Found problems: 3632
1989 AIME Problems, 13
Let $S$ be a subset of $\{1,2,3,\ldots,1989\}$ such that no two members of $S$ differ by $4$ or $7$. What is the largest number of elements $S$ can have?
2007 AIME Problems, 2
Find the number of ordered triple $(a,b,c)$ where $a$, $b$, and $c$ are positive integers, $a$ is a factor of $b$, $a$ is a factor of $c$, and $a+b+c=100$.
2019 AMC 12/AHSME, 18
A sphere with center $O$ has radius $6$. A triangle with sides of length $15, 15,$ and $24$ is situated in space so that each of its sides is tangent to the sphere. What is the distance between $O$ and the plane determined by the triangle?
$
\textbf{(A) }2\sqrt{3}\qquad
\textbf{(B) }4\qquad
\textbf{(C) }3\sqrt{2}\qquad
\textbf{(D) }2\sqrt{5}\qquad
\textbf{(E) }5\qquad
$
2012 AIME Problems, 1
Find the number of ordered pairs of positive integer solutions $(m,n)$ to the equation $20m+12n=2012.$
1994 AMC 12/AHSME, 14
Find the sum of the arithmetic series
\[ 20+20\frac{1}{5}+20\frac{2}{5}+\cdots+40 \]
$ \textbf{(A)}\ 3000 \qquad\textbf{(B)}\ 3030 \qquad\textbf{(C)}\ 3150 \qquad\textbf{(D)}\ 4100 \qquad\textbf{(E)}\ 6000 $
1960 AMC 12/AHSME, 30
Given the line $3x+5y=15$ and a point on this line equidistant from the coordinate axes. Such a point exists in:
$ \textbf{(A)}\ \text{none of the quadrants} \qquad\textbf{(B)}\ \text{quadrant I only} \qquad\textbf{(C)}\ \text{quadrants I, II only} \qquad$
$\textbf{(D)}\ \text{quadrants I, II, III only} \qquad\textbf{(E)}\ \text{each of the quadrants} $
2014 AMC 10, 7
Nonzero real numbers $x$, $y$, $a$, and $b$ satisfy $x < a$ and $y < b$. How many of the following inequalities must be true?
(I) $x+y < a+b$
(II) $x-y < a-b$
(III) $xy < ab$
(IV) $\frac{x}{y} < \frac{a}{b}$
${ \textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}}\ 3\qquad\textbf{(E)}\ 4$
2006 AIME Problems, 11
A sequence is defined as follows $a_1=a_2=a_3=1$, and, for all positive integers $n$, $a_{n+3}=a_{n+2}+a_{n+1}+a_n$. Given that $a_{28}=6090307$, $a_{29}=11201821$, and $a_{30}=20603361$, find the remainder when $\displaystyle \sum^{28}_{k=1} a_k$ is divided by 1000.
2008 National Olympiad First Round, 2
For which value of $A$, does the equation $3m^2n = n^3 + A$ have a solution in natural numbers?
$
\textbf{(A)}\ 301
\qquad\textbf{(B)}\ 403
\qquad\textbf{(C)}\ 415
\qquad\textbf{(D)}\ 427
\qquad\textbf{(E)}\ 481
$
1960 AMC 12/AHSME, 40
Given right triangle $ABC$ with legs $BC=3$, $AC=4$. Find the length of the shorter [i]angle trisector[/i] from $C$ to the hypotenuse:
$ \textbf{(A)}\ \frac{32\sqrt{3}-24}{13}\qquad\textbf{(B)}\ \frac{12\sqrt{3}-9}{13}\qquad\textbf{(C)}\ 6\sqrt{3}-8\qquad\textbf{(D)}\ \frac{5\sqrt{10}}{6} \qquad$
$\textbf{(E)}\ \frac{25}{12}$
2019 AMC 10, 12
Melanie computes the mean $\mu$, the median $M$, and the modes of the $365$ values that are the dates in the months of $2019$. Thus her data consist of $12$ $1\text{s}$, $12$ $2\text{s}$, . . . , $12$ $28\text{s}$, $11$ $29\text{s}$, $11$ $30\text{s}$, and $7$ $31\text{s}$. Let $d$ be the median of the modes. Which of the following statements is true?
$\textbf{(A) } \mu < d < M \qquad\textbf{(B) } M < d < \mu \qquad\textbf{(C) } d = M =\mu \qquad\textbf{(D) } d < M < \mu \qquad\textbf{(E) } d < \mu < M$
2006 AMC 12/AHSME, 19
Circles with centers $ (2,4)$ and $ (14,9)$ have radii 4 and 9, respectively. The equation of a common external tangent to the circles can be written in the form $ y \equal{} mx \plus{} b$ with $ m > 0$. What is $ b$?
[asy]
size(150); defaultpen(linewidth(0.7)+fontsize(8)); draw(circle((2,4),4));draw(circle((14,9),9)); draw((0,-2)--(0,20));draw((-6,0)--(25,0)); draw((2,4)--(2,4)+4*expi(pi*4.5/11)); draw((14,9)--(14,9)+9*expi(pi*6/7)); label("4",(2,4)+2*expi(pi*4.5/11),(-1,0)); label("9",(14,9)+4.5*expi(pi*6/7),(1,1)); label("(2,4)",(2,4),(0.5,-1.5));label("(14,9)",(14,9),(1,-1)); draw((-4,120*-4/119+912/119)--(11,120*11/119+912/119)); dot((2,4)^^(14,9));[/asy]
$ \textbf{(A) } \frac {908}{199}\qquad \textbf{(B) } \frac {909}{119}\qquad \textbf{(C) } \frac {130}{17}\qquad \textbf{(D) } \frac {911}{119}\qquad \textbf{(E) } \frac {912}{119}$
2011 AMC 10, 15
Let @ denote the "averaged with" operation: $a$ @ $b$ = $\frac{a+b}{2}$. Which of the following distributive laws hold for all numbers $x,y$ and $z$?
I. x @ (y+z) = (x @ y) + (x @ z)
II. x + (y @ z) = (x + y) @ (x + z)
III. x @ (y @ z) = (x @ y) @ (x @ z)
$ \textbf{(A)}\ \text{I only} \qquad
\textbf{(B)}\ \text{II only} \qquad
\textbf{(C)}\ \text{III only} \qquad
\textbf{(D)}\ \text{I and III only} \qquad
\textbf{(E)}\ \text{II and III only} $
1972 AMC 12/AHSME, 17
A piece of string is cut in two at a point selected at random. The probability that the longer piece is at least $x$ times as large as the shorter piece is
$\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }\frac{2}{x}\qquad\textbf{(C) }\frac{1}{x+1}\qquad\textbf{(D) }\frac{1}{x}\qquad \textbf{(E) }\frac{2}{x+1}$
2010 AMC 12/AHSME, 5
Halfway through a $ 100$-shot archery tournament, Chelsea leads by $ 50$ points. For each shot a bullseye scores $ 10$ points, with other possible scores being $ 8, 4, 2, 0$ points. Chelsea always scores at least $ 4$ points on each shot. If Chelsea's next $ n$ shots are bulleyes she will be guaranteed victory. What is the minimum value for n?
$ \textbf{(A)}\ 38\qquad \textbf{(B)}\ 40\qquad \textbf{(C)}\ 42\qquad \textbf{(D)}\ 44\qquad \textbf{(E)}\ 46$
2021 AMC 10 Fall, 12
The base-nine representation of the number $N$ is $27{,}006{,}000{,}052_{\rm nine}$. What is the remainder when $N$ is divided by $5?$
$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }4$
2024 AIME, 15
Let $\mathcal{B}$ be the set of rectangular boxes that have volume $23$ and surface area $54$. Suppose $r$ is the least possible radius of a sphere that can fit any element of $\mathcal{B}$ inside it. Then $r^{2}$ can be expressed as $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
2021 AMC 12/AHSME Fall, 19
Regular polygons with $5, 6, 7, $ and $8$ sides are inscribed in the same circle. No two of the polygons share a vertex, and no three of their sides intersect at a common point. At how many points inside the circle do two of their sides intersect?
$\textbf{(A)}\ 52 \qquad\textbf{(B)}\ 56 \qquad\textbf{(C)}\ 60 \qquad\textbf{(D)}\
64 \qquad\textbf{(E)}\ 68$
2013 AIME Problems, 3
A large candle is $119$ centimeters tall. It is designed to burn down more quickly when it is first lit and more slowly as it approaches its bottom. Specifically, the candle takes $10$ seconds to burn down the first centimeter from the top, $20$ seconds to burn down the second centimeter, and $10k$ seconds to burn down the $k$-th centimeter. Suppose it takes $T$ seconds for the candle to burn down completely. Then $\tfrac{T}{2}$ seconds after it is lit, the candle's height in centimeters will be $h$. Find $10h$.
1968 AMC 12/AHSME, 26
Let $S=2+4+6+ \cdots +2N$, where $N$ is the smallest positive integer such that $S>1,000,000$. Then the sum of the digits of $N$ is:
$\textbf{(A)}\ 27 \qquad\textbf{(B)}\ 12 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ 1$
2024 AMC 10, 4
The number $2024$ is written as the sum of not necessarily distinct two-digit numbers. What is the least number of two-digit numbers needed to write this sum?
$\textbf{(A) }20\qquad\textbf{(B) }21\qquad\textbf{(C) }22\qquad\textbf{(D) }23\qquad\textbf{(E) }24$
2020 AMC 12/AHSME, 13
There are integers $a$, $b$, and $c$, each greater than 1, such that $$\sqrt[a]{N \sqrt[b]{N \sqrt[c]{N}}} = \sqrt[36]{N^{25}}$$ for all $N > 1$. What is $b$?
$\textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6$
2016 AMC 10, 14
How many ways are there to write $2016$ as the sum of twos and threes, ignoring order? (For example, $1008\cdot 2 + 0\cdot 3$ and $402\cdot 2 + 404\cdot 3$ are two such ways.)
$\textbf{(A)}\ 236\qquad\textbf{(B)}\ 336\qquad\textbf{(C)}\ 337\qquad\textbf{(D)}\ 403\qquad\textbf{(E)}\ 672$
2014 AMC 10, 5
Doug constructs a square window using $8$ equal-size panes of glass, as shown. The ratio of the height to width for each pane is $5:2$, and the borders around and between the panes are $2$ inches wide. In inches, what is the side length o the square window?
[asy]
fill((0,0)--(25,0)--(25,25)--(0,25)--cycle,grey);
for(int i = 0; i < 4; ++i){
for(int j = 0; j < 2; ++j){
fill((6*i+2,11*j+3)--(6*i+5,11*j+3)--(6*i+5,11*j+11)--(6*i+2,11*j+11)--cycle,white);
}
}[/asy]
$\textbf{(A) }26\qquad\textbf{(B) }28\qquad\textbf{(C) }30\qquad\textbf{(D) }32\qquad\textbf{(E) }34$
1959 AMC 12/AHSME, 20
It is given that $x$ varies directly as $y$ and inversely as the square of $z$, and that $x=10$ when $y=4$ and $z=14$. Then, when $y=16$ and $z=7$, $x$ equals:
$ \textbf{(A)}\ 180\qquad\textbf{(B)}\ 160\qquad\textbf{(C)}\ 154\qquad\textbf{(D)}\ 140\qquad\textbf{(E)}\ 120 $