HMMT November 2023

Four people are playing rock-paper-scissors. They each play one of the three options (rock, paper, or scissors) independently at random, with equal probability of each choice. Compute the probability that someone beats everyone else. (In rock-paper-scissors, a player that plays rock beats a player that plays scissors, a player that plays paper beats a player that plays rock, and a player that plays scissors beats a player that plays paper.)

A regular $n$-gon $P_{1} P_{2} \ldots P_{n}$ satisfies $\angle P_{1} P_{7} P_{8}=178^{\circ}$. Compute $n$.

Compute the number of positive four-digit multiples of 11 whose sum of digits (in base ten) is divisible by 11 .

Suppose that $a$ and $b$ are real numbers such that the line $y=a x+b$ intersects the graph of $y=x^{2}$ at two distinct points $A$ and $B$. If the coordinates of the midpoint of $A B$ are $(5,101)$, compute $a+b$.

On an $8 \times 8$ chessboard, 6 black rooks and $k$ white rooks are placed on different cells so that each rook only attacks rooks of the opposite color. Compute the maximum possible value of $k$. (Two rooks attack each other if they are in the same row or column and no rooks are between them.)

Let $A B C D$ be a square of side length 5. A circle passing through $A$ is tangent to segment $C D$ at $T$ and meets $A B$ and $A D$ again at $X \neq A$ and $Y \neq A$, respectively. Given that $X Y=6$, compute $A T$.

Compute all ordered triples $(x, y, z)$ of real numbers satisfying the following system of equations:

$$\begin{array}{r} x y+z=40 \\ x z+y=51 \\ x+y+z=19 . \end{array}$$

Mark writes the expression $\sqrt{d}$ for each positive divisor $d$ of 8 ! on the board. Seeing that these expressions might not be worth points on HMMT, Rishabh simplifies each expression to the form $a \sqrt{b}$, where $a$ and $b$ are integers such that $b$ is not divisible by the square of a prime number. (For example, $\sqrt{20}, \sqrt{16}$, and $\sqrt{6}$ simplify to $2 \sqrt{5}, 4 \sqrt{1}$, and $1 \sqrt{6}$, respectively.) Compute the sum of $a+b$ across all expressions that Rishabh writes.

An entry in a grid is called a saddle point if it is the largest number in its row and the smallest number in its column. Suppose that each cell in a $3 \times 3$ grid is filled with a real number, each chosen independently and uniformly at random from the interval $[0,1]$. Compute the probability that this grid has at least one saddle point.

Let $A B C D$ be a convex trapezoid such that $\angle A B C=\angle B C D=90^{\circ}, A B=3, B C=6$, and $C D=12$. Among all points $X$ inside the trapezoid satisfying $\angle X B C=\angle X D A$, compute the minimum possible value of $C X$.

Tyler has an infinite geometric series with sum 10 . He increases the first term of his sequence by 4 and swiftly changes the subsequent terms so that the common ratio remains the same, creating a new geometric series with sum 15. Compute the common ratio of Tyler's series.

Suppose rectangle $F O L K$ and square $L O R E$ are on the plane such that $R L=12$ and $R K=11$. Compute the product of all possible areas of triangle $R K L$.

There are 17 people at a party, and each has a reputation that is either $1,2,3,4$, or 5 . Some of them split into pairs under the condition that within each pair, the two people's reputations differ by at most 1. Compute the largest value of $k$ such that no matter what the reputations of these people are, they are able to form $k$ pairs.

Let $L O V E R$ be a convex pentagon such that $L O V E$ is a rectangle. Given that $O V=20$ and $L O=$ $V E=R E=R L=23$, compute the radius of the circle passing through $R, O$, and $V$.

Compute the unique positive integer $n$ such that $\frac{n^{3}-1989}{n}$ is a perfect square.

A function $g$ is ever more than a function $h$ if, for all real numbers $x$, we have $g(x) \geq h(x)$. Consider all quadratic functions $f(x)$ such that $f(1)=16$ and $f(x)$ is ever more than both $(x+3)^{2}$ and $x^{2}+9$. Across all such quadratic functions $f$, compute the minimum value of $f(0)$.

Betty has a $3 \times 4$ grid of dots. She colors each dot either red or maroon. Compute the number of ways Betty can color the grid such that there is no rectangle whose sides are parallel to the grid lines and whose vertices all have the same color.

Call a number feared if it contains the digits 13 as a contiguous substring and fearless otherwise. (For example, 132 is feared, while 123 is fearless.) Compute the smallest positive integer $n$ such that there exists a positive integer $a<100$ such that $n$ and $n+10 a$ are fearless while $n+a, n+2 a, \ldots, n+9 a$ are all feared.

Pentagon $S P E A K$ is inscribed in triangle $N O W$ such that $S$ and $P$ lie on segment $N O, K$ and $A$ lie on segment $N W$, and $E$ lies on segment $O W$. Suppose that $N S=S P=P O$ and $N K=K A=A W$. Given that $E P=E K=5$ and $E A=E S=6$, compute $O W$.

It is midnight on April 29th, and Abigail is listening to a song by her favorite artist while staring at her clock, which has an hour, minute, and second hand. These hands move continuously. Between two consecutive midnights, compute the number of times the hour, minute, and second hands form two equal angles and no two hands overlap.

Let $A B C$ be an equilateral triangle with side length 2 that is inscribed in a circle $\omega$. A chord of $\omega$ passes through the midpoints of sides $A B$ and $A C$. Compute the length of this chord.

A real number $x$ satisfies $9^{x}+3^{x}=6$. Compute the value of $16^{1 / x}+4^{1 / x}$.

Two distinct similar rhombi share a diagonal. The smaller rhombus has area 1 , and the larger rhombus has area 9 . Compute the side length of the larger rhombus.

There are six empty slots corresponding to the digits of a six-digit number. Claire and William take turns rolling a standard six-sided die, with Claire going first. They alternate with each roll until they have each rolled three times. After a player rolls, they place the number from their die roll into a remaining empty slot of their choice. Claire wins if the resulting six-digit number is divisible by 6 , and William wins otherwise. If both players play optimally, compute the probability that Claire wins.

A complex quartic polynomial $Q$ is quirky if it has four distinct roots, one of which is the sum of the other three. There are four complex values of $k$ for which the polynomial $Q(x)=x^{4}-k x^{3}-x^{2}-x-45$ is quirky. Compute the product of these four values of $k$.

The pairwise greatest common divisors of five positive integers are

$$2,3,4,5,6,7,8, p, q, r$$

in some order, for some positive integers $p, q, r$. Compute the minimum possible value of $p+q+r$.

Let $A B C D$ be a convex trapezoid such that $\angle B A D=\angle A D C=90^{\circ}, A B=20, A D=21$, and $C D=28$. Point $P \neq A$ is chosen on segment $A C$ such that $\angle B P D=90^{\circ}$. Compute $A P$.

There are $n \geq 2$ coins, each with a different positive integer value. Call an integer $m$ sticky if some subset of these $n$ coins have total value $m$. We call the entire set of coins a stick if all the sticky numbers form a consecutive range of integers. Compute the minimum total value of a stick across all sticks containing a coin of value 100.

Let $r_{k}$ denote the remainder when $\binom{127}{k}$ is divided by 8 . Compute $r_{1}+2 r_{2}+3 r_{3}+\cdots+63 r_{63}$. Proposed by: Rishabh Das

Compute the number of ways a non-self-intersecting concave quadrilateral can be drawn in the plane such that two of its vertices are $(0,0)$ and $(1,0)$, and the other two vertices are two distinct lattice points $(a, b),(c, d)$ with $0 \leq a, c \leq 59$ and $1 \leq b, d \leq 5$. (A concave quadrilateral is a quadrilateral with an angle strictly larger than $180^{\circ}$. A lattice point is a point with both coordinates integers.)

The formula to convert Celsius to Fahrenheit is

$$F^{\circ}=1.8 \cdot C^{\circ}+32$$

In Celsius, it is $10^{\circ}$ warmer in New York right now than in Boston. In Fahrenheit, how much warmer is it in New York than in Boston?

Compute the number of dates in the year 2023 such that when put in MM/DD/YY form, the three numbers are in strictly increasing order. For example, $06 / 18 / 23$ is such a date since $6<18<23$, while today, $11 / 11 / 23$, is not.

Let $A B C D$ be a rectangle with $A B=20$ and $A D=23$. Let $M$ be the midpoint of $C D$, and let $X$ be the reflection of $M$ across point $A$. Compute the area of triangle $X B D$.

The number 5.6 may be expressed uniquely (ignoring order) as a product $\underline{a} \cdot \underline{b} \times \underline{c} . \underline{d}$ for digits $a, b, c, d$ all nonzero. Compute $\underline{a} \cdot \underline{b}+\underline{c} \cdot \underline{d}$.

Let $A B C D E$ be a convex pentagon such that

$$\begin{aligned} & A B+B C+C D+D E+E A=64 \text { and } \\ & A C+C E+E B+B D+D A=72 . \end{aligned}$$

Compute the perimeter of the convex pentagon whose vertices are the midpoints of the sides of $A B C D E$.

There are five people in a room. They each simultaneously pick two of the other people in the room independently and uniformly at random and point at them. Compute the probability that there exists a group of three people such that each of them is pointing at the other two in the group.

Suppose $a$ and $b$ be positive integers not exceeding 100 such that

$$a b=\left(\frac{\operatorname{lcm}(a, b)}{\operatorname{gcd}(a, b)}\right)^{2}$$

Compute the largest possible value of $a+b$.

Six standard fair six-sided dice are rolled and arranged in a row at random. Compute the expected number of dice showing the same number as the sixth die in the row.

The largest prime factor of 101101101101 is a four-digit number $N$. Compute $N$.

A real number $x$ is chosen uniformly at random from the interval $(0,10)$. Compute the probability that $\sqrt{x}, \sqrt{x+7}$, and $\sqrt{10-x}$ are the side lengths of a non-degenerate triangle.

Let $A B C D$ and $W X Y Z$ be two squares that share the same center such that $W X \| A B$ and $W X<A B$. Lines $C X$ and $A B$ intersect at $P$, and lines $C Z$ and $A D$ intersect at $Q$. If points $P, W$, and $Q$ are collinear, compute the ratio $A B / W X$.

A jar contains 97 marbles that are either red, green, or blue. Neil draws two marbles from the jar without replacement and notes that the probability that they would be the same color is $\frac{5}{12}$. After Neil puts his marbles back, Jerry draws two marbles from the jar with replacement. Compute the probability that the marbles that Jerry draws are the same color.

Suppose $x, y$, and $z$ are real numbers greater than 1 such that

$$\begin{aligned} & x^{\log _{y} z}=2 \\ & y^{\log _{z} x}=4, \text { and } \\ & z^{\log _{x} y}=8 \end{aligned}$$

Compute $\log _{x} y$.

Suppose that point $D$ lies on side $B C$ of triangle $A B C$ such that $A D$ bisects $\angle B A C$, and let $\ell$ denote the line through $A$ perpendicular to $A D$. If the distances from $B$ and $C$ to $\ell$ are 5 and 6 , respectively, compute $A D$.

Lucas writes two distinct positive integers on a whiteboard. He decreases the smaller number by 20 and increases the larger number by 23 , only to discover the product of the two original numbers is equal to the product of the two altered numbers. Compute the minimum possible sum of the original two numbers on the board.

Compute the number of tuples $\left(a_{0}, a_{1}, a_{2}, a_{3}, a_{4}, a_{5}\right)$ of (not necessarily positive) integers such that $a_{i} \leq i$ for all $0 \leq i \leq 5$ and

$$a_{0}+a_{1}+\cdots+a_{5}=6$$

Let $A B C$ be an equilateral triangle of side length 15 . Let $A_{b}$ and $B_{a}$ be points on side $A B, A_{c}$ and $C_{a}$ be points on side $A C$, and $B_{c}$ and $C_{b}$ be points on side $B C$ such that $\triangle A A_{b} A_{c}, \triangle B B_{c} B_{a}$, and $\triangle C C_{a} C_{b}$ are equilateral triangles with side lengths 3, 4 , and 5 , respectively. Compute the radius of the circle tangent to segments $\overline{A_{b} A_{c}}, \overline{B_{a} B_{c}}$, and $\overline{C_{a} C_{b}}$.

Over all real numbers $x$ and $y$ such that

$$x^{3}=3 x+y \text { and } y^{3}=3 y+x$$

compute the sum of all possible values of $x^{2}+y^{2}$.

Suppose $a, b$, and $c$ are real numbers such that

$$\begin{aligned} a^{2}-b c & =14 \\ b^{2}-c a & =14, \text { and } \\ c^{2}-a b & =-3 \end{aligned}$$

Compute $|a+b+c|$.

Let $A B C D$ be a square of side length 10 . Point $E$ is on ray $\overrightarrow{A B}$ such that $A E=17$, and point $F$ is on ray $\overrightarrow{A D}$ such that $A F=14$. The line through $B$ parallel to $C E$ and the line through $D$ parallel to $C F$ meet at $P$. Compute the area of quadrilateral $A E P F$.

An integer $n$ is chosen uniformly at random from the set $\{1,2,3, \ldots, 2023!\}$. Compute the probability that

$$\operatorname{gcd}\left(n^{n}+50, n+1\right)=1$$

There is a $6 \times 6$ grid of lights. There is a switch at the top of each column and on the left of each row. A light will only turn on if the switches corresponding to both its column and its row are in the "on" position. Compute the number of different configurations of lights.

The points $A=\left(4, \frac{1}{4}\right)$ and $B=\left(-5,-\frac{1}{5}\right)$ lie on the hyperbola $x y=1$. The circle with diameter $A B$ intersects this hyperbola again at points $X$ and $Y$. Compute $X Y$.

Compute the smallest positive integer $k$ such that 49 divides $\binom{2 k}{k}$.

A right triangle and a circle are drawn such that the circle is tangent to the legs of the right triangle. The circle cuts the hypotenuse into three segments of lengths 1,24 , and 3 , and the segment of length 24 is a chord of the circle. Compute the area of the triangle.

Compute the smallest multiple of 63 with an odd number of ones in its base two representation.

Compute the number of ways to color the vertices of a regular heptagon red, green, or blue (with rotations and reflections distinct) such that no isosceles triangle whose vertices are vertices of the heptagon has all three vertices the same color.

There is a unique quadruple of positive integers $(a, b, c, k)$ such that $c$ is not a perfect square and $a+\sqrt{b+\sqrt{c}}$ is a root of the polynomial $x^{4}-20 x^{3}+108 x^{2}-k x+9$. Compute $c$.

Let $A_{1} A_{2} \ldots A_{6}$ be a regular hexagon with side length $11 \sqrt{3}$, and let $B_{1} B_{2} \ldots B_{6}$ be another regular hexagon completely inside $A_{1} A_{2} \ldots A_{6}$ such that for all $i \in\{1,2, \ldots, 5\}, A_{i} A_{i+1}$ is parallel to $B_{i} B_{i+1}$. Suppose that the distance between lines $A_{1} A_{2}$ and $B_{1} B_{2}$ is 7 , the distance between lines $A_{2} A_{3}$ and $B_{2} B_{3}$ is 3 , and the distance between lines $A_{3} A_{4}$ and $B_{3} B_{4}$ is 8 . Compute the side length of $B_{1} B_{2} \ldots B_{6}$.

An HMMT party has $m$ MIT students and $h$ Harvard students for some positive integers $m$ and $h$, For every pair of people at the party, they are either friends or enemies. If every MIT student has 16 MIT friends and 8 Harvard friends, and every Harvard student has 7 MIT enemies and 10 Harvard enemies, compute how many pairs of friends there are at the party.

Let $s(n)$ denote the sum of the digits (in base ten) of a positive integer $n$. Compute the number of positive integers $n$ at most $10^{4}$ that satisfy

$$s(11 n)=2 s(n)$$

Compute

$$\sum_{\substack{a+b+c=12 \\ a \geq 6, b, c \geq 0}} \frac{a!}{b!c!(a-b-c)!}$$

where the sum runs over all triples of nonnegative integers $(a, b, c)$ such that $a+b+c=12$ and $a \geq 6$.

Let $\omega_{1}$ and $\omega_{2}$ be two non-intersecting circles. Suppose the following three conditions hold:

• The length of a common internal tangent of $\omega_{1}$ and $\omega_{2}$ is equal to 19 .
• The length of a common external tangent of $\omega_{1}$ and $\omega_{2}$ is equal to 37 .
• If two points $X$ and $Y$ are selected on $\omega_{1}$ and $\omega_{2}$, respectively, uniformly at random, then the expected value of $X Y^{2}$ is 2023 .

Compute the distance between the centers of $\omega_{1}$ and $\omega_{2}$.

Compute the smallest positive integer that does not appear in any problem statement on any round at HMMT November 2023. Submit a positive integer $A$. If the correct answer is $C$, you will receive $\max (0,20-5|A-C|)$ points.

Dorothea has a $3 \times 4$ grid of dots. She colors each dot red, blue, or dark gray. Compute the number of ways Dorothea can color the grid such that there is no rectangle whose sides are parallel to the grid lines and whose vertices all have the same color. Submit a positive integer $A$. If the correct answer is $C$ and your answer is $A$, you will receive $\left\lfloor 20\left(\min \left(\frac{A}{C}, \frac{C}{A}\right)\right)^{2}\right\rfloor$ points.

Isabella writes the expression $\sqrt{d}$ for each positive integer $d$ not exceeding 8 ! on the board. Seeing that these expressions might not be worth points on HMMT, Vidur simplifies each expression to the form $a \sqrt{b}$, where $a$ and $b$ are integers such that $b$ is not divisible by the square of a prime number. (For example, $\sqrt{20}, \sqrt{16}$, and $\sqrt{6}$ simplify to $2 \sqrt{5}, 4 \sqrt{1}$, and $1 \sqrt{6}$, respectively.) Compute the sum of $a+b$ across all expressions that Vidur writes. Submit a positive real number $A$. If the correct answer is $C$ and your answer is $A$, you get $\max \left(0,\left\lceil 20\left(1-|\log (A / C)|^{1 / 5}\right)\right\rceil\right)$ points.