Olympiad problems

2024 HMNT, 13

Tags: guts

Let $f$ and $g$ be two quadratic polynomials with real coefficients such that the equation $f(g(x)) = 0$ has four distinct real solutions: $112, 131, 146,$ and $a.$ Compute the sum of all possible values of $a.$

2025 Harvard-MIT Mathematics Tournament, 21

Tags: guts

Compute the unique five-digit positive integer $\underline{abcde}$ such that $a \neq 0, c \neq 0,$ and $$\underline{abcde}=(\underline{ab}+\underline{cde})^2.$$

2025 Harvard-MIT Mathematics Tournament, 23

Tags: guts

Regular hexagon $ABCDEF$ has side length $2.$ Circle $\omega$ lies inside the hexagon and is tangent to segments $\overline{AB}$ and $\overline{AF}.$ There exist two perpendicular lines tangent to $\omega$ that pass through $C$ and $E,$ respectively. Given that these two lines do not intersect on line $AD,$ compute the radius of $\omega.$

2024 HMNT, 6

Tags: guts

The vertices of a cube are labeled with the integers $1$ through $8,$ with each used exactly once. Let $s$ be the maximum sum of the labels of two edge-adjacent vertices. Compute the minimum possible value of $s$ over all such labelings.

2025 Harvard-MIT Mathematics Tournament, 18

Tags: guts

Let $f: \{1, 2, 3, \ldots, 9\} \to \{1, 2, 3, \ldots, 9\}$ be a permutation chosen uniformly at random from the $9!$ possible permutations. Compute the expected value of $\underbrace{f(f(\cdots f(f(}_{2025 \ f\text{'s}}1))\cdots )).$

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 HMNT, 3

Tags: guts

The graphs of the lines $$y=x+2, \quad y=3x+4, \quad y=5x+6,\quad y=7x+8,\quad y=9x+10,\quad y=11x+12$$ are drawn. These six lines divide the plane into several regions. Compute the number of regions the plane is divided into.

2024 HMNT, 2

Tags: guts

Compute the smallest integer $n > 72$ that has the same set of prime divisors as $72.$

2025 Harvard-MIT Mathematics Tournament, 3

Tags: guts

Jacob rolls two fair six-sided dice. If the outcomes of these dice rolls are the same, he rolls a third fair six-sided die. Compute the probability that the sum of the outcomes of all the dice he rolls is even.

2023 Harvard-MIT Mathematics Tournament, 18

Tags: guts

Elisenda has a piece of paper in the shape of a triangle with vertices $A, B,$ and $C$ such that $AB = 42.$ She chooses a point $D$ on segment $AC,$ and she folds the paper along line $BD$ so that $A$ lands at a point $E$ on segment $BC.$ Then, she folds the paper along line $DE.$ When she does this, $B$ lands at the midpoint of segment $DC.$ Compute the perimeter of the original unfolded triangle.

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.

2023 Harvard-MIT Mathematics Tournament, 26

Tags: guts

Let $PABC$ be a tetrahedron such that $\angle{APB}=\angle{APC}=\angle{BPC}=90^\circ, \angle{ABC}=30^\circ,$ and $AP^2$ equals the area of triangle $ABC.$ Compute $\tan\angle{ACB}.$

2024 HMNT, 21

Tags: guts

Two points are chosen independently and uniformly at random from the interior of the $X$-pentomino shown below. Compute the probability that the line segment between these two points lies entirely within the $X$-pentomino. [center] [img] https://cdn.artofproblemsolving.com/attachments/b/1/17565ba86dbc2358f546fa57145a7726d1b0a9.png [/img] [/center]

2024 Harvard-MIT Mathematics Tournament, 13

Tags: guts

Mark has a cursed six-sided die that never rolls the same number twice in a row, and all other outcomes are equally likely. Compute the expected number of rolls it takes for Mark to roll every number at least once.

2024 HMNT, 1

Tags: guts

A circle of area $1$ is cut by two distinct chords. Compute the maximum possible area of the smallest resulting piece.

2024 Harvard-MIT Mathematics Tournament, 11

Tags: guts

Let $ABCD$ be a rectangle such that $AB = 20$ and $AD = 24.$ Point $P$ lies inside $ABCD$ such that triangles $PAC$ and $PBD$ have areas $20$ and $24,$ respectively. Compute all possible areas of triangle $PAB.$

2024 LMT Fall, 31

Tags: guts

Let $ABC$ be a triangle with circumradius $12$, and denote the orthocenter and circumcenter as $H$ and $O$ respectively. Define $H_A \neq A$ to be the intersection of line $AH$ and the circumcircle of $ABC$. Given that $\overline{OH} \parallel \overline{BC}$ and $\overline{AO} \parallel \overline{BH_A}$, find $AH_A$.

2023 Harvard-MIT Mathematics Tournament, 1

Tags: guts

Suppose $a$ and $b$ are positive integers such that $a^b=2^{2023}.$ Compute the smallest possible value of $b^a.$

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.

2025 Harvard-MIT Mathematics Tournament, 5

Tags: guts

Compute the largest possible radius of a circle contained in the region defined by $|x+|y|| \le 1$ in the coordinate plane.

2023 Harvard-MIT Mathematics Tournament, 15

Tags: guts

Let $A$ and $B$ be points in space such that $AB=1.$ Let $\mathcal{R}$ be the region of points $P$ for which $AP \le 1$ and $BP \le 1.$ Compute the largest possible side length of a cube contained in $\mathcal{R}.$

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 $CX.$

2023 Harvard-MIT Mathematics Tournament, 29

Tags: guts

Let $P_1(x), P_2(x), \ldots, P_k(x)$ be monic polynomials of degree $13$ with integer coefficients. Suppose there are pairwise distinct positive integers $n_1, n_2, \ldots, n_k$ for which, for all positive integers $i$ and $j$ less than or equal to $k,$ the statement "$n_i$ divides $P_j(m)$ for every integer $m$" holds if and only if $i=j.$ Compute the largest possible value of $k.$

2024 Harvard-MIT Mathematics Tournament, 8

Tags: guts

Three points, $A, B,$ and $C,$ are selected independently and uniformly at random from the interior of a unit square. Compute the expected value of $\angle{ABC}.$

2024 LMT Fall, 25

Tags: guts

Define $f(n)$ to be the sum of positive integers $k$ less than or equal to $n$ such that $\gcd(n, k)$ is prime. Find $f(2024)$.