This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

AND:
OR:
NO:

Found problems: 161

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 $BD.$

2023 BMT, 27
Tags: guts , geometry
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 LMT Fall, 7-9
Tags: guts
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 $T$.

2023 Harvard-MIT Mathematics Tournament, 26
Tags: guts
Let $PABC$ be a tetrahedron such that $\angle{APB}=\angle{APC}=\angle{BPC}=90^\circ, \angle{ABC}=30^\circ,$ and $AP^2$ equals the area of triangle $ABC.$ Compute $\tan\angle{ACB}.$

2023 Harvard-MIT Mathematics Tournament, 20
Tags: guts
Five people take a true-or-false test with five questions. Each person randomly guesses on every question. Given that, for each question, a majority of test-takers answered it correctly, let $p$ be the probability that every person answers exactly three questions correctly. Suppose that $p=\tfrac{a}{2^b}$ 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 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 $j\max(x,u)$.

2024 LMT Fall, 2
Tags: guts
A group of nine math team members like to play Survev.io. They noticed that the number of hours each of them played this week forms an arithmetic progression. The person who played the least played for $1$ hour, while the most played for $9.$ Find the total number of hours all nine group members spent playing Survev.io this week.

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.

2023 Harvard-MIT Mathematics Tournament, 7
Tags: guts
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.

2024 LMT Fall, 19
Tags: guts
Given $\sum_{n=1}^{\infty} \tfrac{1}{n^2}=\tfrac{\pi^2}{6}$, find$$\sum_{j=1}^{\infty} \sum_{i=1}^j \frac{1}{ij(i+1)(j+1)}.$$

2023 Harvard-MIT Mathematics Tournament, 25
Tags: guts
The [i]spikiness[/i] of a sequence $a_1, a_2, \ldots, a_n$ of at least two real numbers is the sum $\textstyle\sum_{i=1}^{n-1} |a_{i+1}-a_i|.$ Suppose $x_1, x_2, \ldots, x_9$ are chosen uniformly at random from the set $[0, 1].$ Let $M$ be the largest possible value of the spikiness of a permutation of $x_1, x_2, \ldots, x_9.$ Compute the expected value of $M.$

2024 Harvard-MIT Mathematics Tournament, 12
Tags: guts
Compute the number of quadruples $(a,b,c,d)$ of positive integers satisfying $$12a+21b+28c+84d=2024.$$

2025 Harvard-MIT Mathematics Tournament, 32
Tags: guts
In the coordinate plane, a closed lattice loop of length $2n$ is a sequence of lattice points $P_0, P_1, P_2, \ldots, \ldots, P_{2n}$ such that $P_0$ and $P_{2n}$ are both the origin and $P_{i}P_{i+1}=1$ for each $i.$ A closed lattice loop of length $2026$ is chosen uniformly at random from all such loops. Let $k$ be the maximum integer such that the line $\ell$ with equation $x+y=k$ passes through at least one point of the loop. Compute the expected number of indices $i$ such that $0 \le i \le 2025$ and $P_i$ lies on $\ell.$ (A lattice point is a point with integer coordinates.)

2024 LMT Fall, 15
Tags: guts
Regular hexagon $ABCDEF$ with side length $2$ is inscribed within a sphere of radius $4$. Let point $X$ be on the sphere. Find the maximum value of the volume of the pyramid $ABCDEFX$.

2023 Harvard-MIT Mathematics Tournament, 18
Tags: guts
Elisenda has a piece of paper in the shape of a triangle with vertices $A, B,$ and $C$ such that $AB = 42.$ She chooses a point $D$ on segment $AC,$ and she folds the paper along line $BD$ so that $A$ lands at a point $E$ on segment $BC.$ Then, she folds the paper along line $DE.$ When she does this, $B$ lands at the midpoint of segment $DC.$ Compute the perimeter of the original unfolded triangle.

2023 Harvard-MIT Mathematics Tournament, 10
Tags: guts
The number $$316990099009901=\frac{32016000000000001}{101}$$ is the product of two distinct prime numbers. Compute the smaller of these two primes.

2024 LMT Fall, 22
Tags: guts
Find the number of real numbers $0 \leq \alpha < 50$ such that $\alpha^2 + 2\{\alpha\}$ is an integer. (Here $\{\alpha\}$ denotes the fractional part of $\alpha$.)

2024 HMNT, 6
Tags: guts
The vertices of a cube are labeled with the integers $1$ through $8,$ with each used exactly once. Let $s$ be the maximum sum of the labels of two edge-adjacent vertices. Compute the minimum possible value of $s$ over all such labelings.

2024 HMNT, 17
Tags: guts
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 LMT Fall, 21
Tags: guts
Let $ABC$ be a triangle such that $AB=2$, $BC=3$, and $AC=4$. A circle passing through $A$ intersects $AB$ at $D$, $AC$ at $E$, and $BC$ at $M$ and $N$ such that $BM=MN=NC$. Find $DE$.

2024 HMNT, 21
Tags: guts
Two points are chosen independently and uniformly at random from the interior of the $X$-pentomino shown below. Compute the probability that the line segment between these two points lies entirely within the $X$-pentomino. [center] [img] https://cdn.artofproblemsolving.com/attachments/b/1/17565ba86dbc2358f546fa57145a7726d1b0a9.png [/img] [/center]

2024 LMT Fall, 30
Tags: guts
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 HMNT, 26
Tags: guts
A right rectangular prism of silly powder has dimensions $20 \times 24 \times 25.$ Jerry the wizard applies $10$ bouts of highdroxylation to the box, each of which increases one dimension of the silly powder by $1$ and decreases a different dimension of the silly powder by $1,$ with every possible choice of dimensions equally likely to be chosen and independent of all previous choices. Compute the expected volume of the silly powder after Jerry’s routine.

2025 Harvard-MIT Mathematics Tournament, 30
Tags: guts
Let $a,b,$ and $c$ be real numbers satisfying the system of equations \begin{align*} a\sqrt{1+b^2}+b\sqrt{1+a^2}&=\tfrac{3}{4},\\ b\sqrt{1+c^2}+c\sqrt{1+b^2}&=\tfrac{5}{12}, \ \text{and} \\ c\sqrt{1+a^2}+a\sqrt{1+c^2}&=\tfrac{21}{20}. \end{align*} Compute $a.$