Olympiad problems

2024 HMNT, 12

Tags: guts

A dodecahedron is a polyhedron shown on the left below. One of its nets is shown on the right. Compute the label of the face opposite to $\mathcal{P}.$ [center] [img] https://cdn.artofproblemsolving.com/attachments/a/8/7607ee5d199471fd13b09a41a473c71d5d935b.png [/img] [/center]

2024 LMT Fall, 1

Tags: guts

Find the least prime factor of $2024^{2024}-1$.

2024 Harvard-MIT Mathematics Tournament, 29

Tags: guts

For each prime $p,$ a polynomial $P(x)$ with rational coefficients is called $p$-[i]good[/i] if and only if there exist three integers $a,b,$ and $c$ such that $0 \le a < b < c < \tfrac{p}{3}$ and $p$ divides all the numerators of $P(a), P(b),$ and $P(c),$ when written in simplest form. Compute the number of ordered pairs $(r,s)$ of rational numbers such that the polynomial $x^3+10x^2+rx+s$ is $p$-good for infinitely many primes $p.$

2024 LMT Fall, 28

Tags: guts

Find the number of ways to tile a $2 \times 2 \times 2 \times 2$ four dimensional hypercube with $2 \times 1 \times 1 \times 1$ blocks, with reflections and rotations of the large hypercube distinct.

2024 Harvard-MIT Mathematics Tournament, 22

Tags: guts

Let $x<y$ be positive real numbers such that $$\sqrt{x}+\sqrt{y}=4 \quad \text{and} \quad \sqrt{x+2}+\sqrt{y+2}=5.$$ Compute $x.$

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 $f(2024)$.

2024 LMT Fall, 11

Tags: guts

A Pokemon fan walks into a store. An employee tells them that there are $2$ Pikachus, $3$ Eevees, $4$ Snorlaxes, and $5$ Bulbasaurs remaining inside the gacha machine. Given that this fan cannot see what is inside the Poké Balls before opening them, find the least number of Poké Balls they must buy in order to be sure to get one Pikachu and one Snorlax.

2025 Harvard-MIT Mathematics Tournament, 16

Tags: guts

The [i]Cantor set[/i] is defined as the set of real numbers $x$ such that $0 \le x < 1$ and the digit $1$ does not appear in the base-$3$ expansion of $x.$ Two numbers are uniformly and independently selected at random from the Cantor set. Compute the expected value of their difference. (Formally, one can pick a number $x$ uniformly at random from the Cantor set by first picking a real number $y$ uniformly at random from the interval $[0, 1)$, writing it out in binary, reading its digits as if they were in base-$3,$ and setting $x$ to $2$ times the result.)

2023 Harvard-MIT Mathematics Tournament, 21

Tags: guts

Let $x, y,$ and $N$ be real numbers, with $y$ nonzero, such that the sets $\{(x+y)^2, (x-y)^2, xy, x/y\}$ and $\{4, 12.8, 28.8, N\}$ are equal. Compute the sum of the possible values of $N.$

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 $PAB.$

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 $\omega.$