HMMT 2024

Suppose $r, s$, and $t$ are nonzero reals such that the polynomial $x^{2}+r x+s$ has $s$ and $t$ as roots, and the polynomial $x^{2}+t x+r$ has 5 as a root. Compute $s$.

Suppose $a$ and $b$ are positive integers. Isabella and Vidur both fill up an $a \times b$ table. Isabella fills it up with numbers $1,2, \ldots, a b$, putting the numbers $1,2, \ldots, b$ in the first row, $b+1, b+2, \ldots, 2 b$ in the second row, and so on. Vidur fills it up like a multiplication table, putting $i j$ in the cell in row $i$ and column $j$. (Examples are shown for a $3 \times 4$ table below.)

1	2	3	4
5	6	7	8
9	10	11	12

Isabella's Grid

1	2	3	4
2	4	6	8
3	6	9	12
Vidur's Grid

Isabella sums up the numbers in her grid, and Vidur sums up the numbers in his grid; the difference between these two quantities is 1200 . Compute $a+b$.

Compute the sum of all two-digit positive integers $x$ such that for all three-digit (base 10) positive integers $\underline{a} \underline{b} \underline{c}$, if $\underline{a} \underline{b} \underline{c}$ is a multiple of $x$, then the three-digit (base 10) number $\underline{b} \underline{c} \underline{a}$ is also a multiple of $x$.

Let $f(x)$ be a quotient of two quadratic polynomials. Given that $f(n)=n^{3}$ for all $n \in\{1,2,3,4,5\}$, compute $f(0)$.

Compute the unique ordered pair $(x, y)$ of real numbers satisfying the system of equations

$$\frac{x}{\sqrt{x^{2}+y^{2}}}-\frac{1}{x}=7 \text { and } \frac{y}{\sqrt{x^{2}+y^{2}}}+\frac{1}{y}=4 .$$

Compute the sum of all positive integers $n$ such that $50 \leq n \leq 100$ and $2 n+3$ does not divide $2^{n!}-1$.

Let $P(n)=\left(n-1^{3}\right)\left(n-2^{3}\right) \ldots\left(n-40^{3}\right)$ for positive integers $n$. Suppose that $d$ is the largest positive integer that divides $P(n)$ for every integer $n>2023$. If $d$ is a product of $m$ (not necessarily distinct) prime numbers, compute $m$.

Let $\zeta=\cos \frac{2 \pi}{13}+i \sin \frac{2 \pi}{13}$. Suppose $a>b>c>d$ are positive integers satisfying

$$\left|\zeta^{a}+\zeta^{b}+\zeta^{c}+\zeta^{d}\right|=\sqrt{3}$$

Compute the smallest possible value of $1000 a+100 b+10 c+d$.

Suppose $a, b$, and $c$ are complex numbers satisfying

$$\begin{aligned} a^{2} & =b-c, \\ b^{2} & =c-a, \text { and } \\ c^{2} & =a-b . \end{aligned}$$

Compute all possible values of $a+b+c$.

A polynomial $f \in \mathbb{Z}[x]$ is called splitty if and only if for every prime $p$, there exist polynomials $g_{p}, h_{p} \in \mathbb{Z}[x]$ with $\operatorname{deg} g_{p}, \operatorname{deg} h_{p}<\operatorname{deg} f$ and all coefficients of $f-g_{p} h_{p}$ are divisible by $p$. Compute the sum of all positive integers $n \leq 100$ such that the polynomial $x^{4}+16 x^{2}+n$ is splitty.

Inside an equilateral triangle of side length 6 , three congruent equilateral triangles of side length $x$ with sides parallel to the original equilateral triangle are arranged so that each has a vertex on a side of the larger triangle, and a vertex on another one of the three equilateral triangles, as shown below.

A smaller equilateral triangle formed between the three congruent equilateral triangles has side length 1. Compute $x$.

Let $A B C$ be a triangle with $\angle B A C=90^{\circ}$. Let $D, E$, and $F$ be the feet of altitude, angle bisector, and median from $A$ to $B C$, respectively. If $D E=3$ and $E F=5$, compute the length of $B C$.

Let $\Omega$ and $\omega$ be circles with radii 123 and 61 , respectively, such that the center of $\Omega$ lies on $\omega$. A chord of $\Omega$ is cut by $\omega$ into three segments, whose lengths are in the ratio $1: 2: 3$ in that order. Given that this chord is not a diameter of $\Omega$, compute the length of this chord.

Let $A B C D$ be a square, and let $\ell$ be a line passing through the midpoint of segment $\overline{A B}$ that intersects segment $\overline{B C}$. Given that the distances from $A$ and $C$ to $\ell$ are 4 and 7 , respectively, compute the area of $A B C D$.

Let $A B C D$ be a convex trapezoid such that $\angle D A B=\angle A B C=90^{\circ}, D A=2, A B=3$, and $B C=8$. Let $\omega$ be a circle passing through $A$ and tangent to segment $\overline{C D}$ at point $T$. Suppose that the center of $\omega$ lies on line $B C$. Compute $C T$.

In triangle $A B C$, a circle $\omega$ with center $O$ passes through $B$ and $C$ and intersects segments $\overline{A B}$ and $\overline{A C}$ again at $B^{\prime}$ and $C^{\prime}$, respectively. Suppose that the circles with diameters $B B^{\prime}$ and $C C^{\prime}$ are externally tangent to each other at $T$. If $A B=18, A C=36$, and $A T=12$, compute $A O$.

Let $A B C$ be an acute triangle. Let $D, E$, and $F$ be the feet of altitudes from $A, B$, and $C$ to sides $\overline{B C}$, $\overline{C A}$, and $\overline{A B}$, respectively, and let $Q$ be the foot of altitude from $A$ to line $E F$. Given that $A Q=20$, $B C=15$, and $A D=24$, compute the perimeter of triangle $D E F$.

Let $A B T C D$ be a convex pentagon with area 22 such that $A B=C D$ and the circumcircles of triangles $T A B$ and $T C D$ are internally tangent. Given that $\angle A T D=90^{\circ}, \angle B T C=120^{\circ}, B T=4$, and $C T=5$, compute the area of triangle $T A D$.

Let $A B C$ be a triangle. Let $X$ be the point on side $\overline{A B}$ such that $\angle B X C=60^{\circ}$. Let $P$ be the point on segment $\overline{C X}$ such that $B P \perp A C$. Given that $A B=6, A C=7$, and $B P=4$, compute $C P$.

Suppose point $P$ is inside quadrilateral $A B C D$ such that

$$\begin{aligned} & \angle P A B=\angle P D A \\ & \angle P A D=\angle P D C, \\ & \angle P B A=\angle P C B, \text { and } \\ & \angle P B C=\angle P C D \end{aligned}$$

If $P A=4, P B=5$, and $P C=10$, compute the perimeter of $A B C D$.

Compute the number of ways to divide a $20 \times 24$ rectangle into $4 \times 5$ rectangles. (Rotations and reflections are considered distinct.)

A lame king is a chess piece that can move from a cell to any cell that shares at least one vertex with it, except for the cells in the same column as the current cell. A lame king is placed in the top-left cell of a $7 \times 7$ grid. Compute the maximum number of cells it can visit without visiting the same cell twice (including its starting cell).

Compute the number of ways there are to assemble 2 red unit cubes and 25 white unit cubes into a $3 \times 3 \times 3$ cube such that red is visible on exactly 4 faces of the larger cube. (Rotations and reflections are considered distinct.)

Sally the snail sits on the $3 \times 24$ lattice of points $(i, j)$ for all $1 \leq i \leq 3$ and $1 \leq j \leq 24$. She wants to visit every point in the lattice exactly once. In a move, Sally can move to a point in the lattice exactly one unit away. Given that Sally starts at $(2,1)$, compute the number of possible paths Sally can take.

The country of HMMTLand has 8 cities. Its government decides to construct several two-way roads between pairs of distinct cities. After they finish construction, it turns out that each city can reach exactly 3 other cities via a single road, and from any pair of distinct cities, either exactly 0 or 2 other cities can be reached from both cities by a single road. Compute the number of ways HMMTLand could have constructed the roads.

In each cell of a $4 \times 4$ grid, one of the two diagonals is drawn uniformly at random. Compute the probability that the resulting 32 triangular regions can be colored red and blue so that any two regions sharing an edge have different colors.

There is a grid of height 2 stretching infinitely in one direction. Between any two edge-adjacent cells of the grid, there is a door that is locked with probability $\frac{1}{2}$ independent of all other doors. Philip starts in a corner of the grid (in the starred cell). Compute the expected number of cells that Philip can reach, assuming he can only travel between cells if the door between them is unlocked.

Rishabh has 2024 pairs of socks in a drawer. He draws socks from the drawer uniformly at random, without replacement, until he has drawn a pair of identical socks. Compute the expected number of unpaired socks he has drawn when he stops.

Compute the number of triples $(f, g, h)$ of permutations on $\{1,2,3,4,5\}$ such that

$$\begin{aligned} & f(g(h(x)))=h(g(f(x)))=g(x), \\ & g(h(f(x)))=f(h(g(x)))=h(x), \text { and } \\ & h(f(g(x)))=g(f(h(x)))=f(x) \end{aligned}$$

for all $x \in\{1,2,3,4,5\}$.

A peacock is a ten-digit positive integer that uses each digit exactly once. Compute the number of peacocks that are exactly twice another peacock.

Let $a_{1}, a_{2}, a_{3}, \ldots, a_{100}$ be integers such that

$$\frac{a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+\cdots+a_{100}^{2}}{a_{1}+a_{2}+a_{3}+\cdots+a_{100}}=100 .$$

Determine, with proof, the maximum possible value of $a_{1}$.

Nine distinct positive integers summing to 74 are put into a $3 \times 3$ grid. Simultaneously, the number in each cell is replaced with the sum of the numbers in its adjacent cells. (Two cells are adjacent if they share an edge.) After this, exactly four of the numbers in the grid are 23. Determine, with proof, all possible numbers that could have been originally in the center of the grid.

Let $A B C$ be a scalene triangle and $M$ be the midpoint of $B C$. Let $X$ be the point such that $C X \| A B$ and $\angle A M X=90^{\circ}$. Prove that $A M$ bisects $\angle B A X$.

Each lattice point with nonnegative coordinates is labeled with a nonnegative integer in such a way that the point $(0,0)$ is labeled by 0 , and for every $x, y \geq 0$, the set of numbers labeled on the points $(x, y),(x, y+1)$, and $(x+1, y)$ is $\{n, n+1, n+2\}$ for some nonnegative integer $n$. Determine, with proof, all possible labels for the point $(2000,2024)$.

Determine, with proof, whether there exist positive integers $x$ and $y$ such that $x+y, x^{2}+y^{2}$, and $x^{3}+y^{3}$ are all perfect squares.

Let $\mathbb{Q}$ be the set of rational numbers. Given a rational number $a \neq 0$, find, with proof, all functions $f: \mathbb{Q} \rightarrow \mathbb{Q}$ satisfying the equation

$$f(f(x)+a y)=a f(y)+x$$

for all $x, y \in \mathbb{Q}$.

Let $A B C D E F$ be a regular hexagon with $P$ as a point in its interior. Prove that of the three values $\tan \angle A P D, \tan \angle B P E$, and $\tan \angle C P F$, two of them sum to the third one.

Let $P$ be a point in the interior of quadrilateral $A B C D$ such that the circumcircles of triangles $P D A, P A B$, and $P B C$ are pairwise distinct but congruent. Let the lines $A D$ and $B C$ meet at $X$. If $O$ is the circumcenter of triangle $X C D$, prove that $O P \perp A B$.

On each cell of a $200 \times 200$ grid, we place a car, which faces in one of the four cardinal directions. In a move, one chooses a car that does not have a car immediately in front of it, and slides it one cell forward. If a move would cause a car to exit the grid, the car is removed instead. The cars are placed so that there exists a sequence of moves that eventually removes all the cars from the grid. Across all such starting configurations, determine the maximum possible number of moves to do so.

Across all polynomials $P$ such that $P(n)$ is an integer for all integers $n$, determine, with proof, all possible values of $P(i)$, where $i^{2}=-1$.

Compute the sum of all integers $n$ such that $n^{2}-3000$ is a perfect square.

Jerry and Neil have a 3 -sided die that rolls the numbers 1,2 , and 3 , each with probability $\frac{1}{3}$. Jerry rolls first, then Neil rolls the die repeatedly until his number is at least as large as Jerry's. Compute the probability that Neil's final number is 3 .

Compute the number of even positive integers $n \leq 2024$ such that $1,2, \ldots, n$ can be split into $\frac{n}{2}$ pairs, and the sum of the numbers in each pair is a multiple of 3.

Equilateral triangles $A B F$ and $B C G$ are constructed outside regular pentagon $A B C D E$. Compute $\angle F E G$.

Let $a, b$, and $c$ be real numbers such that

$$\begin{aligned} a+b+c & =100 \\ a b+b c+c a & =20, \text { and } \\ (a+b)(a+c) & =24 \end{aligned}$$

Compute all possible values of $b c$.

In triangle $A B C$, points $M$ and $N$ are the midpoints of $A B$ and $A C$, respectively, and points $P$ and $Q$ trisect $B C$. Given that $A, M, N, P$, and $Q$ lie on a circle and $B C=1$, compute the area of triangle $A B C$.

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 $a+b+c$.

Three points, $A, B$, and $C$, are selected independently and uniformly at random from the interior of a unit square. Compute the expected value of $\angle A B C$.

Compute the sum of all positive integers $n$ such that $n^{2}-3000$ is a perfect square.

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.

Let $A B C D$ be a rectangle such that $A B=20$ and $A D=24$. Point $P$ lies inside $A B C D$ such that triangles $P A C$ and $P B D$ have areas 20 and 24, respectively. Compute all possible areas of triangle $P A B$.

Compute the number of quadruples $(a, b, c, d)$ of positive integers satisfying

$$12 a+21 b+28 c+84 d=2024$$

Mark has a cursed six-sided die that never rolls the same number twice in a row, and all other outcomes are equally likely. Compute the expected number of rolls it takes for Mark to roll every number at least once.

Compute the smallest positive integer such that, no matter how you rearrange its digits (in base ten), the resulting number is a multiple of 63 .

Let $a \star b=a b-2$. Compute the remainder when $(((579 \star 569) \star 559) \star \cdots \star 19) \star 9$ is divided by 100 . Proposed by: Rishabh Das

Let $A B C$ be an acute isosceles triangle with orthocenter $H$. Let $M$ and $N$ be the midpoints of sides $\overline{A B}$ and $\overline{A C}$, respectively. The circumcircle of triangle $M H N$ intersects line $B C$ at two points $X$ and $Y$. Given $X Y=A B=A C=2$, compute $B C^{2}$.

The numbers $1,2, \ldots, 20$ are put into a hat. Claire draws two numbers from the hat uniformly at random, $a<b$, and then puts them back into the hat. Then, William draws two numbers from the hat uniformly at random, $c<d$. Let $N$ denote the number of integers $n$ that satisfy exactly one of $a \leq n \leq b$ and $c \leq n \leq d$. Compute the probability $N$ is even.

An ordered pair $(a, b)$ of positive integers is called spicy if $\operatorname{gcd}(a+b, a b+1)=1$. Compute the probability that both $(99, n)$ and $(101, n)$ are spicy when $n$ is chosen from $\{1,2, \ldots, 2024$ ! $\}$ uniformly at random.

Let $A_{1} A_{2} \ldots A_{19}$ be a regular nonadecagon. Lines $A_{1} A_{5}$ and $A_{3} A_{4}$ meet at $X$. Compute $\angle A_{7} X A_{5}$. Proposed by: Nithid Anchaleenukoon

Compute $\sqrt[4]{5508^{3}+5625^{3}+5742^{3}}$, given that it is an integer.

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 $(6,8)$.

Let $x<y$ be positive real numbers such that

$$\sqrt{x}+\sqrt{y}=4 \text { and } \sqrt{x+2}+\sqrt{y+2}=5$$

Compute $x$.

Let $\ell$ and $m$ be two non-coplanar lines in space, and let $P_{1}$ be a point on $\ell$. Let $P_{2}$ be the point on $m$ closest to $P_{1}, P_{3}$ be the point on $\ell$ closest to $P_{2}, P_{4}$ be the point on $m$ closest to $P_{3}$, and $P_{5}$ be the point on $\ell$ closest to $P_{4}$. Given that $P_{1} P_{2}=5, P_{2} P_{3}=3$, and $P_{3} P_{4}=2$, compute $P_{4} P_{5}$.

A circle is tangent to both branches of the hyperbola $x^{2}-20 y^{2}=24$ as well as the $x$-axis. Compute the area of this circle.

Point $P$ is inside a square $A B C D$ such that $\angle A P B=135^{\circ}, P C=12$, and $P D=15$. Compute the area of this square.

It can be shown that there exists a unique polynomial $P$ in two variables such that for all positive integers $m$ and $n$,

$$P(m, n)=\sum_{i=1}^{m} \sum_{j=1}^{n}(i+j)^{7} .$$

Compute $P(3,-3)$.

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.

Given that the 32-digit integer

$$64312311692944269609355712372657$$

is the product of 6 consecutive primes, compute the sum of these 6 primes.

For each prime $p$, a polynomial $P(x)$ with rational coefficients is called $p$-good if and only if there exist three integers $a, b$, and $c$ such that $0 \leq a<b<c<\frac{p}{3}$ 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 $x^{3}+10 x^{2}+r x+s$ is $p$-good for infinitely many primes $p$.

Let $A B C$ be an equilateral triangle with side length 1 . Points $D, E, F$ lie inside triangle $A B C$ such that $A, E, F$ are collinear, $B, F, D$ are collinear, $C, D, E$ are collinear, and triangle $D E F$ is equilateral. Suppose that there exists a unique equilateral triangle $X Y Z$ with $X$ on side $\overline{B C}, Y$ on side $\overline{A B}$, and $Z$ on side $\overline{A C}$ such that $D$ lies on side $\overline{X Z}, E$ lies on side $\overline{Y Z}$, and $F$ lies on side $\overline{X Y}$. Compute $A Z$.

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 $p$.

Over all pairs of complex numbers $(x, y)$ satisfying the equations

$$x+2 y^{2}=x^{4} \text { and } y+2 x^{2}=y^{4}$$

compute the minimum possible real part of $x$.

Let $p$ denote the proportion of teams, out of all participating teams, who submitted a negative response to problem 5 of the Team round (e.g. "there are no such integers"). Estimate $P=\lfloor 10000 p\rfloor$. An estimate of $E$ earns $\max (0,\lfloor 20-|P-E| / 20\rfloor)$ points. If you have forgotten, problem 5 of the Team round was the following: "Determine, with proof, whether there exist positive integers $x$ and $y$ such that $x+y, x^{2}+y^{2}$, and $x^{3}+y^{3}$ are all perfect squares."

Estimate the number of positive integers $n \leq 10^{6}$ such that $n^{2}+1$ has a prime factor greater than $n$. Submit a positive integer $E$. If the correct answer is $A$, you will receive max $\left(0,\left\lfloor 20 \cdot \min \left(\frac{E}{A}, \frac{10^{6}-E}{10^{6}-A}\right)^{5}+0.5\right\rfloor\right)$ points.

Barry picks infinitely many points inside a unit circle, each independently and uniformly at random, $P_{1}, P_{2}, \ldots$ Compute the expected value of $N$, where $N$ is the smallest integer such that $P_{N+1}$ is inside the convex hull formed by the points $P_{1}, P_{2}, \ldots, P_{N}$.

Submit a positive real number $E$. If the correct answer is $A$, you will receive $\lfloor 100 \cdot \max (0.2099-|E-A|, 0)\rfloor$ points.

Let $A B C$ be a triangle. The following diagram contains points $P_{1}, P_{2}, \ldots, P_{7}$, which are the following triangle centers of triangle $A B C$ in some order:

• the incenter $I$;
• the circumcenter $O$;
• the orthocenter $H$;
• the symmedian point $L$, which is the intersections of the reflections of $B$-median and $C$-median across angle bisectors of $\angle A B C$ and $\angle A C B$, respectively;
• the Gergonne point $G$, which is the intersection of lines from $B$ and $C$ to the tangency points of the incircle with $\overline{A C}$ and $\overline{A B}$, respectively;
• the Nagel point $N$, which is the intersection of line from $B$ to the tangency point between $B$ excircle and $\overline{A C}$, and line from $C$ to the tangency point between $C$-excircle and $\overline{A B}$; and
• the Kosnita point $K$, which is the intersection of lines from $B$ and $C$ to the circumcenters of triangles $A O C$ and $A O B$, respectively.

Note that the triangle $A B C$ is not shown. Compute which triangle centers $\{I, O, H, L, G, N, K\}$ corresponds to $P_{k}$ for $k \in\{1,2,3,4,5,6,7\}$. Your answer should be a seven-character string containing $I, O, H, L, G, N, K$, or $X$ for blank. For instance, if you think $P_{2}=H$ and $P_{6}=L$, you would answer $X H X X X L X$. If you attempt to identify $n>0$ points and get them all correct, then you will receive $\left\lceil(n-1)^{5 / 3}\right\rceil$ points. Otherwise, you will receive 0 points.