Found problems: 161
2024 LMT Fall, 21
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, 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 $AD^2 +AE^2.$
2024 LMT Fall, 5
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 $n$.
2023 Harvard-MIT Mathematics Tournament, 20
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 LMT Fall, 10
David starts at the point $A$ and goes up and right along the grid lines to point $B$. At each of the points $C$, $D$, and $E$ there is a bully. Find the number of paths David can take which make him encounter exactly one bully.
[asy]
size(150);
draw((0,0)--(4,0)--(4,3)--(0,3)--cycle);
draw((0,1)--(4,1));
draw((0,2)--(4,2));
draw((1,0)--(1,3));
draw((2,0)--(2,3));
draw((3,0)--(3,3));
dot((0,0)); label("A", (0,0), W);
dot((4,3)); label("B", (4,3), E);
dot((1,1.5)); label("C", (1,1.5), W);
dot((2,0.5)); label("D", (2,0.5), W);
dot((2.5,2)); label("E", (2.5,2), N);
[/asy]
2024 LMT Fall, 2
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, 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, 22
Let $a_0, a_1, a_2, \ldots$ be an infinite sequence where each term is independently and uniformly at random in the set $\{1, 2, 3, 4\}.$ Define an infinite sequence $b_0, b_1, b_2, \ldots$ recursively by $b_0=1$ and $b_{i+1}=a_i^{b_i}.$ Compute the expected value of the smallest positive integer $k$ such that $b_k \equiv 1 \pmod{5}.$
2025 Harvard-MIT Mathematics Tournament, 19
A subset $S$ of $\{1, 2, 3, \ldots, 2025\}$ is called [i]balanced[/i] if for all elements $a$ and $b$ both in $S,$ there exists an element $c$ in $S$ such that $2025$ divides $a+b-2c.$ Compute the number of [i]nonempty[/i] balanced sets.
2024 LMT Fall, 16
A new meme is circling around social media known as the [i]DaDerek Convertible[/i]. The license plate number of the [i]DaDerek Convertible[/i] is such that the product of its nonzero digits times $5$ is equal to itself. Given that its license plate number has less than or equal to $3$ digits and that it has at least one nonzero digit, find the [i]DaDerek Convertible[/i]'s license plate number.
2025 Harvard-MIT Mathematics Tournament, 30
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 $a.$
2024 Harvard-MIT Mathematics Tournament, 29
For each prime $p,$ a polynomial $P(x)$ with rational coefficients is called $p$-[i]good[/i] if and only if there exist three integers $a,b,$ and $c$ such that $0 \le a < b < c < \tfrac{p}{3}$ and $p$ divides all the numerators of $P(a), P(b),$ and $P(c),$ when written in simplest form. Compute the number of ordered pairs $(r,s)$ of rational numbers such that the polynomial $x^3+10x^2+rx+s$ is $p$-good for infinitely many primes $p.$
2024 Harvard-MIT Mathematics Tournament, 20
Compute $\sqrt[4]{5508^3+5625^3+5742^3},$ given that it is an integer.
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 Harvard-MIT Mathematics Tournament, 27
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 HMNT, 10
Compute the largest prime factor of $3^{12}+3^9+3^5+1.$
2024 LMT Fall, 20
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.
2025 Harvard-MIT Mathematics Tournament, 13
A number is [i]upwards[/i] if its digits in base $10$ are nondecreasing when read from left to right. Compute the number of positive integers less than $10^6$ that are both upwards and multiples of $11.$
2024 LMT Fall, 15
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 $ABCDEFX$.
2024 Harvard-MIT Mathematics Tournament, 3
Compute the number of even positive integers $n \le 2024$ such that $1, 2, \ldots, n$ can be split into $\tfrac{n}{2}$ pairs, and the sum of the numbers in each pair is a multiple of $3.$
2024 LMT Fall, 13
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, 27
Find all positive integer pairs $(a,b)$ that satisfy the equation$$a^2b+ab^2+73=8ab+9a+9b.$$
2024 LMT Fall, 22
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 Harvard-MIT Mathematics Tournament, 4
Equilateral triangles $ABF$ and $BCG$ are constructed outside regular pentagon $ABCDE.$ Compute $\angle{FEG}.$
2024 Harvard-MIT Mathematics Tournament, 9
Compute the sum of all positive integers $n$ such that $n^2-3000$ is a perfect square.