Olympiad problems

2025 Harvard-MIT Mathematics Tournament, 25

Tags: guts

Let $ABCD$ be a trapezoid such that $AB \parallel CD, AD=13, BC=15, AB=20,$ and $CD=34.$ Point $X$ lies inside the trapezoid such that $\angle{XAB}=2\angle{XBA}$ and $\angle{XDC}=2\angle{XCD}.$ Compute $XD-XA.$

2023 Harvard-MIT Mathematics Tournament, 31

Tags: guts

Let $$P=\prod_{i=0}^{2016} (i^3-i-1)^2.$$ The remainder when $P$ is divided by the prime $2017$ is not zero. Compute this remainder.

2025 Harvard-MIT Mathematics Tournament, 1

Tags: guts

Call a $9$-digit number a [i]cassowary[/i] if it uses each of the digits $1$ through $9$ exactly once. Compute the number of cassowaries that are prime.

2024 HMNT, 30

Tags: guts

Compute the number of ways to shade exactly $4$ distinct cells of a $4\times4$ grid such that no two shaded cells share one or more vertices.

2023 Harvard-MIT Mathematics Tournament, 23

Tags: guts

A subset $S$ of the set $\{1, 2, \ldots, 10\}$ is chosen randomly, with all possible subsets being equally likely. Compute the expected number of positive integers which divide the product of the elements of $S.$ (By convention, the product of the elements of the empty set is $1.$)

2024 HMNT, 23

Tags: guts

Consider a quarter-circle with center $O,$ arc $\widehat{AB},$ and radius $2.$ Draw a semicircle with diameter $\overline{OA}$ lying inside the quarter-circle. Points $P$ and $Q$ lie on the semicircle and segment $\overline{OB},$ respectively, such that line $PQ$ is tangent to the semicircle. As $P$ and $Q$ vary, compute the maximum possible area of triangle $BQP.$

2024 HMNT, 26

Tags: guts

A right rectangular prism of silly powder has dimensions $20 \times 24 \times 25.$ Jerry the wizard applies $10$ bouts of highdroxylation to the box, each of which increases one dimension of the silly powder by $1$ and decreases a different dimension of the silly powder by $1,$ with every possible choice of dimensions equally likely to be chosen and independent of all previous choices. Compute the expected volume of the silly powder after Jerry’s routine.

2024 LMT Fall, 17

Tags: guts , algebra

Suppose $x$, $y$, $z$ are pairwise distinct real numbers satisfying \[ x^2+3y =y^2 +3z = z^2+3x. \]Find $(x+y)(y+z)(z+x)$.

2023 Harvard-MIT Mathematics Tournament, 11

Tags: guts

The Fibonacci numbers are defined recursively by $F_0=0, F_1=1,$ and $F_i=F_{i-1}+F_{i-2}$ for $i \ge 2.$ Given $15$ wooden blocks of weights $F_2, F_3, \ldots, F_{16},$ compute the number of ways to paint each block either red or blue such that the total weight of the red blocks equals the total weight of the blue blocks.

2024 HMNT, 18

Tags: guts

Let $ABCD$ be a rectangle whose vertices are labeled in counterclockwise order with $AB=32$ and $AD=60.$ Rectangle $A'B'C'D'$ is constructed by rotating $ABCD$ counterclockwise about $A$ by $60^\circ.$ Given that lines $BB'$ and $DD'$ intersect at point $X,$ compute $CX.$

2025 Harvard-MIT Mathematics Tournament, 11

Tags: guts

Let $f(n)=n^2+100.$ Compute the remainder when $\underbrace{f(f(\cdots f(f(}_{2025 \ f\text{'s}}1))\cdots ))$ is divided by $10^4.$

2023 Harvard-MIT Mathematics Tournament, 17

Tags: guts

An equilateral triangle lies in the Cartesian plane such that the $x$-coordinates of its vertices are pairwise distinct and all satisfy the equation $x^3-9x^2 + 10x + 5 = 0.$ Compute the side length of the triangle.

2024 LMT Fall, 26

Tags: guts

Let $P$ be a point in the interior of square $ABCD$ such that $\angle APB+\angle CPD=180^\circ$ and $\angle APB$ $ <$ $\angle CPD$. If $PC=7$ and $PD=5$, find $\tfrac{PA}{PB}$.

2024 LMT Fall, 4

Tags: guts

A group of $5$ rappers wants to make a song together. They each make their own parts for the song and then arrange the $5$ parts. J Cole wants to be friends with both Drake and Kendrick, so he wants his part to be adjacent to both of theirs. Find the number of possible songs (distinct orders) that can be made.

2024 Harvard-MIT Mathematics Tournament, 26

Tags: guts

It can be shown that there exists a unique polynomial $P$ in two variables such that for all positive integers $m$ and $n,$ $$P(m,n)=\sum_{i=1}^m\sum_{i=1}^n (i+j)^7.$$ Compute $P(3,-3).$

2025 Harvard-MIT Mathematics Tournament, 8

Tags: guts

A [i]checkerboard[/i] is a rectangular grid of cells colored black and white such that the top-left corner is black and no two cells of the same color share an edge. Two checkerboards are [i]distinct[/i] if and only if they have a different number of rows or columns. For example, a $20 \times 25$ checkerboard and a $25 \times 20$ checkerboard are considered distinct. Compute the number of distinct checkerboards that have exactly $41$ distinct black cells.

2024 LMT Fall, 32

Tags: guts

Let $a$ and $b$ be positive integers such that\[a^2+(a+1)^2=b^4.\]Find the least possible value of $a+b$.

2024 Harvard-MIT Mathematics Tournament, 15

Tags: guts

Let $a \star b=ab-2.$ Comute the remainder when $(((579\star569)\star559)\star\cdots\star19)\star9$ is divided by $100.$

2024 HMNT, 27

Tags: guts

For any positive integer $n,$ let $f(n)$ be the number of ordered triples $(a,b,c)$ of positive integers such that [list] [*] $\max(a,b,c)$ divides $n$ and [*] $\gcd(a,b,c)=1.$ [/list] Compute $f(1)+f(2)+\cdots+f(100).$

2024 Harvard-MIT Mathematics Tournament, 4

Tags: guts

Equilateral triangles $ABF$ and $BCG$ are constructed outside regular pentagon $ABCDE.$ Compute $\angle{FEG}.$

2025 Harvard-MIT Mathematics Tournament, 13

Tags: guts

A number is [i]upwards[/i] if its digits in base $10$ are nondecreasing when read from left to right. Compute the number of positive integers less than $10^6$ that are both upwards and multiples of $11.$

2023 Harvard-MIT Mathematics Tournament, 30

Tags: guts

Five pairs of twins are randomly arranged around a circle. Then they perform zero or more swaps, where each swap switches the positions of two adjacent people. They want to reach a state where no one is adjacent to their twin. Compute the expected value of the smallest number of swaps needed to reach such a state.

2023 Harvard-MIT Mathematics Tournament, 4

Tags: guts

A [i]standard $n$-sided die[/i] has $n$ sides labeled $1$ to $n.$ Luis, Luke, and Sean play a game in which they roll a fair standard $4$-sided die, a fair standard $6$-sided die, and a fair standard $8$-sided die, respectively. They lose the game if Luis's roll is less than Luke's roll, and Luke's roll is less than Sean's roll. Compute the probability that they lose the game.

2025 Harvard-MIT Mathematics Tournament, 4

Tags: guts

Let $\triangle{ABC}$ be an equilateral triangle with side length $4.$ Across all points $P$ inside triangle $\triangle{ABC}$ satisfying $[PAB]+[PAC]=[PBC],$ compute the minimal possible length of $PA.$ (Here, $[XYZ]$ denotes the area of triangle $\triangle{XYZ}.$)

2024 HMNT, 7

Tags: guts

Let $\mathcal{P}$ be a regular $10$-gon in the coordinate plane. Mark computes the number of distinct $x$-coordinates that vertices of $\mathcal{P}$ take. Across all possible placements of $\mathcal{P}$ in the plane, compute the sum of all possible answers Mark could get.