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.

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Found problems: 161

2023 Harvard-MIT Mathematics Tournament, 2
Tags: guts
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, 22
Tags: guts
Let $a,b,$ and $c$ be real numbers such that $a^2(b+1)=1, b^2(c+a)=2,$ and $c^2(a+b)=5.$ Given that there are three possible values for $abc,$ compute the minimum possible value of $abc.$

2024 Harvard-MIT Mathematics Tournament, 14
Tags: guts
Compute the smallest positive integer such that, no matter how you rearrange its digits (in base ten), the resulting number is a multiple of $63.$

2025 Harvard-MIT Mathematics Tournament, 6
Tags: guts
Let $\triangle{ABC}$ be an equilateral triangle. Point $D$ is on segment $\overline{BC}$ such that $BD=1$ and $DC=4.$ Points $E$ and $F$ lie on rays $\overrightarrow{AC}$ and $\overrightarrow{AB},$ respectively, such that $D$ is the midpoint of $\overline{EF}.$ Compute $EF.$

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 $T$.

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 HMNT, 28
Tags: guts
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 HMNT, 8
Tags: guts
Derek is bored in math class and is drawing a flower. He first draws $8$ points $A_1, A_2, \ldots, A_8$ equally spaced around an enormous circle. He then draws $8$ arcs outside the circle where the $i$th arc for $i = 1, 2, \ldots, 8$ has endpoints $A_i, A_{i+1}$ with $A_9 = A_1,$ such that all of the arcs have radius $1$ and any two consecutive arcs are tangent. Compute the perimeter of Derek’s $8$-petaled flower (not including the central circle). [center] [img] https://cdn.artofproblemsolving.com/attachments/8/4/e8b23c587762c089adb77b29cae155209f5db5.png [/img] [/center]

2024 Harvard-MIT Mathematics Tournament, 20
Tags: guts
Compute $\sqrt[4]{5508^3+5625^3+5742^3},$ given that it is an integer.

2025 Harvard-MIT Mathematics Tournament, 9
Tags: guts
Let $P$ and $Q$ be points selected uniformly and independently at random inside a regular hexagon $ABCDEF.$ Compute the probability that segment $\overline{PQ}$ is entirely contained in at least one of the quadrilaterals $ABCD,$ $BCDE,$ $CDEF,$ $DEFA,$ $EFAB,$ or $FABC.$

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 $BD.$

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 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 $a,b.$

2023 Harvard-MIT Mathematics Tournament, 5
Tags: guts
If $a$ and $b$ are positive real numbers such that $a \cdot 2^b=8$ and $a^b=2,$ compute $a^{\log_2 a} 2^{b^2}.$

2024 LMT Fall, 20
Tags: guts
A base $9$ number [i]probably places[/i] if it has a $7$ as one of its digits. Find the number of base $9$ numbers less than or equal to $100$ in base $10$ that probably place.

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 $\overline{450}=54$).

2024 HMNT, 14
Tags: guts
Let $ABCD$ be a trapezoid with $AB \parallel CD.$ Point $X$ is placed on segment $BC$ such that $\angle{BAX} = \angle{XDC}.$ Given that $AB = 5, BX =3, CX =4,$ and $CD =12,$ compute $AX.$

2024 Harvard-MIT Mathematics Tournament, 24
Tags: guts
A circle is tangent to both branches of the hyperbola $x^2 - 20y^2 = 24$ as well as the $x$-axis. Compute the area of this circle.

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 $100a+b.$

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, 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, 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 $a.$

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 Harvard-MIT Mathematics Tournament, 19
Tags: guts
let $A_1A_2\ldots A_{19}$ be a regular nonadecagon. Lines $A_1A_5$ and $A_3A_4$ meet at $X.$ Compute $\angle A_7 X A_5.$

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 $DE$.