Found problems: 161
2023 Harvard-MIT Mathematics Tournament, 13
Suppose $a, b, c,$ and $d$ are pairwise distinct positive perfect squares such that $a^b = c^d.$ Compute the smallest possible value of $a + b + c + d.$
2024 HMNT, 23
Consider a quarter-circle with center $O,$ arc $\widehat{AB},$ and radius $2.$ Draw a semicircle with diameter $\overline{OA}$ lying inside the quarter-circle. Points $P$ and $Q$ lie on the semicircle and segment $\overline{OB},$ respectively, such that line $PQ$ is tangent to the semicircle. As $P$ and $Q$ vary, compute the maximum possible area of triangle $BQP.$
2025 Harvard-MIT Mathematics Tournament, 14
A parallelogram $P$ can be folded over a straight line so that the resulting shape is a regular pentagon with side length $1.$ Compute the perimeter of $P.$
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 $\triangle{XYZ}.$)
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 LMT Fall, 7-9
Let $L$ be the answer to problem $9$. Find the solution to the equation $4x+\sqrt{L}=0$.
Let $M$ be the answer to problem $7$. Let $f(x)=x^4+4x^3+6x^2+1$. Find $f(M)$.
Let $T$ be the answer to problem $8$. Find the area of a square with side length $T$.
2024 Harvard-MIT Mathematics Tournament, 26
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 $P(3,-3).$
2024 LMT Fall, 14
Find the number of trailing $0$s in the base $12$ expression of $99!$ (Note: $99$ is in base $10$).
2024 HMNT, 17
Compute the number of ways to shade in some subset of the $16$ cells in a $4 \times 4$ grid such that each of the $25$ vertices of the grid is a corner of at least one shaded cell.
2024 HMNT, 11
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.
2023 BMT, 27
Let $\omega$ be a circle with positive integer radius $r$. Suppose that it is possible to draw isosceles triangle with integer side lengths inscribed in $\omega$. Compute the number of possible values of $r$ where $1 \le r \le 2023^2$.
Submit your answer as a positive integer $E$. If the correct answer is $A$, your score for this question will be $\max \left( 0, 25\left(3 - 2 \max \left( \frac{A}{E} , \frac{E}{A}\right)\right)\right)$, rounded to the nearest integer.
2024 HMNT, 28
The graph of the equation $\tan(x+y) = \tan(x)+2\tan(y),$ with its pointwise holes filled in, partitions the coordinate plane into congruent regions. Compute the perimeter of one of these regions.
2024 LMT Fall, 24
Let $ABC$ be a triangle with $AB=13, BC=15, AC=14$. Let $P$ be the point such that $AP$ $=$ $CP$ $=$ $\tfrac12 BP$. Find $AP^2$.
2024 HMNT, 31
Positive integers $a, b,$ and $c$ have the property that $\text{lcm}(a,b), \text{lcm}(b,c),$ and $\text{lcm}(c,a)$ end in $4, 6,$ and $7,$ respectively, when written in base $10.$ Compute the minimum possible value of $a + b + c.$
2024 LMT Fall, 1
Find the least prime factor of $2024^{2024}-1$.
2024 LMT Fall, 29
Let $P(x)$ be a quartic polynomial with integer coefficients and leading coefficient $1$ such that $P(\sqrt 2+\sqrt 3+\sqrt 6)=0$. Find $P(1)$.
2023 Harvard-MIT Mathematics Tournament, 7
Let $\Omega$ be a sphere of radius $4$ and $\Gamma$ be a sphere of radius $2.$ Suppose that the center of $\Gamma$ lies on the surface of $\Omega.$ The intersection of the surfaces of $\Omega$ and $\Gamma$ is a circle. Compute this circle's circumfrence.
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.
2024 LMT Fall, 30
Find
\[\sum_{n=1}^{\infty} \frac{\varphi(n)}{(-4)^n-1},\]where $\varphi(n)$ is the number of positive integers $k \le n$ relatively prime to $n$. (Note $\varphi(1)=1$.)
2024 Harvard-MIT Mathematics Tournament, 15
Let $a \star b=ab-2.$ Comute the remainder when $(((579\star569)\star559)\star\cdots\star19)\star9$ is divided by $100.$
2024 LMT Fall, 18
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 $M$.
2025 Harvard-MIT Mathematics Tournament, 2
Compute $$\frac{20+\frac{1}{25-\frac{1}{20}}}{25+\frac{1}{20-\frac{1}{25}}}.$$
2025 Harvard-MIT Mathematics Tournament, 27
Compute the number of ordered pairs $(m,n)$ of [i]odd[/i] positive integers both less than $80$ such that $$\gcd(4^m+2^m+1, 4^n+2^n+1)>1.$$
2025 Harvard-MIT Mathematics Tournament, 1
Call a $9$-digit number a [i]cassowary[/i] if it uses each of the digits $1$ through $9$ exactly once. Compute the number of cassowaries that are prime.
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 $(6,8).$