12},\frac{2}{3}\Big]$
1963 AMC 12/AHSME, 12
Three vertices of parallelogram $PQRS$ are $P(-3,-2)$, $Q(1,-5)$, $R(9,1)$ with $P$ and $R$ diagonally opposite. The sum of the coordinates of vertex $S$ is:
$\textbf{(A)}\ 13 \qquad
\textbf{(B)}\ 12 \qquad
\textbf{(C)}\ 11 \qquad
\textbf{(D)}\ 10 \qquad
\textbf{(E)}\ 9$
2001 AMC 12/AHSME, 13
The parabola with equation $ y \equal{} ax^2 \plus{} bx \plus{} c$ and vertex $ (h,k)$ is reflected about the line $ y \equal{} k$. This results in the parabola with equation $ y \equal{} dx^2 \plus{} ex \plus{} f$. Which of the following equals $ a \plus{} b \plus{} c \plus{} d \plus{} e \plus{} f$?
$ \textbf{(A)} \ 2b \qquad \textbf{(B)} \ 2c \qquad \textbf{(C)} \ 2a \plus{} 2b \qquad \textbf{(D)} \ 2h \qquad \textbf{(E)} \ 2k$
2017 AMC 12/AHSME, 18
Let $S(n)$ equal the sum of the digits of positive integer $n$. For example, $S(1507) = 13$. For a particular positive integer $n$, $S(n) = 1274$. Which of the following could be the value of $S(n+1)$?
$\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 1239\qquad\textbf{(E)}\ 1265$
2011 AMC 12/AHSME, 19
A lattice point in an $xy$-coordinate system is any point $(x,y)$ where both $x$ and $y$ are integers. The graph of $y=mx+2$ passes through no lattice point with $0<x \le 100$ for all $m$ such that $\frac{1}{2}<m<a$. What is the maximum possible value of $a$?
$ \textbf{(A)}\ \frac{51}{101} \qquad
\textbf{(B)}\ \frac{50}{99} \qquad
\textbf{(C)}\ \frac{51}{100} \qquad
\textbf{(D)}\ \frac{52}{101} \qquad
\textbf{(E)}\ \frac{13}{25} $
2019 AIME Problems, 10
For distinct complex numbers $z_1,z_2,\dots,z_{673}$, the polynomial
\[ (x-z_1)^3(x-z_2)^3 \cdots (x-z_{673})^3 \]
can be expressed as $x^{2019} + 20x^{2018} + 19x^{2017}+g(x)$, where $g(x)$ is a polynomial with complex coefficients and with degree at most $2016$. The value of
\[ \left| \sum_{1 \le j <k \le 673} z_jz_k \right| \]
can be expressed in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
2008 AMC 12/AHSME, 13
Vertex $ E$ of equilateral $ \triangle{ABE}$ is in the interior of unit square $ ABCD$. Let $ R$ be the region consisting of all points inside $ ABCD$ and outside $ \triangle{ABE}$ whose distance from $ \overline{AD}$ is between $ \frac{1}{3}$ and $ \frac{2}{3}$. What is the area of $ R$?
$ \textbf{(A)}\ \frac{12\minus{}5\sqrt3}{72} \qquad
\textbf{(B)}\ \frac{12\minus{}5\sqrt3}{36} \qquad
\textbf{(C)}\ \frac{\sqrt3}{18} \qquad
\textbf{(D)}\ \frac{3\minus{}\sqrt3}{9} \qquad
\textbf{(E)}\ \frac{\sqrt3}{12}$
1969 AMC 12/AHSME, 35
Let $L(m)$ be the $x$-coordinate of the left end point of the intersection of the graphs of $y=x^2-6$ and $y=m$, where $-6<m<6$. Let $r=[L(-m)-L(m)]/m$. Then, as $m$ is made arbitrarily close to zero, the value of $r$ is:
$\textbf{(A) }\text{arbitrarily close to zero}\qquad
\textbf{(B) }\text{arbitrarily close to }\tfrac1{\sqrt6}\qquad$
$\textbf{(C) }\text{arbitrarily close to }\tfrac2{\sqrt6}\qquad\,\,\,
\textbf{(D) }\text{arbitrarily large}\qquad$
$\textbf{(E) }\text{undetermined}$
2010 AMC 12/AHSME, 7
Shelby drives her scooter at a speed of 30 miles per hour if it is not raining, and 20 miles per hour if it is raining. Today she drove in the sun in the morning and in the rain in the evening, for a total of 16 miles in 40 minutes. How many minutes did she drive in the rain?
$ \textbf{(A)}\ 18\qquad\textbf{(B)}\ 21\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 30$
2008 AMC 12/AHSME, 15
Let $ k\equal{}2008^2\plus{}2^{2008}$. What is the units digit of $ k^2\plus{}2^k$?
$ \textbf{(A)}\ 0 \qquad
\textbf{(B)}\ 2 \qquad
\textbf{(C)}\ 4 \qquad
\textbf{(D)}\ 6 \qquad
\textbf{(E)}\ 8$
2020 CHMMC Winter (2020-21), 3
For any nonnegative integer $n$, let $S(n)$ be the sum of the digits of $n$. Let $K$ be the number of nonnegative integers $n \le 10^{10}$ that satisfy the equation
\[
S(n) = (S(S(n)))^2.
\]
Find the remainder when $K$ is divided by $1000$.
2011 AMC 10, 14
A rectangular parking lot has a diagonal of $25$ meters and an area of $168$ square meters. In meters, what is the perimeter of the parking lot?
$ \textbf{(A)}\ 52 \qquad
\textbf{(B)}\ 58 \qquad
\textbf{(C)}\ 62 \qquad
\textbf{(D)}\ 68 \qquad
\textbf{(E)}\ 70 $
2020 AMC 12/AHSME, 2
The acronym AMC is shown in the rectangular grid below with grid lines spaced $1$ unit apart. In units, what is the sum of the lengths of the line segments that form the acronym AMC$?$
[asy]
import olympiad;
unitsize(25);
for (int i = 0; i < 3; ++i) {
for (int j = 0; j < 9; ++j) {
pair A = (j,i);
}
}
for (int i = 0; i < 3; ++i) {
for (int j = 0; j < 9; ++j) {
if (j != 8) {
draw((j,i)--(j+1,i), gray(0.6)+dashed);
}
if (i != 2) {
draw((j,i)--(j,i+1), gray(0.6)+dashed);
}
}
}
draw((0,0)--(2,2),linewidth(2));
draw((2,0)--(2,2),linewidth(2));
draw((1,1)--(2,1),linewidth(2));
draw((3,0)--(3,2),linewidth(2));
draw((5,0)--(5,2),linewidth(2));
draw((4,1)--(3,2),linewidth(2));
draw((4,1)--(5,2),linewidth(2));
draw((6,0)--(8,0),linewidth(2));
draw((6,2)--(8,2),linewidth(2));
draw((6,0)--(6,2),linewidth(2));
[/asy]
$\textbf{(A) } 17 \qquad \textbf{(B) } 15 + 2\sqrt{2} \qquad \textbf{(C) } 13 + 4\sqrt{2} \qquad \textbf{(D) } 11 + 6\sqrt{2} \qquad \textbf{(E) } 21$
1959 AMC 12/AHSME, 28
In triangle $ABC$, $AL$ bisects angle $A$ and $CM$ bisects angle $C$. Points $L$ and $M$ are on $BC$ and $AB$, respectively. The sides of triangle $ABC$ are $a,b,$ and $c$. Then $\frac{\overline{AM}}{\overline{MB}}=k\frac{\overline{CL}}{\overline{LB}}$ where $k$ is:
$ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ \frac{bc}{a^2}\qquad\textbf{(C)}\ \frac{a^2}{bc}\qquad\textbf{(D)}\ \frac{c}{b}\qquad\textbf{(E)}\ \frac{c}{a} $
1963 AMC 12/AHSME, 28
Given the equation $3x^2 - 4x + k = 0$ with real roots. The value of $k$ for which the product of the roots of the equation is a maximum is:
$\textbf{(A)}\ \dfrac{16}{9} \qquad
\textbf{(B)}\ \dfrac{16}{3}\qquad
\textbf{(C)}\ \dfrac{4}{9} \qquad
\textbf{(D)}\ \dfrac{4}{3} \qquad
\textbf{(E)}\ -\dfrac{4}{3}$