Olympiad problems

2024 Harvard-MIT Mathematics Tournament, 23

Tags: guts

Let $\ell$ and $m$ be two non-coplanar lines in space, and let $P_1$ be a point on $\ell.$ Let $P_2$ be the point on $,m$ closest to $P_1,$ $P_3$ be the point on $\ell$ closest to $P_3,$ $P_4$ be the point on $m$ closest to $P_3,$ and $P_5$ be the point on $\ell$ closest to $P_4.$ Given that $P_1P_2=5, P_2P_3=3,$ and $P_3P_4=2,$ compute $P_4P_5.$

2024 HMNT, 24

Tags: guts

Let $f(x) = x^2 +6x+6.$ Compute the greatest real number $x$ such that $f(f(f(f(f(f(x)))))) = 0.$

2024 Harvard-MIT Mathematics Tournament, 18

Tags: guts

An ordered pair $(a,b)$ of positive integers is called [i]spicy[/i] if $\gcd(a+b, ab+1)=1.$ Compute the probability that both $(99, n)$ and $(101,n)$ are spicy when $n$ is chosen from $\{1, 2, \ldots, 2024!\}$ uniformly at random.

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 LMT Fall, 17

Tags: algebra , guts

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, 6

Tags: guts

Let $A, E, H, L, T,$ and $V$ be chosen independently and at random from the set $\{0, \tfrac{1}{2}, 1\}.$ Compute the probability that $\lfloor T \cdot H \cdot E \rfloor = L \cdot A \cdot V \cdot A.$

2024 Harvard-MIT Mathematics Tournament, 17

Tags: guts

The numbers $1, 2, \ldots, 20$ are put into a hat. Claire draws two numbers from the hat uniformly at random, $a<b,$ and then puts them back into the hat. Then, William draws two numbers from the hat uniformly at random, $c<d.$ Let $N$ denote the number of integers $n$ that satisfy exactly one of $a \le n \le b$ and $c \le n \le d.$ Compute the probability that $N$ is even.

2023 Harvard-MIT Mathematics Tournament, 2

Tags: guts

Let $n$ be a positive integer, and let $s$ be the sum of the digits of the base-four representation of $2^n-1.$ If $s=2023$ (in base ten), compute $n$ (in base ten).

2023 Harvard-MIT Mathematics Tournament, 12

Tags: guts

The number $770$ is written on a blackboard. Melody repeatedly performs moves, where a move consists of subtracting either $40$ or $41$ from the number on the board. She performs moves until the number is not positive, and then she stops. Let $N$ be the number of sequences of moves that Melody could perform. Suppose $N = a\cdot 2^b$ where $a$ is an odd positive integer and $b$ is a nonnegative integer. Compute $100a+b.$

2023 Harvard-MIT Mathematics Tournament, 3

Tags: guts

Let $ABCD$ be a convex quadrilateral such that $\angle{ABD}=\angle{BCD}=90^\circ,$ and let $M$ be the midpoint of segment $BD.$ Suppose that $CM=2$ and $AM=3.$ Compute $AD.$

2024 Harvard-MIT Mathematics Tournament, 2

Tags: guts

Jerry and Neil have a $3$-sided die that rolls the numbers $1,2,$ and $3,$ each with probability $\tfrac{1}{3}.$ Jerry rolls first, then Neil rolls the die repeatedly until his number is at least as large as Jerry's. Compute the probability that Neil's final number is $3.$

2024 LMT Fall, 33

Tags: guts

Let $a$ and $b$ be positive real numbers that satisfy \begin{align*} \sqrt{a-ab}+\sqrt{b-ab}=\frac{\sqrt{6}+\sqrt{2}}{4} \,\,\, \text{and}\,\,\, \sqrt{a-a^2}+\sqrt{b-b^2}=\left(\frac{\sqrt{6}+\sqrt{2}}{4}\right)^2. \end{align*} Find the ordered pair $(a, b)$ such that $a>b$ and $a+b$ is maximal.

2023 Harvard-MIT Mathematics Tournament, 9

Tags: guts

One hundred points labeled $1$ to $100$ are arranged in a $10\times 10$ grid such that adjacent points are one unit apart. The labels are increasing left to right, top to bottom (so the first row has labels $1$ to $10,$ the second row has labels $11$ to $20,$ and so on). Convex polygon $\mathcal{P}$ has the property that every point with a label divisible by $7$ is either on the boundary or in the interior of $\mathcal{P}.$ Compute the smallest possible area of $\mathcal{P}.$

2023 Harvard-MIT Mathematics Tournament, 16

Tags: guts

The graph of the equation $x+y=\lfloor x^2+y^2 \rfloor$ consists of several line segments. Compute the sum of their lengths.

2025 Harvard-MIT Mathematics Tournament, 29

Tags: guts

Points $A$ and $B$ lie on circle $\omega$ with center $O.$ Let $X$ be a point inside $\omega.$ Suppose that $XO=2\sqrt{2}, XA=1, XB=3,$ and $\angle{AXB}=90^\circ.$ Points $Y$ and $Z$ are on $\omega$ such that $Y \neq A$ and triangles $\triangle{AXB}$ and $\triangle{YXZ}$ are similar with the same orientation. Compute $XY.$

2024 Harvard-MIT Mathematics Tournament, 1

Tags: guts

Compute the sum of all integers $n$ such that $n^2-3000$ is a perfect square.

2024 LMT Fall, 3

Tags: guts

Two distinct positive even integers sum to $8$. Find the larger of the two integers.

2025 Harvard-MIT Mathematics Tournament, 31

Tags: guts

There exists a unique circle that is both tangent to the parabola $y=x^2$ at two points and tangent to the curve $x=\sqrt{\tfrac{y^3}{1-y}}.$ Compute the radius of this circle.

2023 Harvard-MIT Mathematics Tournament, 14

Tags: guts

Acute triangle $ABC$ has circumcenter $O.$ The bisector of $ABC$ and the altitude from $C$ to side $AB$ intersect at $X.$ Suppose that there is a circle passing through $B, O, X,$ and $C.$ If $\angle BAC = n^\circ,$ where $n$ is a positive integer, compute the largest possible value of $n.$

2024 Harvard-MIT Mathematics Tournament, 10

Tags: guts

Alice, Bob, and Charlie are playing a game with $6$ cards numbered $1$ through $6.$ Each player is dealt $2$ cards uniformly at random. On each player’s turn, they play one of their cards, and the winner is the person who plays the median of the three cards played. Charlie goes last, so Alice and Bob decide to tell their cards to each other, trying to prevent him from winning whenever possible. Compute the probability that Charlie wins regardless.

2024 Harvard-MIT Mathematics Tournament, 31

Tags: guts

Ash and Gary independently come up with their own lineups of $15$ fire, grass, and water monsters. Then, the first monster of both lineups will fight, with fire beating grass, grass beating water, and water beating fire. The defeated monster is then substituted with the next one from their team’s lineup; if there is a draw, both monsters get defeated. Gary completes his lineup randomly, with each monster being equally likely to be any of the three types. Without seeing Gary’s lineup, Ash chooses a lineup that maximizes the probability p that his monsters are the last ones standing. Compute $p.$

2024 HMNT, 22

Tags: guts

Suppose that $a$ and $b$ are positive integers such that $\gcd(a^3 - b^3,(a-b)^3)$ is not divisible by any perfect square except $1.$ Given that $1 \le a-b \le 50,$ compute the number of possible values of $a-b$ across all such $a,b.$

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 $XD-XA.$

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.)

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.