Found problems: 161
2024 Harvard-MIT Mathematics Tournament, 9
Compute the sum of all positive integers $n$ such that $n^2-3000$ is a perfect square.
2024 Harvard-MIT Mathematics Tournament, 4
Equilateral triangles $ABF$ and $BCG$ are constructed outside regular pentagon $ABCDE.$ Compute $\angle{FEG}.$
2025 Harvard-MIT Mathematics Tournament, 20
Compute the $100$th smallest positive multiple of $7$ whose digits in base $10$ are all strictly less than $3.$
2024 Harvard-MIT Mathematics Tournament, 10
Alice, Bob, and Charlie are playing a game with $6$ cards numbered $1$ through $6.$ Each player is dealt $2$ cards uniformly at random. On each player’s turn, they play one of their cards, and the winner is the person who plays the median of the three cards played. Charlie goes last, so Alice and Bob decide to tell their cards to each other, trying to prevent him from winning whenever possible. Compute the probability that Charlie wins regardless.
2023 Harvard-MIT Mathematics Tournament, 24
Let $AXBY$ be a cyclic quadrilateral, and let line $AB$ and line $XY$ intersect at $C.$ Suppose $AX \cdot AY = 6, BX \cdot BY=5,$ and $CX \cdot CY=4.$ Compute $AB^2.$
2024 HMNT, 33
A grid is called [i]groovy[/i] if each cell of the grid is labeled with the smallest positive integer that does not appear below it in the same column or to the left of it in the same row. Compute the sum of the entries of a groovy $14 \times 14$ grid whose bottom left entry is $1.$
2024 LMT Fall, 4
A group of $5$ rappers wants to make a song together. They each make their own parts for the song and then arrange the $5$ parts. J Cole wants to be friends with both Drake and Kendrick, so he wants his part to be adjacent to both of theirs. Find the number of possible songs (distinct orders) that can be made.
2024 Harvard-MIT Mathematics Tournament, 7
Positive integers $a, b,$ and $c$ have the property that $a^b, b^c,$ and $c^a$ end in $4, 2,$ and $9,$ respectively. Compute the minimum possible value of $a+b+c.$
2024 LMT Fall, 26
Let $P$ be a point in the interior of square $ABCD$ such that $\angle APB+\angle CPD=180^\circ$ and $\angle APB$ $ <$ $\angle CPD$. If $PC=7$ and $PD=5$, find $\tfrac{PA}{PB}$.
2025 Harvard-MIT Mathematics Tournament, 3
Jacob rolls two fair six-sided dice. If the outcomes of these dice rolls are the same, he rolls a third fair six-sided die. Compute the probability that the sum of the outcomes of all the dice he rolls is even.
2023 Harvard-MIT Mathematics Tournament, 1
Suppose $a$ and $b$ are positive integers such that $a^b=2^{2023}.$ Compute the smallest possible value of $b^a.$
2025 Harvard-MIT Mathematics Tournament, 15
Right triangle $\triangle{DEF}$ with $\angle{D}=90^\circ$ and $\angle{F}=30^\circ$ is inscribed in equilateral triangle $\triangle{ABC}$ such that $D, E,$ and $F$ lie on segments $\overline{BC}, \overline{CA},$ and $\overline{AB},$ respectively. Given that $BD=7$ and $DC=4,$ compute $DE.$
2025 Harvard-MIT Mathematics Tournament, 6
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, 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 HMNT, 14
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 LMT Fall, 17
Suppose $x$, $y$, $z$ are pairwise distinct real numbers satisfying
\[
x^2+3y =y^2 +3z = z^2+3x.
\]Find $(x+y)(y+z)(z+x)$.
2024 Harvard-MIT Mathematics Tournament, 5
Let $a,b,$ and $c$ be real numbers such that
\begin{align*}
a+b+c &= 100 \\
ab+bc+ca &= 20, \text{ and} \\
(a+b)(a+c) &=24.
\end{align*}
Compute all possible values of $bc.$
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.
2023 Harvard-MIT Mathematics Tournament, 17
An equilateral triangle lies in the Cartesian plane such that the $x$-coordinates of its vertices are pairwise distinct and all satisfy the equation $x^3-9x^2 + 10x + 5 = 0.$ Compute the side length of the triangle.
2025 Harvard-MIT Mathematics Tournament, 28
Let $f$ be a function from nonnegative integers to nonnegative integers such that $f(0)=0$ and $$f(m)=f\left(\left\lfloor \frac{m}{2}\right\rfloor\right)+\left\lceil\frac{m}{2}\right\rceil^2$$ for all positive integers $m.$ Compute $$\frac{f(1)}{1\cdot2}+\frac{f(2)}{2\cdot3}+\frac{f(3)}{3\cdot4}+\cdots+\frac{f(31)}{31\cdot32}.$$(Here, $\lfloor z \rfloor$ is the greatest integer less than or equal to $z,$ and $\lceil z \rceil$ is the least positive integer greater than or equal to $z.$)
2023 Harvard-MIT Mathematics Tournament, 29
Let $P_1(x), P_2(x), \ldots, P_k(x)$ be monic polynomials of degree $13$ with integer coefficients. Suppose there are pairwise distinct positive integers $n_1, n_2, \ldots, n_k$ for which, for all positive integers $i$ and $j$ less than or equal to $k,$ the statement "$n_i$ divides $P_j(m)$ for every integer $m$" holds if and only if $i=j.$ Compute the largest possible value of $k.$
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.
2025 Harvard-MIT Mathematics Tournament, 7
The number $$\frac{9^9-8^8}{1001}$$ is an integer. Compute the sum of its prime factors.
2024 HMNT, 2
Compute the smallest integer $n > 72$ that has the same set of prime divisors as $72.$
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 $BQP.$