HMMT November 2021

A domino has a left end and a right end, each of a certain color. Alice has four dominos, colored red-red, red-blue, blue-red, and blue-blue. Find the number of ways to arrange the dominos in a row end-to-end such that adjacent ends have the same color. The dominos cannot be rotated.

Suppose $a$ and $b$ are positive integers for which $8 a^{a} b^{b}=27 a^{b} b^{a}$. Find $a^{2}+b^{2}$.

Let $A B C D$ be a unit square. A circle with radius $\frac{32}{49}$ passes through point $D$ and is tangent to side $A B$ at point $E$. Then $D E=\frac{m}{n}$, where $m, n$ are positive integers and $\operatorname{gcd}(m, n)=1$. Find $100 m+n$.

The sum of the digits of the time 19 minutes ago is two less than the sum of the digits of the time right now. Find the sum of the digits of the time in 19 minutes. (Here, we use a standard 12-hour clock of the form hh:mm.)

A chord is drawn on a circle by choosing two points uniformly at random along its circumference. This is done two more times to obtain three total random chords. The circle is cut along these three lines, splitting it into pieces. The probability that one of the pieces is a triangle is $\frac{m}{n}$, where $m, n$ are positive integers and $\operatorname{gcd}(m, n)=1$. Find $100 m+n$.

Mario has a deck of seven pairs of matching number cards and two pairs of matching Jokers, for a total of 18 cards. He shuffles the deck, then draws the cards from the top one by one until he holds a pair of matching Jokers. The expected number of complete pairs that Mario holds at the end (including the Jokers) is $\frac{m}{n}$, where $m, n$ are positive integers and $\operatorname{gcd}(m, n)=1$. Find $100 m+n$.

Define the function $f: \mathbb{R} \rightarrow \mathbb{R}$ by

$$f(x)= \begin{cases}\frac{1}{x^{2}+\sqrt{x^{4}+2 x}} & \text { if } x \notin(-\sqrt[3]{2}, 0] \\ 0 & \text { otherwise }\end{cases}$$

The sum of all real numbers $x$ for which $f^{10}(x)=1$ can be written as $\frac{a+b \sqrt{c}}{d}$, where $a, b, c, d$ are integers, $d$ is positive, $c$ is square-free, and $\operatorname{gcd}(a, b, d)=1$. Find $1000 a+100 b+10 c+d$. (Here, $f^{n}(x)$ is the function $f(x)$ iterated $n$ times. For example, $f^{3}(x)=f(f(f(x)))$.)

Eight points are chosen on the circumference of a circle, labelled $P_{1}, P_{2}, \ldots, P_{8}$ in clockwise order. A route is a sequence of at least two points $P_{a_{1}}, P_{a_{2}}, \ldots, P_{a_{n}}$ such that if an ant were to visit these points in their given order, starting at $P_{a_{1}}$ and ending at $P_{a_{n}}$, by following $n-1$ straight line segments (each connecting each $P_{a_{i}}$ and $P_{a_{i+1}}$ ), it would never visit a point twice or cross its own path. Find the number of routes.

$A B C D E$ is a cyclic convex pentagon, and $A C=B D=C E . A C$ and $B D$ intersect at $X$, and $B D$ and $C E$ intersect at $Y$. If $A X=6, X Y=4$, and $Y E=7$, then the area of pentagon $A B C D E$ can be written as $\frac{a \sqrt{b}}{c}$, where $a, b, c$ are integers, $c$ is positive, $b$ is square-free, and $\operatorname{gcd}(a, c)=1$. Find $100 a+10 b+c$.

Real numbers $x, y, z$ satisfy

$$x+x y+x y z=1, y+y z+x y z=2, z+x z+x y z=4$$

The largest possible value of $x y z$ is $\frac{a+b \sqrt{c}}{d}$, where $a, b, c, d$ are integers, $d$ is positive, $c$ is square-free, and $\operatorname{gcd}(a, b, d)=1$. Find $1000 a+100 b+10 c+d$.

Let $n$ be the answer to this problem. In acute triangle $A B C$, point $D$ is located on side $B C$ so that $\angle B A D=\angle D A C$ and point $E$ is located on $A C$ so that $B E \perp A C$. Segments $B E$ and $A D$ intersect at $X$ such that $\angle B X D=n^{\circ}$. Given that $\angle X B A=16^{\circ}$, find the measure of $\angle B C A$.

Let $n$ be the answer to this problem. An urn contains white and black balls. There are $n$ white balls and at least two balls of each color in the urn. Two balls are randomly drawn from the urn without replacement. Find the probability, in percent, that the first ball drawn is white and the second is black.

Let $n$ be the answer to this problem. Hexagon $A B C D E F$ is inscribed in a circle of radius 90 . The area of $A B C D E F$ is $8 n, A B=B C=D E=E F$, and $C D=F A$. Find the area of triangle $A B C$.

Let $n$ be the answer to this problem. We define the digit sum of a date as the sum of its 4 digits when expressed in mmdd format (e.g. the digit sum of 13 May is $0+5+1+3=9$ ). Find the number of dates in the year 2021 with digit sum equal to the positive integer $n$.

Let $n$ be the answer to this problem. The polynomial $x^{n}+a x^{2}+b x+c$ has real coefficients and exactly $k$ real roots. Find the sum of the possible values of $k$.

Let $n$ be the answer to this problem. $a$ and $b$ are positive integers satisfying

$$\begin{aligned} & 3 a+5 b \equiv 19 (\bmod n+1) \\ & 4 a+2 b \equiv 25 (\bmod n+1) \end{aligned}$$

Find $2 a+6 b$.

Let $n$ be the answer to this problem. Box $B$ initially contains $n$ balls, and Box $A$ contains half as many balls as Box $B$. After 80 balls are moved from Box $A$ to Box $B$, the ratio of balls in Box $A$ to Box $B$ is now $\frac{p}{q}$, where $p, q$ are positive integers with $\operatorname{gcd}(p, q)=1$. Find $100 p+q$.

Let $n$ be the answer to this problem. Given $n>0$, find the number of distinct (i.e. non-congruent), non-degenerate triangles with integer side lengths and perimeter $n$.

Let $n$ be the answer to this problem. Find the minimum number of colors needed to color the divisors of $(n-24)$ ! such that no two distinct divisors $s, t$ of the same color satisfy $s \mid t$.

Let $n$ be the answer to this problem. Suppose square $A B C D$ has side-length 3 . Then, congruent non-overlapping squares $E H G F$ and $I H J K$ of side-length $\frac{n}{6}$ are drawn such that $A, C$, and $H$ are collinear, $E$ lies on $B C$ and $I$ lies on $C D$. Given that $A J G$ is an equilateral triangle, then the area of $A J G$ is $a+b \sqrt{c}$, where $a, b, c$ are positive integers and $c$ is not divisible by the square of any prime. Find $a+b+c$.

Let $A B C D$ be a parallelogram. Let $E$ be the midpoint of $A B$ and $F$ be the midpoint of $C D$. Points $P$ and $Q$ are on segments $E F$ and $C F$, respectively, such that $A, P$, and $Q$ are collinear. Given that $E P=5, P F=3$, and $Q F=12$, find $C Q$.

Joey wrote a system of equations on a blackboard, where each of the equations was of the form $a+b=c$ or $a \cdot b=c$ for some variables or integers $a, b, c$. Then Sean came to the board and erased all of the plus signs and multiplication signs, so that the board reads:

$$\begin{array}{ll} x & z=15 \\ x & y=12 \\ x & x=36 \end{array}$$

If $x, y, z$ are integer solutions to the original system, find the sum of all possible values of $100 x+10 y+z$. Proposed by: David Vulakh

Suppose $m$ and $n$ are positive integers for which

• the sum of the first $m$ multiples of $n$ is 120 , and
• the sum of the first $m^{3}$ multiples of $n^{3}$ is 4032000 .

Determine the sum of the first $m^{2}$ multiples of $n^{2}$.

Find the number of 10-digit numbers $\overline{a_{1} a_{2} \cdots a_{10}}$ which are multiples of 11 such that the digits are non-increasing from left to right, i.e. $a_{i} \geq a_{i+1}$ for each $1 \leq i \leq 9$.

How many ways are there to place 31 knights in the cells of an $8 \times 8$ unit grid so that no two attack one another? (A knight attacks another knight if the distance between the centers of their cells is exactly $\sqrt{5}$.)

The taxicab distance between points $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$ is $\left|x_{2}-x_{1}\right|+\left|y_{2}-y_{1}\right|$. A regular octagon is positioned in the $x y$ plane so that one of its sides has endpoints $(0,0)$ and $(1,0)$. Let $S$ be the set of all points inside the octagon whose taxicab distance from some octagon vertex is at most $\frac{2}{3}$. The area of $S$ can be written as $\frac{m}{n}$, where $m, n$ are positive integers and $\operatorname{gcd}(m, n)=1$. Find $100 m+n$.

Let $f(x)=x^{3}+3 x-1$ have roots $a, b, c$. Given that

$$\frac{1}{a^{3}+b^{3}}+\frac{1}{b^{3}+c^{3}}+\frac{1}{c^{3}+a^{3}}$$

can be written as $\frac{m}{n}$, where $m, n$ are positive integers and $\operatorname{gcd}(m, n)=1$, find $100 m+n$.

Paul and Sara are playing a game with integers on a whiteboard, with Paul going first. When it is Paul's turn, he can pick any two integers on the board and replace them with their product; when it is Sara's turn, she can pick any two integers on the board and replace them with their sum. Play continues until exactly one integer remains on the board. Paul wins if that integer is odd, and Sara wins if it is even.

Initially, there are 2021 integers on the board, each one sampled uniformly at random from the set $\{0,1,2,3, \ldots, 2021\}$. Assuming both players play optimally, the probability that Paul wins is $\frac{m}{n}$, where $m, n$ are positive integers and $\operatorname{gcd}(m, n)=1$. Find the remainder when $m+n$ is divided by 1000 .

Let $N$ be the smallest positive integer for which

$$x^{2}+x+1 \text { divides } 166-\sum_{d \mid N, d>0} x^{d}$$

Find the remainder when $N$ is divided by 1000.

Three faces $\mathcal{X}, \mathcal{Y}, \mathcal{Z}$ of a unit cube share a common vertex. Suppose the projections of $\mathcal{X}, \mathcal{Y}, \mathcal{Z}$ onto a fixed plane $\mathcal{P}$ have areas $x, y, z$, respectively. If $x: y: z=6: 10: 15$, then $x+y+z$ can be written as $\frac{m}{n}$, where $m, n$ are positive integers and $\operatorname{gcd}(m, n)=1$. Find $100 m+n$.

The graphs of the equations

$$\begin{aligned} y & =-x+8 \\ 173 y & =-289 x+2021 \end{aligned}$$

on the Cartesian plane intersect at $(a, b)$. Find $a+b$.

There are 8 lily pads in a pond numbered $1,2, \ldots, 8$. A frog starts on lily pad 1. During the $i$-th second, the frog jumps from lily pad $i$ to $i+1$, falling into the water with probability $\frac{1}{i+1}$. The probability that the frog lands safely on lily pad 8 without having fallen into the water at any point can be written as $\frac{m}{n}$, where $m, n$ are positive integers and $\operatorname{gcd}(m, n)=1$. Find $100 m+n$.

Suppose

$$h \cdot a \cdot r \cdot v \cdot a \cdot r \cdot d=m \cdot i \cdot t=h \cdot m \cdot m \cdot t=100 .$$

Find $(r \cdot a \cdot d) \cdot(t \cdot r \cdot i \cdot v \cdot i \cdot a)$.

Find the number of ways in which the letters in "HMMTHMMT" can be rearranged so that each letter is adjacent to another copy of the same letter. For example, "MMMMTTHH" satisfies this property, but "HHTMMMTM" does not.

A perfect power is an integer $n$ that can be represented as $a^{k}$ for some positive integers $a \geq 1$ and $k \geq 2$. Find the sum of all prime numbers $0<p<50$ such that $p$ is 1 less than a perfect power.

Let $A B C D$ be a parallelogram with $A B=480, A D=200$, and $B D=625$. The angle bisector of $\angle B A D$ meets side $C D$ at point $E$. Find $C E$.

Two unit squares $S_{1}$ and $S_{2}$ have horizontal and vertical sides. Let $x$ be the minimum distance between a point in $S_{1}$ and a point in $S_{2}$, and let $y$ be the maximum distance between a point in $S_{1}$ and a point in $S_{2}$. Given that $x=5$, the difference between the maximum and minimum possible values for $y$ can be written as $a+b \sqrt{c}$, where $a, b$, and $c$ are integers and $c$ is positive and square-free. Find $100 a+10 b+c$.

Let $p, q, r$ be primes such that $2 p+3 q=6 r$. Find $p+q+r$.

Let $n$ be an integer and

$$m=(n-1001)(n-2001)(n-2002)(n-3001)(n-3002)(n-3003) .$$

Given that $m$ is positive, find the minimum number of digits of $m$.

Squares $A B C D$ and $D E F G$ have side lengths 1 and $\frac{1}{3}$, respectively, where $E$ is on $\overline{C D}$ and points $A, D, G$ lie on a line in that order. Line $C F$ meets line $A G$ at $X$. The length $A X$ can be written as $\frac{m}{n}$, where $m, n$ are positive integers and $\operatorname{gcd}(m, n)=1$. Find $100 m+n$.

Let $n$ be a positive integer. Given that $n^{n}$ has 861 positive divisors, find $n$.

Alice draws three cards from a standard 52 -card deck with replacement. Ace through 10 are worth 1 to 10 points respectively, and the face cards King, Queen, and Jack are each worth 10 points. The probability that the sum of the point values of the cards drawn is a multiple of 10 can be written as $\frac{m}{n}$, where $m, n$ are positive integers and $\operatorname{gcd}(m, n)=1$. Find $100 m+n$.

Find the number of ways in which the nine numbers

$$1,12,123,1234, \ldots, 123456789$$

can be arranged in a row so that adjacent numbers are relatively prime.

In a $k \times k$ chessboard, a set $S$ of 25 cells that are in a $5 \times 5$ square is chosen uniformly at random. The probability that there are more black squares than white squares in $S$ is $48 \%$. Find $k$.

Tetrahedron $A B C D$ has side lengths $A B=6, B D=6 \sqrt{2}, B C=10, A C=8, C D=10$, and $A D=6$. The distance from vertex $A$ to face $B C D$ can be written as $\frac{a \sqrt{b}}{c}$, where $a, b, c$ are positive integers, $b$ is square-free, and $\operatorname{gcd}(a, c)=1$. Find $100 a+10 b+c$.

A counter begins at 0 . Then, every second, the counter either increases by 1 or resets back to 0 with equal probability. The expected value of the counter after ten seconds can be written as $\frac{m}{n}$, where $m, n$ are positive integers and $\operatorname{gcd}(m, n)=1$. Find $100 m+n$.

Let $A B C D E F G H$ be an equilateral octagon with $\angle A \cong \angle C \cong \angle E \cong \angle G$ and $\angle B \cong \angle D \cong \angle F \cong$ $\angle H$. If the area of $A B C D E F G H$ is three times the area of $A C E G$, then $\sin B$ can be written as $\frac{m}{n}$, where $m, n$ are positive integers and $\operatorname{gcd}(m, n)=1$. Find $100 m+n$.

Let $x, y, z$ be real numbers satisfying

$$\frac{1}{x}+y+z=x+\frac{1}{y}+z=x+y+\frac{1}{z}=3 .$$

The sum of all possible values of $x+y+z$ can be written as $\frac{m}{n}$, where $m, n$ are positive integers and $\operatorname{gcd}(m, n)=1$. Find $100 m+n$.

Integers $0 \leq a, b, c, d \leq 9$ satisfy

$$\begin{gathered} 6 a+9 b+3 c+d=88 \\ a-b+c-d=-6 \\ a-9 b+3 c-d=-46 \end{gathered}$$

Find $1000 a+100 b+10 c+d$.

On a chessboard, a queen attacks every square it can reach by moving from its current square along a row, column, or diagonal without passing through a different square that is occupied by a chess piece. Find the number of ways in which three indistinguishable queens can be placed on an $8 \times 8$ chess board so that each queen attacks both others.

Circle $\omega$ is inscribed in rhombus $H M_{1} M_{2} T$ so that $\omega$ is tangent to $\overline{H M_{1}}$ at $A, \overline{M_{1} M_{2}}$ at $I, \overline{M_{2} T}$ at $M$, and $\overline{T H}$ at $E$. Given that the area of $H M_{1} M_{2} T$ is 1440 and the area of $E M T$ is 405 , find the area of $A I M E$.

Two distinct squares on a $4 \times 4$ chessboard are chosen, with each pair of squares equally likely to be chosen. A knight is placed on one of the squares. The expected value of the minimum number of moves it takes for the knight to reach the other squarecan be written as $\frac{m}{n}$, where $m, n$ are positive integers and $\operatorname{gcd}(m, n)=1$. Find $100 m+n$.

Side $\overline{A B}$ of $\triangle A B C$ is the diameter of a semicircle, as shown below. If $A B=3+\sqrt{3}, B C=3 \sqrt{2}$, and $A C=2 \sqrt{3}$, then the area of the shaded region can be written as $\frac{a+(b+c \sqrt{d}) \pi}{e}$, where $a, b, c, d, e$ are integers, $e$ is positive, $d$ is square-free, and $\operatorname{gcd}(a, b, c, e)=1$. Find $10000 a+1000 b+100 c+10 d+e$.

Find the number of subsets $S$ of $\{1,2, \ldots, 48\}$ satisfying both of the following properties:

• For each integer $1 \leq k \leq 24$, exactly one of $2 k-1$ and $2 k$ is in $S$.
• There are exactly nine integers $1 \leq m \leq 47$ so that both $m$ and $m+1$ are in $S$.

Let $x, y, z$ be real numbers satisfying

$$\begin{aligned} 2 x+y+4 x y+6 x z & =-6 \\ y+2 z+2 x y+6 y z & =4 \\ x-z+2 x z-4 y z & =-3 \end{aligned}$$

Find $x^{2}+y^{2}+z^{2}$.

Let $X$ be the number of sequences of integers $a_{1}, a_{2}, \ldots, a_{2047}$ that satisfy all of the following properties:

• Each $a_{i}$ is either 0 or a power of 2 .
• $a_{i}=a_{2 i}+a_{2 i+1}$ for $1 \leq i \leq 1023$.
• $a_{1}=1024$.

Find the remainder when $X$ is divided by 100 .

$O$ is the center of square $A B C D$, and $M$ and $N$ are the midpoints of $\overline{B C}$ and $\overline{A D}$, respectively. Points $A^{\prime}, B^{\prime}, C^{\prime}, D^{\prime}$ are chosen on $\overline{A O}, \overline{B O}, \overline{C O}, \overline{D O}$, respectively, so that $A^{\prime} B^{\prime} M C^{\prime} D^{\prime} N$ is an equiangular hexagon. The ratio $\frac{\left[A^{\prime} B^{\prime} M C^{\prime} D^{\prime} N\right]}{[A B C D]}$ can be written as $\frac{a+b \sqrt{c}}{d}$, where $a, b, c, d$ are integers, $d$ is positive, $c$ is square-free, and $\operatorname{gcd}(a, b, d)=1$. Find $1000 a+100 b+10 c+d$.

Find the smallest positive integer $n$ such that the divisors of $n$ can be partitioned into three sets with equal sums.

Kevin writes down the positive integers $1,2, \ldots, 15$ on a blackboard. Then, he repeatedly picks two random integers $a, b$ on the blackboard, erases them, and writes down $\operatorname{gcd}(a, b)$ and $\operatorname{lcm}(a, b)$. He does this until he is no longer able to change the set of numbers written on the board. Find the maximum sum of the numbers on the board after this process.

The function $f: \mathbb{Z}^{2} \rightarrow \mathbb{Z}$ satisfies

• $f(x, 0)=f(0, y)=0$, and
• $f(x, y)=f(x-1, y)+f(x, y-1)+x+y$ for all nonnegative integers $x$ and $y$. Find $f(6,12)$.

For positive integers $n$, let $f(n)$ be the product of the digits of $n$. Find the largest positive integer $m$ such that

$$\sum_{n=1}^{\infty} \frac{f(n)}{m^{\left\lfloor\log _{10} n\right\rfloor}}$$

is an integer.

There are $N$ lockers, labeled from 1 to $N$, placed in clockwise order around a circular hallway. Initially, all lockers are open. Ansoon starts at the first locker and always moves clockwise. When she is at locker $n$ and there are more than $n$ open lockers, she keeps locker $n$ open and closes the next $n$ open lockers, then repeats the process with the next open locker. If she is at locker $n$ and there are at most $n$ lockers still open, she keeps locker $n$ open and closes all other lockers. She continues this process until only one locker is left open. What is the smallest integer $N>2021$ such that the last open locker is locker 1?

Point $P$ lies inside equilateral triangle $A B C$ so that $\angle B P C=120^{\circ}$ and $A P \sqrt{2}=B P+C P$. $\frac{A P}{A B}$ can be written as $\frac{a \sqrt{b}}{c}$, where $a, b, c$ are integers, $c$ is positive, $b$ is square-free, and $\operatorname{gcd}(a, c)=1$. Find $100 a+10 b+c$.

Suppose two distinct competitors of the HMMT 2021 November contest are chosen uniformly at random. Let $p$ be the probability that they can be labelled $A$ and $B$ so that $A$ 's score on the General round is strictly greater than $B$ 's, and $B$ 's score on the theme round is strictly greater than $A$ 's. Estimate $P=\lfloor 10000 p\rfloor$.

An estimate of $E$ will earn $\left\lfloor 20 \min \left(\frac{A}{E}, \frac{E}{A}\right)^{6}\right\rfloor$ points.

The following image is 1024 pixels by 1024 pixels, and each pixel is either black or white. The border defines the boundaries of the image, but is not part of the image. Let $a$ be the proportion of pixels that are black. Estimate $A=\lfloor 10000 a\rfloor$. An estimate of $E$ will earn $\left\lfloor 20 \min \left(\frac{A}{E}, \frac{E}{A}\right)^{12}\right\rfloor$ points.

Let $N$ be the number of ways in which the letters in "HMMTHMMTHMMTHMMTHMMTHMMT" ("HMMT" repeated six times) can be rearranged so that each letter is adjacent to another copy of the same letter. For example, "MMMMMMTTTTTTHHHHHHHHHHHH" satisfies this property, but "HMMMMMTTTTTTHННННННННННМ" does not. Estimate $N$. An estimate of $E$ will earn $\left\lfloor 20 \min \left(\frac{N}{E}, \frac{E}{N}\right)^{4}\right\rfloor$ points.