Found problems: 634
2010 AIME Problems, 12
Let $ M \ge 3$ be an integer and let $ S \equal{} \{3,4,5,\ldots,m\}$. Find the smallest value of $ m$ such that for every partition of $ S$ into two subsets, at least one of the subsets contains integers $ a$, $ b$, and $ c$ (not necessarily distinct) such that $ ab \equal{} c$.
[b]Note[/b]: a partition of $ S$ is a pair of sets $ A$, $ B$ such that $ A \cap B \equal{} \emptyset$, $ A \cup B \equal{} S$.
1990 AIME Problems, 1
The increasing sequence $2,3,5,6,7,10,11,\ldots$ consists of all positive integers that are neither the square nor the cube of a positive integer. Find the 500th term of this sequence.
2004 AIME Problems, 7
Let $C$ be the coefficient of $x^2$ in the expansion of the product \[(1 - x)(1 + 2x)(1 - 3x)\cdots(1 + 14x)(1 - 15x).\] Find $|C|$.
2018 AIME Problems, 5
Suppose that $x$, $y$, and $z$ are complex numbers such that $xy = -80-320i$, $yz = 60$, and $zx = -96+24i$, where $i = \sqrt{-1}$. Then there are real numbers $a$ and $b$ such that $x+y+z = a+bi$. Find $a^2 + b^2$.
2006 AIME Problems, 11
A collection of 8 cubes consists of one cube with edge-length $k$ for each integer $k,\thinspace 1 \le k \le 8.$ A tower is to be built using all 8 cubes according to the rules:
$\bullet$ Any cube may be the bottom cube in the tower.
$\bullet$ The cube immediately on top of a cube with edge-length $k$ must have edge-length at most $k+2.$
Let $T$ be the number of different towers than can be constructed. What is the remainder when $T$ is divided by 1000?
2019 AIME Problems, 15
In acute triangle $ABC$ points $P$ and $Q$ are the feet of the perpendiculars from $C$ to $\overline{AB}$ and from $B$ to $\overline{AC}$, respectively. Line $PQ$ intersects the circumcircle of $\triangle ABC$ in two distinct points, $X$ and $Y$. Suppose $XP=10$, $PQ=25$, and $QY=15$. The value of $AB\cdot AC$ can be written in the form $m\sqrt n$ where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$.
2001 AIME Problems, 6
A fair die is rolled four times. The probability that each of the final three rolls is at least as large as the roll preceding it may be expressed in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
2024 AIME, 1
Every morning, Aya does a $9$ kilometer walk, and then finishes at the coffee shop. One day, she walks at $s$ kilometers per hour, and the walk takes $4$ hours, including $t$ minutes at the coffee shop. Another morning, she walks at $s+2$ kilometers per hour, and the walk takes $2$ hours and $24$ minutes, including $t$ minutes at the coffee shop. This morning, if she walks at $s+\frac12$ kilometers per hour, how many minutes will the walk take, including the $t$ minutes at the coffee shop?
2008 AIME Problems, 15
Find the largest integer $ n$ satisfying the following conditions:
(i) $ n^2$ can be expressed as the difference of two consecutive cubes;
(ii) $ 2n\plus{}79$ is a perfect square.
1986 AIME Problems, 11
The polynomial $1-x+x^2-x^3+\cdots+x^{16}-x^{17}$ may be written in the form $a_0+a_1y+a_2y^2+\cdots +a_{16}y^{16}+a_{17}y^{17}$, where $y=x+1$ and thet $a_i$'s are constants. Find the value of $a_2$.
2001 AIME Problems, 1
Find the sum of all positive two-digit integers that are divisible by each of their digits.
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 $m+n+p$.
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?
1994 AIME Problems, 14
A beam of light strikes $\overline{BC}$ at point $C$ with angle of incidence $\alpha=19.94^\circ$ and reflects with an equal angle of reflection as shown. The light beam continues its path, reflecting off line segments $\overline{AB}$ and $\overline{BC}$ according to the rule: angle of incidence equals angle of reflection. Given that $\beta=\alpha/10=1.994^\circ$ and $AB=AC,$ determine the number of times the light beam will bounce off the two line segments. Include the first reflection at $C$ in your count.
[asy]
size(250);defaultpen(linewidth(0.7));
real alpha=24, beta=32;
pair B=origin, C=(1,0), A=dir(beta), D=C+0.5*dir(alpha);
pair EE=2*dir(180-alpha), E=intersectionpoint(C--EE, A--B);
pair EEE=reflect(B,A)*EE, EEEE=reflect(C,B)*EEE, F=intersectionpoint(E--EEE, B--C), G=intersectionpoint(F--EEEE, A--B);
draw((1.4,0)--B--1.4*dir(beta));
draw(D--C, linetype("4 4"),EndArrow(5));
draw(C--E, linetype("4 4"),EndArrow(5));
draw(E--F, linetype("4 4"),EndArrow(5));
draw(F--G, linetype("4 4"),EndArrow(5));
markscalefactor=0.01;
draw(anglemark(C,B,A));
draw(anglemark((1.4,0), C,D));
label("$\beta$", 0.07*dir(beta/2), dir(beta/2), fontsize(10));
label("$\alpha$", C+0.07*dir(alpha/2), dir(alpha/2), fontsize(10));
label("$A$", A, dir(90)*dir(A));
label("$B$", B, dir(beta/2+180));
label("$C$", C, S);[/asy]
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 $m+n$.
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 $1000$.
1997 AIME Problems, 12
The function $f$ defined by $\displaystyle f(x)= \frac{ax+b}{cx+d}$. where $a,b,c$ and $d$ are nonzero real numbers, has the properties $f(19)=19, f(97)=97$ and $f(f(x))=x$ for all values except $\displaystyle \frac{-d}{c}$. Find the unique number that is not in the range of $f$.
2024 AIME, 2
A list of positive integers has the following properties:
- The sum of the items in the list is $30$.
- The unique mode of the list is $9$.
- The median of the list is a positive integer that does not appear in the list itself.
Find the sum of the squares of all the items in the list.
2023 AIME, 4
The sum of all positive integers $m$ for which $\tfrac{13!}{m}$ is a perfect square can be written as $2^{a}3^{b}5^{c}7^{d}11^{e}13^{f}$, where $a, b, c, d, e,$ and $f$ are positive integers. Find $a+b+c+d+e+f$.
1992 AIME Problems, 15
Define a positive integer $ n$ to be a factorial tail if there is some positive integer $ m$ such that the decimal representation of $ m!$ ends with exactly $ n$ zeroes. How many positive integers less than $ 1992$ are not factorial tails?
2015 AIME Problems, 4
Point $B$ lies on line segment $\overline{AC}$ with $AB=16$ and $BC=4$. Points $D$ and $E$ lie on the same side of line $AC$ forming equilateral triangles $\triangle ABD$ and $\triangle BCE$. Let $M$ be the midpoint of $\overline{AE}$, and $N$ be the midpoint of $\overline{CD}$. The area of $\triangle BMN$ is $x$. Find $x^2$.
2014 USAMTS Problems, 3a:
A group of people is lined up in [i]almost-order[/i] if, whenever person $A$ is to the left of person $B$ in the line, $A$ is not more than $8$ centimeters taller than $B$. For example, five people with heights $160, 165, 170, 175$, and $180$ centimeters could line up in almost-order with heights (from left-to-right) of $160, 170, 165, 180, 175$ centimeters.
(a) How many different ways are there to line up $10$ people in [i]almost-order[/i] if their heights are $140, 145, 150, 155,$ $160,$ $165,$ $170,$ $175,$ $180$, and $185$ centimeters?
2014 AMC 10, 24
A sequence of natural numbers is constructed by listing the first $4$, then skipping one, listing the next $5$, skipping $2$, listing $6$, skipping $3$, and, on the $n$th iteration, listing $n+3$ and skipping $n$. The sequence begins $1,2,3,4,6,7,8,9,10,13$. What is the $500,000$th number in the sequence?
$ \textbf{(A)}\ 996,506\qquad\textbf{(B)}\ 996507\qquad\textbf{(C)}\ 996508\qquad\textbf{(D)}\ 996509\qquad\textbf{(E)}\ 996510 $
2014 AIME Problems, 11
A token starts at the point $(0,0)$ of an $xy$-coordinate grid and them makes a sequence of six moves. Each move is $1$ unit in a direction parallel to one of the coordinate axes. Each move is selected randomly from the four possible directions and independently of the other moves. The probability the token ends at a point on the graph of $|y|=|x|$ is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
1995 AIME Problems, 4
Circles of radius 3 and 6 are externally tangent to each other and are internally tangent to a circle of radius 9. The circle of radius 9 has a chord that is a common external tangent of the other two circles. Find the square of the length of this chord.