Found problems: 161
2023 Harvard-MIT Mathematics Tournament, 5
If $a$ and $b$ are positive real numbers such that $a \cdot 2^b=8$ and $a^b=2,$ compute $a^{\log_2 a} 2^{b^2}.$
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 $DE.$
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.
2023 Harvard-MIT Mathematics Tournament, 11
The Fibonacci numbers are defined recursively by $F_0=0, F_1=1,$ and $F_i=F_{i-1}+F_{i-2}$ for $i \ge 2.$ Given $15$ wooden blocks of weights $F_2, F_3, \ldots, F_{16},$ compute the number of ways to paint each block either red or blue such that the total weight of the red blocks equals the total weight of the blue blocks.
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 $z.$)
2024 HMNT, 8
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).
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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 LMT Fall, 11
A Pokemon fan walks into a store. An employee tells them that there are $2$ Pikachus, $3$ Eevees, $4$ Snorlaxes, and $5$ Bulbasaurs remaining inside the gacha machine. Given that this fan cannot see what is inside the Poké Balls before opening them, find the least number of Poké Balls they must buy in order to be sure to get one Pikachu and one Snorlax.
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 HMNT, 26
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, 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 $\tfrac{g_2}{g_1}.$
2024 HMNT, 20
There exists a unique line tangent to the graph of $y=x^4-20x^3+24x^2-20x+25$ at two distinct points. Compute the product of the $x$-coordinates of the two tangency points.
2024 Harvard-MIT Mathematics Tournament, 6
In triangle $ABC,$ points $M$ and $N$ are the midpoints of $AB$ and $AC,$ respectively, and points $P$ and $Q$ trisect $BC.$ Given that $A, M, P, N,$ and $Q$ lie on a circle and $BC=1,$ compute the area of triangle $ABC.$
2023 Harvard-MIT Mathematics Tournament, 21
Let $x, y,$ and $N$ be real numbers, with $y$ nonzero, such that the sets $\{(x+y)^2, (x-y)^2, xy, x/y\}$ and $\{4, 12.8, 28.8, N\}$ are equal. Compute the sum of the possible values of $N.$
2023 Harvard-MIT Mathematics Tournament, 28
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, 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 $BD.$
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 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 $1.$
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.)
2024 HMNT, 15
Compute the sum of the three smallest positive integers $n$ for which $$\frac{1+2+3+\cdots+(2024n-1)+2024n}{1+2+3+\cdots+(4n-1)+4n}$$ is an integer.
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 $a+b+c.$
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
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[*]$x_1^k+x_2^k+\ldots+x_n^k=1$ for $k=1, 2, \ldots, n-1;$
[*]$x_1^n+x_2^n+\ldots+x_n^n=2;$ and
[*]$x_1^m+x_2^m+\ldots+x_n^m= 4.$
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Compute the smallest possible value of $m+n.$
2024 Harvard-MIT Mathematics Tournament, 12
Compute the number of quadruples $(a,b,c,d)$ of positive integers satisfying $$12a+21b+28c+84d=2024.$$
2025 Harvard-MIT Mathematics Tournament, 10
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.
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 $10000.$