Found problems: 3632
1995 USAMO, 2
A calculator is broken so that the only keys that still work are the $ \sin$, $ \cos$, and $ \tan$ buttons, and their inverses (the $ \arcsin$, $ \arccos$, and $ \arctan$ buttons). The display initially shows $ 0$. Given any positive rational number $ q$, show that pressing some finite sequence of buttons will yield the number $ q$ on the display. Assume that the calculator does real number calculations with infinite precision. All functions are in terms of radians.
2010 AMC 12/AHSME, 13
In $ \triangle ABC, \ \cos(2A \minus{} B) \plus{} \sin(A\plus{}B) \equal{} 2$ and $ AB\equal{}4.$ What is $ BC?$
$ \textbf{(A)}\ \sqrt{2} \qquad \textbf{(B)}\ \sqrt{3} \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 2\sqrt{2} \qquad \textbf{(E)}\ 2\sqrt{3}$
2021 AMC 10 Fall, 9
When a certain unfair die is rolled, an even number is $3$ times as likely to appear as an odd number. The die is rolled twice. What is the probability that the sum of the numbers rolled is even?
$\textbf{(A) }\dfrac38\qquad\textbf{(B) }\dfrac49\qquad\textbf{(C) }\dfrac59\qquad\textbf{(D) }\dfrac9{16}\qquad\textbf{(E) }\dfrac58$
2023 AIME, 10
Let $N$ be the number of ways to place the integers $1$ through $12$ in the $12$ cells of a $2\times 6$ grid so that for any two cells sharing a side, the difference between the numbers in those cells is not divisible by $3$. One way to do this is shown below. Find the number of positive integer divisors of $N$.
[asy]
size(160);
defaultpen(linewidth(0.6));
for(int j=0;j<=6;j=j+1)
{
draw((j,0)--(j,2));
}
for(int i=0;i<=2;i=i+1)
{
draw((0,i)--(6,i));
}
for(int k=1;k<=12;k=k+1)
{
label("$"+((string) k)+"$",(floor((k-1)/2)+0.5,k%2+0.5));
}
[/asy]
2023 AMC 10, 21
Each of $2023$ balls is placed in on of $3$ bins. Which of the following is closest to the probability that each of the bins will contain an odd number of balls?
$\textbf{(A) } \frac{2}{3} \qquad \textbf{(B) } \frac{3}{10} \qquad \textbf{(C) } \frac{1}{2} \qquad \textbf{(D) } \frac{1}{3} \qquad \textbf{(E) } \frac{1}{4}$
2024 AMC 12/AHSME, 22
Let $\triangle{ABC}$ be a triangle with integer side lengths and the property that $\angle{B} = 2\angle{A}$. What is the least possible perimeter of such a triangle?
$
\textbf{(A) }13 \qquad
\textbf{(B) }14 \qquad
\textbf{(C) }15 \qquad
\textbf{(D) }16 \qquad
\textbf{(E) }17 \qquad
$
2009 AMC 12/AHSME, 3
What number is one third of the way from $ \frac14$ to $ \frac34$?
$ \textbf{(A)}\ \frac{1}{3} \qquad
\textbf{(B)}\ \frac{5}{12} \qquad
\textbf{(C)}\ \frac{1}{2} \qquad
\textbf{(D)}\ \frac{7}{12} \qquad
\textbf{(E)}\ \frac{2}{3}$
2021 AMC 10 Fall, 18
A farmer's rectangular field is partitioned into $2$ by $2$ grid of $4$ rectangular sections as shown in the figure. In each section the farmer will plant one crop: corn, wheat, soybeans, or potatoes. The farmer does not want to grow corn and wheat in any two sections that share a border, and the farmer does not want to grow soybeans and potatoes in any two sections that share a border. Given these restrictions, in how many ways can the farmer choose crops to plant in each of the four sections of the field?
[asy]
draw((0,0)--(100,0)--(100,50)--(0,50)--cycle);
draw((50,0)--(50,50));
draw((0,25)--(100,25));
[/asy]
$\textbf{(A)}\ 12 \qquad \textbf{(B)}\ 64 \qquad \textbf{(C)}\ 84 \qquad \textbf{(D)}\ 90 \qquad \textbf{(E)}\ 144$
2017 USAMO, 6
Find the minimum possible value of \[\frac{a}{b^3+4}+\frac{b}{c^3+4}+\frac{c}{d^3+4}+\frac{d}{a^3+4}\] given that $a$, $b$, $c$, $d$ are nonnegative real numbers such that $a+b+c+d=4$.
[i]Proposed by Titu Andreescu[/i]
1991 USAMO, 4
Let $a = \frac{m^{m+1} + n^{n+1}}{m^m + n^n}$, where $m$ and $n$ are positive integers. Prove that $a^m + a^n \geq m^m + n^n$.
2019 AMC 10, 2
What is the hundreds digit of $(20!-15!)?$
$\textbf{(A) } 0 \qquad\textbf{(B) } 1 \qquad\textbf{(C) } 2 \qquad\textbf{(D) } 4 \qquad\textbf{(E) } 5$
1967 AMC 12/AHSME, 28
Given the two hypotheses: $\text{I}$ Some Mems are not Ens and $\text{II}$ No Ens are Veens. If "some" means "at least one," we can conclude that:
$\textbf{(A)}\ \text{Some Mems are not Veens}\qquad
\textbf{(B)}\ \text{Some Vees are not Mems}\\
\textbf{(C)}\ \text{No Mem is a Vee}\qquad
\textbf{(D)}\ \text{Some Mems are Vees}\\
\textbf{(E)}\ \text{Neither} \; \textbf{(A)} \; \text{nor} \; \textbf{(B)} \; \text{nor} \; \textbf{(C)} \; \text{nor} \; \textbf{(D)} \; \text{is deducible from the given statements}$
1991 AMC 12/AHSME, 19
Triangle $ABC$ has a right angle at $C$, $AC = 3$ and $BC = 4$. Triangle $ABD$ has a right angle at $A$ and $AD = 12$. Points $C$ and $D$ are on opposite sides of $\overline{AB}$. The line through $D$ parallel to $\overline{AC}$ meets $\overline{CB}$ extended at $E$. If $\frac{DE}{DB} = \frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, then $m + n = $
[asy]
size(170);
defaultpen(fontsize(10pt)+linewidth(.8pt));
pair C=origin, A=(0,3), B=(4,0), D=(7.2,12.6), E=(7.2,0);
draw(A--C--B--A--D--B--E--D);
label("$A$",A,W);
label("$B$",B,S);
label("$C$",C,SW);
label("$D$",D,NE);
label("$E$",E,SE);
[/asy]
$ \textbf{(A)}\ 25\qquad\textbf{(B)}\ 128\qquad\textbf{(C)}\ 153\qquad\textbf{(D)}\ 243\qquad\textbf{(E)}\ 256 $
1961 AMC 12/AHSME, 8
Let the two base angles of a triangle be $A$ and $B$, with $B$ larger than $A$. The altitude to the base divides the vertex angle $C$ into two parts, $C_1$ and $C_2$, with $C_2$ adjacent to side $a$. Then:
${{{ \textbf{(A)}\ C_1+C_2=A+B \qquad\textbf{(B)}\ C_1-C_2=B-A \qquad\textbf{(C)}\ C_1-C_2=A-B} \qquad\textbf{(D)}\ C_1+C_2=B-A}\qquad\textbf{(E)}\ C_1-C_2=A+B} $
2015 AMC 10, 23
Let $n$ be a positive integer greater than 4 such that the decimal representation of $n!$ ends in $k$ zeros and the decimal representation of $(2n)!$ ends in $3k$ zeros. Let $s$ denote the sum of the four least possible values of $n$. What is the sum of the digits of $s$?
$ \textbf{(A) }7\qquad\textbf{(B) }8\qquad\textbf{(C) }9\qquad\textbf{(D) }10\qquad\textbf{(E) }11 $
2014 AMC 12/AHSME, 11
David drives from his home to the airport to catch a flight. He drives $35$ miles in the first hour, but realizes that he will be $1$ hour late if he continues at this speed. He increases his speed by $15$ miles per hour for the rest of the way to the airport and arrives $30$ minutes early. How many miles is the airport from his home?
$\textbf{(A) }140\qquad
\textbf{(B) }175\qquad
\textbf{(C) }210\qquad
\textbf{(D) }245\qquad
\textbf{(E) }280\qquad$
2018 AMC 10, 5
Alice, Bob, and Charlie were on a hike and were wondering how far away the nearest town was. When Alice said, "We are at least 6 miles away," Bob replied, "We are at most 5 miles away." Charlie then remarked, "Actually the nearest town is at most 4 miles away." It turned out that none of the three statements were true. Let $d$ be the distance in miles to the nearest town. Which of the following intervals is the set of all possible values of $d$?
$\textbf{(A) } (0,4) \qquad \textbf{(B) } (4,5) \qquad \textbf{(C) } (4,6) \qquad \textbf{(D) } (5,6) \qquad \textbf{(E) } (5,\infty) $
2015 AMC 12/AHSME, 5
The Tigers beat the Sharks $2$ out of the first $3$ times they played. They then played $N$ more times, and the Sharks ended up winning at least $95\%$ of all the games played. What is the minimum possible value for $N$?
$\textbf{(A) }35\qquad\textbf{(B) }37\qquad\textbf{(C) }39\qquad\textbf{(D) }41\qquad\textbf{(E) }43$
2016 AMC 10, 1
What is the value of $\frac{2a^{-1}+\frac{a^{-1}}{2}}{a}$ when $a= \frac{1}{2}$?
$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ \frac{5}{2}\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 20$
1994 AMC 12/AHSME, 20
Suppose $x,y,z$ is a geometric sequence with common ratio $r$ and $x \neq y$. If $x, 2y, 3z$ is an arithmetic sequence, then $r$ is
$ \textbf{(A)}\ \frac{1}{4} \qquad\textbf{(B)}\ \frac{1}{3} \qquad\textbf{(C)}\ \frac{1}{2} \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ 4$
1971 AMC 12/AHSME, 13
If $(1.0025)^{10}$ is evaluated correct to $5$ decimal places, then the digit in the fifth decimal place is
$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }5\qquad \textbf{(E) }8$
2011 AMC 12/AHSME, 24
Consider all quadrilaterals $ABCD$ such that $AB=14$, $BC=9$, $CD=7$, $DA=12$. What is the radius of the largest possible circle that fits inside or on the boundary of such a quadrilateral?
$ \textbf{(A)}\ \sqrt{15} \qquad\textbf{(B)}\ \sqrt{21} \qquad\textbf{(C)}\ 2\sqrt{6} \qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 2\sqrt{7} $
1997 AMC 8, 1
$\dfrac{1}{10} + \dfrac{9}{100} + \dfrac{9}{1000} + \dfrac{7}{10000} = $
$\textbf{(A)}\ 0.0026 \qquad \textbf{(B)}\ 0.0197 \qquad \textbf{(C)}\ 0.1997 \qquad \textbf{(D)}\ 0.26 \qquad \textbf{(E)}\ 1.997$
1969 AMC 12/AHSME, 4
Let a binary operation $*$ on ordered pairs of integers be defined by $(a,b)*(c,d)=(a-c,b+d)$. Then, if $(3,2)*(0,0)$ and $(x,y)*(3,2)$ represent idential pairs, $x$ equals:
$\textbf{(A) }-3\qquad
\textbf{(B) }0\qquad
\textbf{(C) }2\qquad
\textbf{(D) }3\qquad
\textbf{(E) }6$
2014 AIME Problems, 9
Let $x_1<x_2<x_3$ be three real roots of equation $\sqrt{2014}x^3-4029x^2+2=0$. Find $x_2(x_1+x_3)$.