Olympiad problems

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 LMT Fall, 27

Tags: guts

Find all positive integer pairs $(a,b)$ that satisfy the equation$$a^2b+ab^2+73=8ab+9a+9b.$$

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, 28

Tags: guts

Suppose $ABCD$ is a convex quadrilateral with $\angle{ABD}=105^\circ, \angle{ADB}=15^\circ, AC=7,$ and $BC=CD=5.$ Compute the sum of all possible values of $BD.$

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, 22

Tags: guts

Let $a_0, a_1, a_2, \ldots$ be an infinite sequence where each term is independently and uniformly at random in the set $\{1, 2, 3, 4\}.$ Define an infinite sequence $b_0, b_1, b_2, \ldots$ recursively by $b_0=1$ and $b_{i+1}=a_i^{b_i}.$ Compute the expected value of the smallest positive integer $k$ such that $b_k \equiv 1 \pmod{5}.$

2025 Harvard-MIT Mathematics Tournament, 19

Tags: guts

A subset $S$ of $\{1, 2, 3, \ldots, 2025\}$ is called [i]balanced[/i] if for all elements $a$ and $b$ both in $S,$ there exists an element $c$ in $S$ such that $2025$ divides $a+b-2c.$ Compute the number of [i]nonempty[/i] balanced sets.

2024 Harvard-MIT Mathematics Tournament, 16

Tags: guts

Let $ABC$ be an isosceles triangle with orthocenter $H.$ Let $M$ and $N$ be the midpoints of sides $\overline{AB}$ and $\overline{AC},$ respectively. The circumcircle of triangle $MHN$ intersects line $BC$ at two points $X$ and $Y.$ Given $XY=AB=AC=2,$ compute $BC^2.$

2024 HMNT, 25

Tags: guts

Let $ABC$ be an equilateral triangle. A regular hexagon $PXQYRZ$ of side length $2$ is placed so that $P, Q,$ and $R$ lie on segments $\overline{BC}, \overline{CA},$ and $\overline{AB}$, respectively. If points $A, X,$ and $Y$ are collinear, compute $BC.$

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, 23

Tags: guts

Define $\overline{a}$ of a positive integer $a$ to be the number $a$ with its digits reversed. For example, $\overline{31564} = 46513.$ Find the sum of all positive integers $n \leq 100$ such that $(\overline{n})^2=\overline{n^2}.$ (Note: For a number that ends with a zero, like 450, the reverse would exclude the zero, so $\overline{450}=54$).

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 LMT Fall, 5

Tags: guts

Ben Y's favorite number $p$ is prime, and his second favorite number is some integer $n$. Given that $p$ divides $n$ and $n$ divides $3p+91$, find the maximum possible value of $n$.

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 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, 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 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 $a,b.$

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.

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.

2024 HMNT, 20

Tags: guts

There exists a unique line tangent to the graph of $y=x^4-20x^3+24x^2-20x+25$ at two distinct points. Compute the product of the $x$-coordinates of the two tangency points.

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 $P(3,-3).$

2023 Harvard-MIT Mathematics Tournament, 8

Tags: guts

Suppose $a,b,c$ are distinct positive integers such that $\sqrt{a\sqrt{b\sqrt{c}}}$ is an integer. Compute the least possible value of $a+b+c.$

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 $P_4P_5.$

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 $\mathcal{P}.$

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 $AH_A$.