Olympiad problems

2025 Harvard-MIT Mathematics Tournament, 27

Tags: guts

Compute the number of ordered pairs $(m,n)$ of [i]odd[/i] positive integers both less than $80$ such that $$\gcd(4^m+2^m+1, 4^n+2^n+1)>1.$$

2024 LMT Fall, 14

Tags: guts

Find the number of trailing $0$s in the base $12$ expression of $99!$ (Note: $99$ is in base $10$).

2024 HMNT, 24

Tags: guts

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

2025 Harvard-MIT Mathematics Tournament, 28

Tags: guts

Let $f$ be a function from nonnegative integers to nonnegative integers such that $f(0)=0$ and $$f(m)=f\left(\left\lfloor \frac{m}{2}\right\rfloor\right)+\left\lceil\frac{m}{2}\right\rceil^2$$ for all positive integers $m.$ Compute $$\frac{f(1)}{1\cdot2}+\frac{f(2)}{2\cdot3}+\frac{f(3)}{3\cdot4}+\cdots+\frac{f(31)}{31\cdot32}.$$(Here, $\lfloor z \rfloor$ is the greatest integer less than or equal to $z,$ and $\lceil z \rceil$ is the least positive integer greater than or equal to $z.$)

2025 Harvard-MIT Mathematics Tournament, 10

Tags: guts

A square of side length $1$ is dissected into two congruent pentagons. Compute the least upper bound of the perimeter of one of these pentagons.

2024 LMT Fall, 19

Tags: guts

Given $\sum_{n=1}^{\infty} \tfrac{1}{n^2}=\tfrac{\pi^2}{6}$, find$$\sum_{j=1}^{\infty} \sum_{i=1}^j \frac{1}{ij(i+1)(j+1)}.$$

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.

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.

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 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 Harvard-MIT Mathematics Tournament, 27

Tags: guts

A deck of $100$ cards is labeled $1,2,\ldots,100$ from top to bottom. The top two cards are drawn; one of them is discarded at random, and the other is inserted back at the bottom of the deck. This process is repeated until only one card remains in the deck. Compute the expected value of the label of the remaining card.

2024 Harvard-MIT Mathematics Tournament, 6

Tags: guts

In triangle $ABC,$ points $M$ and $N$ are the midpoints of $AB$ and $AC,$ respectively, and points $P$ and $Q$ trisect $BC.$ Given that $A, M, P, N,$ and $Q$ lie on a circle and $BC=1,$ compute the area of triangle $ABC.$

2024 HMNT, 5

Tags: guts

Let $ABCD$ be a trapezoid with $AB \parallel CD, AB=20, CD=24,$ and area $880.$ Compute the area of the triangle formed by the midpoints of $AB, AC,$ and $BD.$

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, 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.

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 $n.$

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 $AD.$

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 HMNT, 19

Tags: guts

An equilateral triangle is inscribed in a circle $\omega.$ A chord of $\omega$ is cut by the perimeter of the triangle into three segments of lengths $55, 121,$ and $55,$ in that order. Compute the sum of all possible side lengths of the triangle.

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 HMNT, 9

Tags: guts

Compute the remainder when $$1002003004005006007008009$$ is divided by $13.$

2024 HMNT, 11

Tags: guts

A four-digit integer in base $10$ is [i]friendly[/i] if its digits are four consecutive digits in any order. A four-digit integer is [i]shy[/i] if there exist two adjacent digits in its representation that differ by $1.$ Compute the number of four-digit integers that are both friendly and shy.

2024 LMT Fall, 18

Tags: guts

In the electoral college, each of $51$ places get some positive number of electoral votes for a nationwide total of $538$. Thus, $270$ electoral votes guarantees a win. Across all distributions of electoral votes to each place, let $M$ be the maximum number of sets of places that combine to have at least $270$ electoral votes. Find $M$.

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.

2024 LMT Fall, 13

Tags: guts

Suppose $j$, $x$, and $u$ are positive real numbers such that $jxu=20$ and $x+u=24$. Find the minimum possible value of $j\max(x,u)$.