Found problems: 161
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 $(6,8).$
2024 LMT Fall, 33
Let $a$ and $b$ be positive real numbers that satisfy
\begin{align*}
\sqrt{a-ab}+\sqrt{b-ab}=\frac{\sqrt{6}+\sqrt{2}}{4} \,\,\, \text{and}\,\,\,
\sqrt{a-a^2}+\sqrt{b-b^2}=\left(\frac{\sqrt{6}+\sqrt{2}}{4}\right)^2.
\end{align*}
Find the ordered pair $(a, b)$ such that $a>b$ and $a+b$ is maximal.
2025 Harvard-MIT Mathematics Tournament, 26
Isabella has a bag with $20$ blue diamonds and $25$ purple diamonds. She repeats the following process $44$ times: she removes a diamond from the bag uniformly at random, then puts one blue diamond and one purple diamond into the bag. Compute the expected number of blue diamonds in the bag after all $44$ repetitions.
2023 Harvard-MIT Mathematics Tournament, 18
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.
2024 HMNT, 10
Compute the largest prime factor of $3^{12}+3^9+3^5+1.$
2024 Harvard-MIT Mathematics Tournament, 31
Ash and Gary independently come up with their own lineups of $15$ fire, grass, and water monsters. Then, the first monster of both lineups will fight, with fire beating grass, grass beating water, and water beating fire. The defeated monster is then substituted with the next one from their team’s lineup; if there is a draw, both monsters get defeated.
Gary completes his lineup randomly, with each monster being equally likely to be any of the three types. Without seeing Gary’s lineup, Ash chooses a lineup that maximizes the probability p that his monsters are the last ones standing. Compute $p.$
2023 Harvard-MIT Mathematics Tournament, 26
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 $\tan\angle{ACB}.$
2024 LMT Fall, 12
Snorlax's weight is modeled by the function $w(t)=t2^t$ where $w(t)$ is Snorlax's weight at time $t$ minutes. Find the smallest integer time $t$ such that Snorlax's weight is greater than $10000.$
2025 Harvard-MIT Mathematics Tournament, 17
Let $f$ be a quadratic polynomial with real coefficients, and let $g_1, g_2, g_3, \ldots$ be a geometric progression of real numbers. Define $a_n=f(n)+g_n.$ Given that $a_1, a_2, a_3, a_4,$ and $a_5$ are equal to $1, 2, 3, 14,$ and $16,$ respectively, compute $\tfrac{g_2}{g_1}.$
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 $3.$
2023 Harvard-MIT Mathematics Tournament, 8
Suppose $a,b,c$ are distinct positive integers such that $\sqrt{a\sqrt{b\sqrt{c}}}$ is an integer. Compute the least possible value of $a+b+c.$
2024 Harvard-MIT Mathematics Tournament, 25
Point $P$ is inside a square $ABCD$ such that $\angle APB = 135^\circ, PC=12,$ and $PD=15.$ Compute the area of this square.
2025 Harvard-MIT Mathematics Tournament, 32
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.)
2023 Harvard-MIT Mathematics Tournament, 25
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 HMNT, 1
A circle of area $1$ is cut by two distinct chords. Compute the maximum possible area of the smallest resulting piece.
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 $bc.$
2024 HMNT, 32
Let $ABC$ be an acute triangle and $D$ be the foot of altitude from $A$ to $BC.$ Let $X$ and $Y$ be points on the segment $BC$ such that $\angle{BAX} = \angle{YAC}, BX = 2, XY = 6,$ and $YC = 3.$ Given that $AD = 12,$ compute $BD.$
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 Harvard-MIT Mathematics Tournament, 1
Compute the sum of all integers $n$ such that $n^2-3000$ is a perfect square.
2024 Harvard-MIT Mathematics Tournament, 32
Over all pairs of complex numbers $(x,y)$ satisfying the equations $$x+2y^2=x^4 \quad \text{and} \quad y+2x^2=y^2,$$ compute the minimum possible real part of $x.$
2025 Harvard-MIT Mathematics Tournament, 24
For any integer $x,$ let $$f(x)=100!\left(1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots+\frac{x^{100}}{100!}\right).$$ A positive integer $a$ is chosen such that $f(a)-20$ is divisible by $101^2.$ Compute the remainder when $f(x+101)$ is divided by $101^2.$
2024 LMT Fall, 31
Let $ABC$ be a triangle with circumradius $12$, and denote the orthocenter and circumcenter as $H$ and $O$ respectively. Define $H_A \neq A$ to be the intersection of line $AH$ and the circumcircle of $ABC$. Given that $\overline{OH} \parallel \overline{BC}$ and $\overline{AO} \parallel \overline{BH_A}$, find $AH_A$.
2023 Harvard-MIT Mathematics Tournament, 27
Suppose $m>n>1$ are positive integers such that there exist $n$ complex numbers $x_1, x_2, \ldots, x_n$ for which
[list]
[*]$x_1^k+x_2^k+\ldots+x_n^k=1$ for $k=1, 2, \ldots, n-1;$
[*]$x_1^n+x_2^n+\ldots+x_n^n=2;$ and
[*]$x_1^m+x_2^m+\ldots+x_n^m= 4.$
[/list]
Compute the smallest possible value of $m+n.$
2023 Harvard-MIT Mathematics Tournament, 19
Compute the number of ways to select $99$ cells in a $19 \times 19$ square grid such that no two selected cells share an edge or a vertex.
2024 Harvard-MIT Mathematics Tournament, 30
Let $ABC$ be an equilateral triangle with side length $1.$ Points $D, E,$ and $F$ lie inside triangle $ABC$ such that $A, E, F$ are collinear, $B, F, D$ are collinear, $C, D, E$ are collinear, and triangle $DEF$ is equilateral. Suppose that there exists a unique equilateral triangle $XYZ$ with $X$ on side $\overline{BC},$ $Y$ is on side $\overline{AB},$ and $Z$ is on side $\overline{AC}$ such that $D$ lies on side $\overline{XZ},$ $E$ lies on side $\overline{YZ},$ and $F$ lies on side $\overline{XY}.$ Compute $AZ.$