Found problems: 161
2024 HMNT, 10
Compute the largest prime factor of $3^{12}+3^9+3^5+1.$
2024 HMNT, 30
Compute the number of ways to shade exactly $4$ distinct cells of a $4\times4$ grid such that no two shaded cells share one or more vertices.
2024 LMT Fall, 6
Let $P$ be a point in rectangle $ABCD$ such that the area of $PAB$ is $20$ and the area of $PCD$ is $24$. Find the area of $ABCD$.
2024 Harvard-MIT Mathematics Tournament, 25
Point $P$ is inside a square $ABCD$ such that $\angle APB = 135^\circ, PC=12,$ and $PD=15.$ Compute the area of this square.
2024 HMNT, 13
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 $a.$
2024 HMNT, 7
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.
2025 Harvard-MIT Mathematics Tournament, 8
A [i]checkerboard[/i] is a rectangular grid of cells colored black and white such that the top-left corner is black and no two cells of the same color share an edge. Two checkerboards are [i]distinct[/i] if and only if they have a different number of rows or columns. For example, a $20 \times 25$ checkerboard and a $25 \times 20$ checkerboard are considered distinct.
Compute the number of distinct checkerboards that have exactly $41$ distinct black cells.
2023 Harvard-MIT Mathematics Tournament, 32
Let $ABC$ be a triangle with $\angle{BAC}>90^\circ.$ Let $D$ be the foot of the perpendicular from $A$ to side $BC.$ Let $M$ and $N$ be the midpoints of segments $BC$ and $BD,$ respectively. Suppose that $AC=2, \angle{BAN}=\angle{MAC},$ and $AB \cdot BC = AM.$ Compute the distance from $B$ to line $AM.$
2024 HMNT, 29
Let $ABC$ be a triangle such that $AB = 3, AC = 4,$ and $\angle{BAC} = 75^\circ.$ Square $BCDE$ is constructed outside triangle $ABC.$ Compute $AD^2 +AE^2.$
2023 Harvard-MIT Mathematics Tournament, 14
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 HMNT, 18
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 $CX.$
2024 HMNT, 9
Compute the remainder when $$1002003004005006007008009$$ is divided by $13.$
2024 LMT Fall, 12
Snorlax's weight is modeled by the function $w(t)=t2^t$ where $w(t)$ is Snorlax's weight at time $t$ minutes. Find the smallest integer time $t$ such that Snorlax's weight is greater than $10000.$
2025 Harvard-MIT Mathematics Tournament, 24
For any integer $x,$ let $$f(x)=100!\left(1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots+\frac{x^{100}}{100!}\right).$$ A positive integer $a$ is chosen such that $f(a)-20$ is divisible by $101^2.$ Compute the remainder when $f(x+101)$ is divided by $101^2.$
2024 Harvard-MIT Mathematics Tournament, 21
Kelvin the frog currently sits at $(0,0)$ in the coordinate plane. If Kelvin is at $(x,y),$ either he can walk to any of $(x,y + 1),$ $(x + 1,y),$ or $(x + 1,y + 1),$ or he can jump to any of $(x,y + 2), (x + 2,y),$ or $(x+1,y+1).$ Walking and jumping from $(x,y)$ to $(x+1,y+1)$ are considered distinct actions. Compute the number of ways Kelvin can reach $(6,8).$
2024 Harvard-MIT Mathematics Tournament, 32
Over all pairs of complex numbers $(x,y)$ satisfying the equations $$x+2y^2=x^4 \quad \text{and} \quad y+2x^2=y^2,$$ compute the minimum possible real part of $x.$
2023 Harvard-MIT Mathematics Tournament, 30
Five pairs of twins are randomly arranged around a circle. Then they perform zero or more swaps, where each swap switches the positions of two adjacent people. They want to reach a state where no one is adjacent to their twin. Compute the expected value of the smallest number of swaps needed to reach such a state.
2024 HMNT, 1
A circle of area $1$ is cut by two distinct chords. Compute the maximum possible area of the smallest resulting piece.
2023 Harvard-MIT Mathematics Tournament, 6
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, 19
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.
2024 LMT Fall, 28
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 LMT Fall, 3
Two distinct positive even integers sum to $8$. Find the larger of the two integers.
2025 Harvard-MIT Mathematics Tournament, 12
Holden has a collection of polygons. He writes down a list containing the measure of each interior angle of each of his polygons. He writes down the list $30^\circ, 50^\circ, 60^\circ, 70^\circ, 90^\circ, 100^\circ, 120^\circ, 160^\circ,$ and $x^\circ,$ in some order. Compute $x.$
2023 Harvard-MIT Mathematics Tournament, 2
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).
2025 Harvard-MIT Mathematics Tournament, 4
Let $\triangle{ABC}$ be an equilateral triangle with side length $4.$ Across all points $P$ inside triangle $\triangle{ABC}$ satisfying $[PAB]+[PAC]=[PBC],$ compute the minimal possible length of $PA.$
(Here, $[XYZ]$ denotes the area of triangle $\triangle{XYZ}.$)