HMMT 2025

Compute the sum of the positive divisors (including 1) of 9! that have units digit 1.

Mark writes the expression $\sqrt{abcd}$ on the board, where $abcd$ is a four-digit number and $a \neq 0$ . Derek, a toddler, decides to move the $a$ , changing Mark's expression to $a\sqrt{bc}$ . Surprisingly, these two expressions are equal. Compute the only possible four-digit number $abcd$ .

Given that $x$ , $y$ , and $z$ are positive real numbers such that

$$x^{\log_{2}(yz)} = 2^{8} \cdot 3^{4}, y^{\log_{2}(zx)} = 2^{9} \cdot 3^{6}, \text{and} z^{\log_{2}(xy)} = 2^{5} \cdot 3^{10},$$

compute the smallest possible value of $xyz$ .

Let $\lfloor z\rfloor$ denote the greatest integer less than or equal to $z$ . Compute

$$\sum_{j = -1000}^{1000} \left\lfloor \frac{2025}{j + 0.5} \right\rfloor .$$

Let $\mathcal{S}$ be the set of all nonconstant monic polynomials $P$ with integer coefficients satisfying $P\left(\sqrt{3} + \sqrt{2}\right) = P\left(\sqrt{3} - \sqrt{2}\right)$ . If $Q$ is an element of $\mathcal{S}$ with minimal degree, compute the only possible value of $Q(10) - Q(0)$ .

Let $r$ be the remainder when $2017^{2025!} - 1$ is divided by 2025!. Compute $\frac{r}{2025!}$ . (Note that 2017 is prime.)

There exists a unique triple $(a,b,c)$ of positive real numbers that satisfies the equations

$$2(a^{2} + 1) = 3(b^{2} + 1) = 4(c^{2} + 1) \mathrm{and} ab + bc + ca = 1.$$

Compute $a + b + c$ .

Define $\operatorname {sgn}(x)$ to be 1 when $x$ is positive, $-1$ when $x$ is negative, and 0 when $x$ is 0. Compute

$$\sum_{n = 1}^{\infty}\frac{\operatorname{sgn}(\sin(2^{n}))}{2^{n}}.$$

(The arguments to sin are in radians.)

Let $f$ be the unique polynomial of degree at most 2026 such that for all $n\in \{1,2,3,\ldots ,2027\}$

Suppose that $\frac{a}{b}$ is the coefficient of $x^{2025}$ in $f$ , where $a$ and $b$ are integers such that $\gcd (a,b) = 1$ . Compute the unique integer $r$ between 0 and 2026 (inclusive) such that $a - rb$ is divisible by 2027. (Note that 2027 is prime.)

Let $a$ , $b$ , and $c$ be pairwise distinct complex numbers such that

$$a^{2} = b + 6, b^{2} = c + 6, \mathrm{and} c^{2} = a + 6.$$

Compute the two possible values of $a + b + c$

Equilateral triangles $\triangle ABC$ and $\triangle DEF$ are drawn such that points $B$ , $E$ , $F$ , and $C$ lie on a line in this order, and point $D$ lies inside triangle $\triangle ABC$ . If $BE = 14$ , $EF = 15$ , and $FC = 16$ , compute $AD$ .

In a two-dimensional cave with a parallel floor and ceiling, two stalactites of lengths 16 and 36 hang perpendicularly from the ceiling, while two stalagmites of heights 25 and 49 grow perpendicularly from the ground. If the tips of these four structures form the vertices of a square in some order, compute the height of the cave.

Point $P$ lies inside square $ABCD$ such that the areas of $\triangle PAB$ , $\triangle PBC$ , $\triangle PCD$ , and $\triangle PDA$ are 1, 2, 3, and 4, in some order. Compute $PA \cdot PB \cdot PC \cdot PD$ .

A semicircle is inscribed in another semicircle if the smaller semicircle's diameter is a chord of the larger semicircle, and the smaller semicircle's arc is tangent to the diameter of the larger semicircle.

Semicircle $S_{1}$ is inscribed in a semicircle $S_{2}$ , which is inscribed in another semicircle $S_{3}$ . The radii of $S_{1}$ and $S_{3}$ are 1 and 10, respectively, and the diameters of $S_{1}$ and $S_{3}$ are parallel. The endpoints of the diameter of $S_{3}$ are $A$ and $B$ , and $S_{2}$ 's arc is tangent to $AB$ at $C$ . Compute $AC \cdot CB$ .

Let $\triangle ABC$ be an equilateral triangle with side length 6. Let $P$ be a point inside triangle $\triangle ABC$ such that $\angle BPC = 120^{\circ}$ . The circle with diameter $\overline{AP}$ meets the circumcircle of $\triangle ABC$ again at $X \neq A$ . Given that $AX = 5$ , compute $XP$ .

Trapezoid $A B C D$ , with $A B\parallel C D$ , has side lengths $A B = 11$ , $B C = 8$ , $C D = 19$ , and $D A = 4$ . Compute the area of the convex quadrilateral whose vertices are the circumcenters of $\triangle A B C$ , $\triangle B C D$ , $\triangle C D A$ , and $\triangle D A B$ .

Point $P$ is inside triangle $\triangle A B C$ such that $\angle A B P = \angle A C P$ . Given that $A B = 6$ , $A C = 8$ , $B C = 7$ , and $\frac{B P}{P C} = \frac{1}{2}$ , compute $\frac{|B P C|}{|A B C|}$ .

(Here, $[X Y Z]$ denotes the area of $\triangle X Y Z$ ).

Let $A B C D$ be an isosceles trapezoid such that $C D > A B = 4$ . Let $E$ be a point on line $C D$ such that $D E = 2$ and $D$ lies between $E$ and $C$ . Let $M$ be the midpoint of $\overline{{A E}}$ . Given that points $A$ , $B$ , $C$ , $D$ , and $M$ lie on a circle with radius 5, compute $M D$ .

Let $A B C D$ be a rectangle with $B C = 24$ . Point $X$ lies inside the rectangle such that $\angle A X B = 90^{\circ}$ . Given that triangles $\triangle A X D$ and $\triangle B X C$ are both acute and have circumradii 13 and 15, respectively, compute $A B$ .

A plane $\mathcal{P}$ intersects a rectangular prism at a hexagon which has side lengths 45, 66, 63, 55, 54, and 77, in that order. Compute the distance from the center of the rectangular prism to $\mathcal{P}$ .

Compute the number of ways to arrange the numbers 1, 2, 3, 4, 5, 6, and 7 around a circle such that the product of every pair of adjacent numbers on the circle is at most 20. (Rotations and reflections count as different arrangements.)

Kelvin the frog is on the bottom-left lily pad of a $3\times 3$ grid of lily pads, and his home is at the topright lily pad. He can only jump between two lily pads which are horizontally or vertically adjacent. Compute the number of ways to remove 4 of the lily pads so that the bottom-left and top-right lily pads both remain, but Kelvin cannot get home.

Ben has 16 balls labeled 1, 2, 3, ..., 16, as well as 4 indistinguishable boxes. Two balls are neighbors if their labels differ by 1. Compute the number of ways for him to put 4 balls in each box such that each ball is in the same box as at least one of its neighbors. (The order in which the balls are placed does not matter.)

Sophie is at $(0,0)$ on a coordinate grid and would like to get to $(3,3)$ . If Sophie is at $(x,y)$ , in a single step she can move to one of $(x + 1,y)$ , $(x,y + 1)$ , $(x - 1,y + 1)$ , or $(x + 1,y - 1)$ . She cannot revisit any points along her path, and neither her $x$ - coordinate nor her $y$ - coordinate can ever be less than 0 or greater than 3. Compute the number of ways for Sophie to reach $(3,3)$ .

In an $11 \times 11$ grid of cells, each pair of edge-adjacent cells is connected by a door. Karthik wants to walk a path in this grid. He can start in any cell, but he must end in the same cell he started in, and he cannot go through any door more than once (not even in opposite directions). Compute the maximum number of doors he can go through in such a path.

Compute the number of ways to pick two rectangles in a $5 \times 5$ grid of squares such that the edges of the rectangles lie on the lines of the grid and the rectangles do not overlap at their interiors, edges, or vertices. The order in which the rectangles are chosen does not matter.

Compute the number of ways to arrange 3 copies of each of the 26 lowercase letters of the English alphabet such that for any two distinct letters $x_{1}$ and $x_{2}$ , the number of $x_{2}$ 's between the first and second occurrences of $x_{1}$ equals the number of $x_{2}$ 's between the second and third occurrences of $x_{1}$ .

Albert writes 2025 numbers $a_{1}$ , ..., $a_{2025}$ in a circle on a blackboard. Initially, each of the numbers is uniformly and independently sampled at random from the interval $[0,1]$ . Then, each second, he simultaneously replaces $a_{i}$ with $\max (a_{i - 1}, a_{i}, a_{i + 1})$ for all $i = 1, 2, \ldots , 2025$ (where $a_{0} = a_{2025}$ and $a_{2026} = a_{1}$ ). Compute the expected value of the number of distinct values remaining after 100 seconds.

Two points are selected independently and uniformly at random inside a regular hexagon. Compute the probability that a line passing through both of the points intersects a pair of opposite edges of the hexagon.

The circumference of a circle is divided into 45 arcs, each of length 1. Initially, there are 15 snakes, each of length 1, occupying every third arc. Every second, each snake independently moves either one arc left or one arc right, each with probability $\frac{1}{2}$ . If two snakes ever touch, they merge to form a single snake occupying the arcs of both of the previous snakes, and the merged snake moves as one snake. Compute the expected number of seconds until there is only one snake left.

Let $a$ , $b$ , and $c$ be pairwise distinct positive integers such that $\frac{1}{a}$ , $\frac{1}{b}$ , $\frac{1}{c}$ is an increasing arithmetic sequence in that order. Prove that $\gcd (a,b) > 1$ .

A polyomino is a connected figure constructed by joining one or more unit squares edge-to-edge. Determine, with proof, the number of non-congruent polyominoes with no holes, perimeter 180, and area 2024.

Let $\omega_{1}$ and $\omega_{2}$ be two circles intersecting at distinct points $A$ and $B$ . Point $X$ varies along $\omega_{1}$ , and point $Y$ on $\omega_{2}$ is chosen such that $AB$ bisects the angle $\angle XAY$ . Prove that as $X$ varies along $\omega_{1}$ , the circumcenter of $\triangle AXY$ (if it exists) varies along a fixed line.

Jerry places at most one rook in each cell of a $2025 \times 2025$ grid of cells. A rook attacks another rook if the two rooks are in the same row or column and there are no other rooks between them.

Determine, with proof, the maximum number of rooks Jerry can place on the grid such that no rook attacks 4 other rooks.

Let $\triangle ABC$ be an acute triangle with orthocenter $H$ . Points $E$ and $F$ are on segments $\overline{AC}$ and $\overline{AB}$ , respectively, such that $\angle EHF = 90^{\circ}$ . Let $X$ be the foot of the altitude from $H$ to $\overline{EF}$ . Prove that $\angle BXC = 90^{\circ}$ .

Complex numbers $\omega_{1}$ , ..., $\omega_{n}$ each have magnitude 1. Let $z$ be a complex number distinct from $\omega_{1}$ , ..., $\omega_{n}$ such that

$$\frac{z + \omega_{1}}{z - \omega_{1}} +\dots +\frac{z + \omega_{n}}{z - \omega_{n}} = 0.$$

Prove that $|z| = 1$

Determine, with proof, whether a square can be dissected into finitely many (not necessarily congruent) triangles, each of which has interior angles $30^{\circ}$ , $75^{\circ}$ , and $75^{\circ}$ .

Let $\triangle ABC$ be a triangle with incenter $I$ . The incircle of triangle $\triangle ABC$ touches $\overline{BC}$ at $D$ . Let $M$ be the midpoint of $\overline{BC}$ , and let line $AI$ meet the circumcircle of triangle $\triangle ABC$ again at $L \neq A$ . Let $\omega$ be the circle centered at $L$ tangent to $AB$ and $AC$ . If $\omega$ intersects segment $\overline{AD}$ at point $P$ , prove that $\angle IPM = 90^{\circ}$ .

Let $\mathbb{Z}$ be the set of integers. Determine, with proof, all primes $p$ for which there exists a function $f\colon \mathbb{Z}\to \mathbb{Z}$ such that for any integer $x$ ,

$f(x + p) = f(x)$ and $p$ divides $f(x + f(x)) - x$ .

Determine, with proof, all possible values of $\gcd (a^{2} + b^{2} + c^{2}, abc)$ across all triples of positive integers $(a, b, c)$ .

Call a 9-digit number a cassowary if it uses each of the digits 1 through 9 exactly once. Compute the number of cassowaries that are prime.

Compute

$$\frac{20 + \frac{1}{25 - \frac{1}{20}}}{25 + \frac{1}{20 - \frac{1}{25}}}.$$

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 outcomes of all the dice he rolls is even.

Let $\triangle ABC$ be an equilateral triangle with side length 4. Across all points $P$ inside triangle $\triangle ABC$ satisfying $[PAB] + [PAC] = [PBC]$ , compute the minimum possible length of $PA$ .

(Here, $[XYZ]$ denotes the area of triangle $\triangle XYZ$ .)

Compute the largest possible radius of a circle contained in the region defined by $|x + |y||\leq 1$ in the coordinate plane.

Let $\triangle ABC$ be an equilateral triangle. Point $D$ lies 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 $EF$ .

The number

$$\frac{9^{9} - 8^{8}}{1001}$$

is an integer. Compute the sum of its prime factors.

A checkerboard is a rectangular grid of cells colored black and white such that the top-left corner is black and no two cells of the same color share an edge. Two checkerboards are distinct if and only if they have a different number of rows or columns. For example, a $20 \times 25$ checkerboard and a $25 \times 20$ checkerboard are considered distinct.

Compute the number of distinct checkerboards that have exactly 41 black cells.

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 $FABC$ .

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.

Let $f(n) = n^{2} + 100$ . Compute the remainder when $f(f(\dots f(f(1))\dots))$ is divided by $10^{4}$ .

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 $x$ .

A number is upwards 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.

A parallelogram $P$ can be folded over a straight line so that the resulting shape is a regular pentagon with side length 1. Compute the perimeter of $P$ .

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 $DE$ .

The Cantor set is defined as the set of real numbers $x$ such that $0 \leq x < 1$ and the digit 1 does not appear in the base-3 expansion of $x$ . Two numbers are uniformly and independently selected at random from the Cantor set. Compute the expected value of their absolute difference.

(Formally, one can pick a number $x$ uniformly at random from the Cantor set by first picking a real number $y$ uniformly at random from the interval $[0,1)$ , writing it out in binary, reading its digits as if they were in base- 3, and setting $x$ to 2 times the result.)

Let $f$ be a quadratic polynomial with real coefficients, and let $g_{1}$ , $g_{2}$ , $g_{3}$ , ... 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 $\frac{g_{2}}{g_{1}}$ .

Let $f:\{1,2,3,\ldots ,9\} \to \{1,2,3,\ldots ,9\}$ be a permutation chosen uniformly at random from the 9! possible permutations. Compute the expected value of $\underbrace{f(f(\cdots f(f(1))\cdots))}_{2025 f s}$ .

A subset $S$ of $\{1,2,3,\ldots ,2025\}$ is called balanced 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 nonempty balanced subsets.

Compute the 100th smallest positive multiple of 7 whose digits in base 10 are all strictly less than 3.

Compute the unique 5-digit positive integer $abcde$ such that $a \neq 0$ , $c \neq 0$ , and

$$abcde = (ab + cde)^2.$$

Let $a$ , $b$ , and $c$ be real numbers such that $a^2(b + c) = 1$ , $b^2(c + a) = 2$ , and $c^2(a + b) = 5$ . Given that there are three possible values for $abc$ , compute the minimum possible value of $abc$ .

Regular hexagon $ABCDEF$ has side length 2. Circle $\omega$ lies inside the hexagon and is tangent to segments $\overline{AB}$ and $\overline{AF}$ . There exist two perpendicular lines tangent to $\omega$ that pass through $C$ and $E$ , respectively. Given that these two lines do not intersect on line $AD$ , compute the radius of $\omega$ .

For any integer $x$ , let

$$f(x) = 100! \left(1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \dots + \frac{x^{100}}{100!}\right).$$

A positive integer $a$ is chosen such that $f(a) - 20$ is divisible by $101^{2}$ . Compute the remainder when $f(a + 101)$ is divided by $101^{2}$ .

Let $A B C D$ be a trapezoid such that $A B \parallel C D$ , $A D = 13$ , $B C = 15$ , $A B = 20$ , and $C D = 34$ . Point $X$ lies inside the trapezoid such that $\angle X A B = 2\angle X B A$ and $\angle X D C = 2\angle X C D$ . Compute $X D - X A$ .

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.

Compute the number of ordered pairs $(m, n)$ of odd positive integers both less than 80 such that

$$\gcd (4^{m} + 2^{m} + 1,4^{n} + 2^{n} + 1) > 1.$$

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

$$\frac{f(1)}{1\cdot2} +\frac{f(2)}{2\cdot3} +\frac{f(3)}{3\cdot4} +\dots +\frac{f(31)}{31\cdot32}.$$

(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 $z$ .)

Points $A$ and $B$ lie on circle $\omega$ with center $O$ . Let $X$ be a point inside $\omega$ . Suppose that $XO = 2\sqrt{2}$ , $XA = 1$ , $XB = 3$ , and $\angle AXB = 90^{\circ}$ . Points $Y$ and $Z$ are on $\omega$ such that $Y \neq A$ and triangles $\triangle AXB$ and $\triangle YXZ$ are similar with the same orientation. Compute $XY$ .

Let $a$ , $b$ , and $c$ be real numbers satisfying the system of equations

$$a\sqrt{1 + b^{2}} +b\sqrt{1 + a^{2}} = \frac{3}{4},$$ $$b\sqrt{1 + c^{2}} +c\sqrt{1 + b^{2}} = \frac{5}{12},\mathrm{and}$$ $$c\sqrt{1 + a^{2}} +a\sqrt{1 + c^{2}} = \frac{21}{20}.$$

Compute $a$ .

There exists a unique circle that is both tangent to the parabola $y = x^{2}$ at two points and tangent to the curve $x = \sqrt{\frac{y^{3}}{1 - y}}$ . Compute the radius of this circle.

In the coordinate plane, a closed lattice loop of length $2n$ is a sequence of lattice points $P_{0}, P_{1}, P_{2}, \ldots , P_{2n}$ 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 $0 \leq i \leq 2025$ and $P_{i}$ lies on $\ell$ .

(A lattice point is a point with integer coordinates.)

Estimate the total number of pages that teams submitted to the Team Round this year. (All pages associated to at least one problem number count as submitted pages, even blank cover sheets for a problem.)

Submit a positive integer $E$ . If the correct answer is $A$ , you will receive $\max \left(0, \left[20 \left(1 - \left(\frac{|E - A|}{100}\right)^{2 / 3}\right)\right]\right)$ points.

On the perimeter of a unit circle, 12 points are chosen uniformly and independently at random. Estimate the expected value of the area of the convex 12-gon formed by these points.

Submit a positive number $E$ written in decimal. If the correct answer is $A$ , you will receive round $\left(20e^{-15|E - A|}\right)$ points.

Call an 8-digit number a flamingo if it uses each of the digits 2 through 9 exactly once. Estimate the number of flamingos that are prime.

Submit a positive integer $E$ . If the correct answer is $A$ , you will receive round $\left(20 \cdot \min \left(\frac{A}{E}, \frac{E}{A}\right)^{21}\right)$ points.

Ethan initially writes some numbers on a blackboard, each of which is either a 3 or a 5. He then repeatedly picks two numbers and replaces them with their sum, difference, product, or quotient (if the divisor is nonzero). Let $f(n)$ denote the minimum number of numbers Ethan must initially write for him to be able to eventually write the number $n$ . For example, $f(2025) \leq 6$ because Ethan could start with 3, 3, 3, 3, 5, and 5 on the board, then repeatedly multiply two numbers at a time to eventually get 2025.

Submit a comma- separated ordered 8- tuple of integers corresponding to the values of $f(164)$ , $f(187)$ , $f(191)$ , $f(224)$ , $f(255)$ , $f(286)$ , $f(374)$ , and $f(479)$ , in that order, or an X for any value you wish to leave blank. For instance, if you think $f(164) = 9$ and $f(224) = 8$ , you should submit "9, X, X, 8, X,

X, X, X". You will earn $\left|0.6^{W} \cdot \frac{(C + 1)^{2}}{4}\right|$ points, where $C$ is the number of correct answers you submit and $W$ is the number of incorrect (non- blank) answers.