HMMT 2021

Compute the sum of all positive integers $n$ for which the expression

$$\frac{n+7}{\sqrt{n-1}}$$

is an integer.

Compute the number of ordered pairs of integers $(a, b)$, with $2 \leq a, b \leq 2021$, that satisfy the equation

$$a^{\log _{b}\left(a^{-4}\right)}=b^{\log _{a}\left(b a^{-3}\right)} .$$

Among all polynomials $P(x)$ with integer coefficients for which $P(-10)=145$ and $P(9)=164$, compute the smallest possible value of $|P(0)|$.

Suppose that $P(x, y, z)$ is a homogeneous degree 4 polynomial in three variables such that $P(a, b, c)=$ $P(b, c, a)$ and $P(a, a, b)=0$ for all real $a, b$, and $c$. If $P(1,2,3)=1$, compute $P(2,4,8)$. Note: $P(x, y, z)$ is a homogeneous degree 4 polynomial if it satisfies $P(k a, k b, k c)=k^{4} P(a, b, c)$ for all real $k, a, b, c$.

Suppose that $m$ and $n$ are positive integers with $m<n$ such that the interval $[m, n)$ contains more multiples of 2021 than multiples of 2000 . Compute the maximum possible value of $n-m$.

Suppose that $x, y$, and $z$ are complex numbers of equal magnitude that satisfy

$$x+y+z=-\frac{\sqrt{3}}{2}-i \sqrt{5}$$

and

$$x y z=\sqrt{3}+i \sqrt{5} .$$

If $x=x_{1}+i x_{2}, y=y_{1}+i y_{2}$, and $z=z_{1}+i z_{2}$ for real $x_{1}, x_{2}, y_{1}, y_{2}, z_{1}$, and $z_{2}$, then

$$\left(x_{1} x_{2}+y_{1} y_{2}+z_{1} z_{2}\right)^{2}$$

can be written as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $100 a+b$.

For positive integers $a$ and $b$, let $M(a, b)=\frac{\operatorname{lcm}(a, b)}{\operatorname{gcd}(a, b)}$, and for each positive integer $n \geq 2$, define

$$x_{n}=M(1, M(2, M(3, \ldots, M(n-2, M(n-1, n)) \ldots)))$$

Compute the number of positive integers $n$ such that $2 \leq n \leq 2021$ and $5 x_{n}^{2}+5 x_{n+1}^{2}=26 x_{n} x_{n+1}$.

Let $f$ be a monic cubic polynomial satisfying $f(x)+f(-x)=0$ for all real numbers $x$. For all real numbers $y$, define $g(y)$ to be the number of distinct real solutions $x$ to the equation $f(f(x))=y$. Suppose that the set of possible values of $g(y)$ over all real numbers $y$ is exactly $\{1,5,9\}$. Compute the sum of all possible values of $f(10)$.

Let $S$ be a set of positive integers satisfying the following two conditions:

• For each positive integer $n$, at least one of $n, 2 n, \ldots, 100 n$ is in $S$.
• If $a_{1}, a_{2}, b_{1}, b_{2}$ are positive integers such that $\operatorname{gcd}\left(a_{1} a_{2}, b_{1} b_{2}\right)=1$ and $a_{1} b_{1}, a_{2} b_{2} \in S$, then $a_{2} b_{1}, a_{1} b_{2} \in S$.

Suppose that $S$ has natural density $r$. Compute the minimum possible value of $\left\lfloor 10^{5} r\right\rfloor$. Note: $S$ has natural density $r$ if $\frac{1}{n}|S \cap\{1, \ldots, n\}|$ approaches $r$ as $n$ approaches $\infty$.

A circle contains the points $(0,11)$ and $(0,-11)$ on its circumference and contains all points $(x, y)$ with $x^{2}+y^{2}<1$ in its interior. Compute the largest possible radius of the circle.

Let $X_{0}$ be the interior of a triangle with side lengths 3,4 , and 5 . For all positive integers $n$, define $X_{n}$ to be the set of points within 1 unit of some point in $X_{n-1}$. The area of the region outside $X_{20}$ but inside $X_{21}$ can be written as $a \pi+b$, for integers $a$ and $b$. Compute $100 a+b$.

Triangle $A B C$ has a right angle at $C$, and $D$ is the foot of the altitude from $C$ to $A B$. Points $L$, $M$, and $N$ are the midpoints of segments $A D, D C$, and $C A$, respectively. If $C L=7$ and $B M=12$, compute $B N^{2}$.

Let $A B C D$ be a trapezoid with $A B \| C D, A B=5, B C=9, C D=10$, and $D A=7$. Lines $B C$ and $D A$ intersect at point $E$. Let $M$ be the midpoint of $C D$, and let $N$ be the intersection of the circumcircles of $\triangle B M C$ and $\triangle D M A$ (other than $M$ ). If $E N^{2}=\frac{a}{b}$ for relatively prime positive integers $a$ and $b$, compute $100 a+b$.

Let $A E F$ be a triangle with $E F=20$ and $A E=A F=21$. Let $B$ and $D$ be points chosen on segments $A E$ and $A F$, respectively, such that $B D$ is parallel to $E F$. Point $C$ is chosen in the interior of triangle $A E F$ such that $A B C D$ is cyclic. If $B C=3$ and $C D=4$, then the ratio of areas $\frac{[A B C D]}{[A E F]}$ can be written as $\frac{a}{b}$ for relatively prime positive integers $a, b$. Compute $100 a+b$.

In triangle $A B C$, let $M$ be the midpoint of $B C, H$ be the orthocenter, and $O$ be the circumcenter. Let $N$ be the reflection of $M$ over $H$. Suppose that $O A=O N=11$ and $O H=7$. Compute $B C^{2}$.

Let $O$ and $A$ be two points in the plane with $O A=30$, and let $\Gamma$ be a circle with center $O$ and radius $r$. Suppose that there exist two points $B$ and $C$ on $\Gamma$ with $\angle A B C=90^{\circ}$ and $A B=B C$. Compute the minimum possible value of $\lfloor r\rfloor$.

Two circles with radii 71 and 100 are externally tangent. Compute the largest possible area of a right triangle whose vertices are each on at least one of the circles.

Let $A B C D$ be a trapezoid with $A B \| C D$ and $A D=B D$. Let $M$ be the midpoint of $A B$, and let $P \neq C$ be the second intersection of the circumcircle of $\triangle B C D$ and the diagonal $A C$. Suppose that $B C=27, C D=25$, and $A P=10$. If $M P=\frac{a}{b}$ for relatively prime positive integers $a$ and $b$, compute $100 a+b$.

Acute triangle $A B C$ has circumcircle $\Gamma$. Let $M$ be the midpoint of $B C$. Points $P$ and $Q$ lie on $\Gamma$ so that $\angle A P M=90^{\circ}$ and $Q \neq A$ lies on line $A M$. Segments $P Q$ and $B C$ intersect at $S$. Suppose that $B S=1, C S=3, P Q=8 \sqrt{\frac{7}{37}}$, and the radius of $\Gamma$ is $r$. If the sum of all possible values of $r^{2}$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$, compute $100 a+b$.

Leo the fox has a 5 by 5 checkerboard grid with alternating red and black squares. He fills in the grid with the numbers $1,2,3, \ldots, 25$ such that any two consecutive numbers are in adjacent squares (sharing a side) and each number is used exactly once. He then computes the sum of the numbers in the 13 squares that are the same color as the center square. Compute the maximum possible sum Leo can obtain.

Ava and Tiffany participate in a knockout tournament consisting of a total of 32 players. In each of 5 rounds, the remaining players are paired uniformly at random. In each pair, both players are equally likely to win, and the loser is knocked out of the tournament. The probability that Ava and Tiffany play each other during the tournament is $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $100 a+b$.

Let $N$ be a positive integer. Brothers Michael and Kylo each select a positive integer less than or equal to $N$, independently and uniformly at random. Let $p_{N}$ denote the probability that the product of these two integers has a units digit of 0 . The maximum possible value of $p_{N}$ over all possible choices of $N$ can be written as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $100 a+b$.

Let $S=\{1,2, \ldots, 9\}$. Compute the number of functions $f: S \rightarrow S$ such that, for all $s \in S, f(f(f(s)))=$ $s$ and $f(s)-s$ is not divisible by 3.

Teresa the bunny has a fair 8 -sided die. Seven of its sides have fixed labels $1,2, \ldots, 7$, and the label on the eighth side can be changed and begins as 1 . She rolls it several times, until each of $1,2, \ldots, 7$ appears at least once. After each roll, if $k$ is the smallest positive integer that she has not rolled so far, she relabels the eighth side with $k$. The probability that 7 is the last number she rolls is $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $100 a+b$.

A light pulse starts at a corner of a reflective square. It bounces around inside the square, reflecting off of the square's perimeter $n$ times before ending in a different corner. The path of the light pulse, when traced, divides the square into exactly 2021 regions. Compute the smallest possible value of $n$.

Let $S=\{1,2, \ldots, 2021\}$, and let $\mathcal{F}$ denote the set of functions $f: S \rightarrow S$. For a function $f \in \mathcal{F}$, let

$$T_{f}=\left\{f^{2021}(s): s \in S\right\}$$

where $f^{2021}(s)$ denotes $f(f(\cdots(f(s)) \cdots))$ with 2021 copies of $f$. Compute the remainder when

$$\sum_{f \in \mathcal{F}}\left|T_{f}\right|$$

is divided by the prime 2017, where the sum is over all functions $f$ in $\mathcal{F}$.

Compute the number of ways to fill each cell in a $8 \times 8$ square grid with one of the letters $H, M$, or $T$ such that every $2 \times 2$ square in the grid contains the letters $H, M, M, T$ in some order.

An up-right path between two lattice points $P$ and $Q$ is a path from $P$ to $Q$ that takes steps of length 1 unit either up or to the right.

How many up-right paths from $(0,0)$ to $(7,7)$, when drawn in the plane with the line $y=x-2.021$, enclose exactly one bounded region below that line?

Jude repeatedly flips a coin. If he has already flipped $n$ heads, the coin lands heads with probability $\frac{1}{n+2}$ and tails with probability $\frac{n+1}{n+2}$. If Jude continues flipping forever, let $p$ be the probability that he flips 3 heads in a row at some point. Compute $\lfloor 180 p\rfloor$.

Let $a$ and $b$ be positive integers with $a>b$. Suppose that

$$\sqrt{\sqrt{a}+\sqrt{b}}+\sqrt{\sqrt{a}-\sqrt{b}}$$

is a integer. (a) Must $\sqrt{a}$ be an integer? (b) Must $\sqrt{b}$ be an integer?

Let $A B C$ be a right triangle with $\angle A=90^{\circ}$. A circle $\omega$ centered on $B C$ is tangent to $A B$ at $D$ and $A C$ at $E$. Let $F$ and $G$ be the intersections of $\omega$ and $B C$ so that $F$ lies between $B$ and $G$. If lines $D G$ and $E F$ intersect at $X$, show that $A X=A D$.

Let $m$ be a positive integer. Show that there exists a positive integer $n$ such that each of the $2 m+1$ integers

$$2^{n}-m, 2^{n}-(m-1), \ldots, 2^{n}+(m-1), 2^{n}+m$$

is positive and composite.

Let $k$ and $n$ be positive integers and let

$$S=\left\{\left(a_{1}, \ldots, a_{k}\right) \in \mathbb{Z}^{k} \mid 0 \leq a_{k} \leq \cdots \leq a_{1} \leq n, a_{1}+\cdots+a_{k}=k\right\}$$

Determine, with proof, the value of

$$\sum_{\left(a_{1}, \ldots, a_{k}\right) \in S}\binom{n}{a_{1}}\binom{a_{1}}{a_{2}} \cdots\binom{a_{k-1}}{a_{k}}$$

in terms of $k$ and $n$, where the sum is over all $k$-tuples $\left(a_{1}, \ldots, a_{k}\right)$ in $S$.

A convex polyhedron has $n$ faces that are all congruent triangles with angles $36^{\circ}, 72^{\circ}$, and $72^{\circ}$. Determine, with proof, the maximum possible value of $n$.

Let $f(x)=x^{2}+x+1$. Determine, with proof, all positive integers $n$ such that $f(k)$ divides $f(n)$ whenever $k$ is a positive divisor of $n$.

In triangle $A B C$, let $M$ be the midpoint of $B C$ and $D$ be a point on segment $A M$. Distinct points $Y$ and $Z$ are chosen on rays $\overrightarrow{C A}$ and $\overrightarrow{B A}$, respectively, such that $\angle D Y C=\angle D C B$ and $\angle D B C=\angle D Z B$. Prove that the circumcircle of $\triangle D Y Z$ is tangent to the circumcircle of $\triangle D B C$.

For each positive real number $\alpha$, define

$$\lfloor\alpha \mathbb{N}\rfloor:=\{\lfloor\alpha m\rfloor \mid m \in \mathbb{N}\}$$

Let $n$ be a positive integer. A set $S \subseteq\{1,2, \ldots, n\}$ has the property that: for each real $\beta>0$,

$$\text { if } S \subseteq\lfloor\beta \mathbb{N}\rfloor \text {, then }\{1,2, \ldots, n\} \subseteq\lfloor\beta \mathbb{N}\rfloor$$

Determine, with proof, the smallest possible size of $S$.

Let scalene triangle $A B C$ have circumcenter $O$ and incenter $I$. Its incircle $\omega$ is tangent to sides $B C, C A$, and $A B$ at $D, E$, and $F$, respectively. Let $P$ be the foot of the altitude from $D$ to $E F$, and let line $D P$ intersect $\omega$ again at $Q \neq D$. The line $O I$ intersects the altitude from $A$ to $B C$ at $T$. Given that $O I \| B C$, show that $P Q=P T$.

Let $n>1$ be a positive integer. Each unit square in an $n \times n$ grid of squares is colored either black or white, such that the following conditions hold:

• Any two black squares can be connected by a sequence of black squares where every two consecutive squares in the sequence share an edge;
• Any two white squares can be connected by a sequence of white squares where every two consecutive squares in the sequence share an edge;
• Any $2 \times 2$ subgrid contains at least one square of each color.

Determine, with proof, the maximum possible difference between the number of black squares and white squares in this grid (in terms of $n$ ).

Amelia wrote down a sequence of consecutive positive integers, erased one integer, and scrambled the rest, leaving the sequence below. What integer did she erase?

$$6,12,1,3,11,10,8,15,13,9,7,4,14,5,2$$

Suppose there exists a convex $n$-gon such that each of its angle measures, in degrees, is an odd prime number. Compute the difference between the largest and smallest possible values of $n$.

A semicircle with radius 2021 has diameter $A B$ and center $O$. Points $C$ and $D$ lie on the semicircle such that $\angle A O C<\angle A O D=90^{\circ}$. A circle of radius $r$ is inscribed in the sector bounded by $O A$ and $O C$ and is tangent to the semicircle at $E$. If $C D=C E$, compute $\lfloor r\rfloor$.

In a 3 by 3 grid of unit squares, an up-right path is a path from the bottom left corner to the top right corner that travels only up and right in steps of 1 unit. For such a path $p$, let $A_{p}$ denote the number of unit squares under the path $p$. Compute the sum of $A_{p}$ over all up-right paths $p$.

Let $m, n>2$ be integers. One of the angles of a regular $n$-gon is dissected into $m$ angles of equal size by $(m-1)$ rays. If each of these rays intersects the polygon again at one of its vertices, we say $n$ is $m$-cut. Compute the smallest positive integer $n$ that is both 3 -cut and 4 -cut.

In a group of 50 children, each of the children in the group have all of their siblings in the group. Each child with no older siblings announces how many siblings they have; however, each child with an older sibling is too embarrassed, and says they have 0 siblings. If the average of the numbers everyone says is $\frac{12}{25}$, compute the number of different sets of siblings represented in the group.

Milan has a bag of 2020 red balls and 2021 green balls. He repeatedly draws 2 balls out of the bag uniformly at random. If they are the same color, he changes them both to the opposite color and returns them to the bag. If they are different colors, he discards them. Eventually the bag has 1 ball left. Let $p$ be the probability that it is green. Compute $\lfloor 2021 p\rfloor$.

Compute the product of all positive integers $b \geq 2$ for which the base $b$ number $111111_{b}$ has exactly $b$ distinct prime divisors.

Let $A D, B E$, and $C F$ be segments sharing a common midpoint, with $A B<A E$ and $B C<B F$. Suppose that each pair of segments forms a $60^{\circ}$ angle, and that $A D=7, B E=10$, and $C F=18$. Let $K$ denote the sum of the areas of the six triangles $\triangle A B C, \triangle B C D, \triangle C D E, \triangle D E F, \triangle E F A$, and $\triangle F A B$. Compute $K \sqrt{3}$.

Let $a_{1}, a_{2}, \ldots, a_{n}$ be a sequence of distinct positive integers such that $a_{1}+a_{2}+\cdots+a_{n}=2021$ and $a_{1} a_{2} \cdots a_{n}$ is maximized. If $M=a_{1} a_{2} \cdots a_{n}$, compute the largest positive integer $k$ such that $2^{k} \mid M$. Proposed by: Sheldon Kieren Tan

For each positive integer $1 \leq m \leq 10$, Krit chooses an integer $0 \leq a_{m}<m$ uniformly at random. Let $p$ be the probability that there exists an integer $n$ for which $n \equiv a_{m}(\bmod m)$ for all $m$. If $p$ can be written as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$, compute $100 a+b$.

Compute the number of labelings $f:\{0,1\}^{3} \rightarrow\{0,1, \ldots, 7\}$ of the vertices of the unit cube such that

$$\left|f\left(v_{i}\right)-f\left(v_{j}\right)\right| \geq d\left(v_{i}, v_{j}\right)^{2}$$

for all vertices $v_{i}, v_{j}$ of the unit cube, where $d\left(v_{i}, v_{j}\right)$ denotes the Euclidean distance between $v_{i}$ and $v_{j}$.

A tournament among 2021 ranked teams is played over 2020 rounds. In each round, two teams are selected uniformly at random among all remaining teams to play against each other. The better ranked team always wins, and the worse ranked team is eliminated. Let $p$ be the probability that the second best ranked team is eliminated in the last round. Compute $\lfloor 2021 p\rfloor$.

In triangle $A B C, \angle A=2 \angle C$. Suppose that $A C=6, B C=8$, and $A B=\sqrt{a}-b$, where $a$ and $b$ are positive integers. Compute $100 a+b$.

Two circles $\Gamma_{1}$ and $\Gamma_{2}$ of radius 1 and 2, respectively, are centered at the origin. A particle is placed at $(2,0)$ and is shot towards $\Gamma_{1}$. When it reaches $\Gamma_{1}$, it bounces off the circumference and heads back towards $\Gamma_{2}$. The particle continues bouncing off the two circles in this fashion. If the particle is shot at an acute angle $\theta$ above the $x$-axis, it will bounce 11 times before returning to $(2,0)$ for the first time. If $\cot \theta=a-\sqrt{b}$ for positive integers $a$ and $b$, compute $100 a+b$.

Let $f: \mathbb{Z}^{2} \rightarrow \mathbb{Z}$ be a function such that, for all positive integers $a$ and $b$,

$$f(a, b)= \begin{cases}b & \text { if } a>b \\ f(2 a, b) & \text { if } a \leq b \text { and } f(2 a, b)<a \\ f(2 a, b)-a & \text { otherwise }\end{cases}$$

Compute $f\left(1000,3^{2021}\right)$.

Let $k$ be the answer to this problem. The probability that an integer chosen uniformly at random from $\{1,2, \ldots, k\}$ is a multiple of 11 can be written as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $100 a+b$.

Triangle $A B C$ has side lengths $A B=19, B C=20$, and $C A=21$. Points $X$ and $Y$ are selected on sides $A B$ and $A C$, respectively, such that $A Y=X Y$ and $X Y$ is tangent to the incircle of $\triangle A B C$. If the length of segment $A X$ can be written as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers, compute $100 a+b$.

Almondine has a bag with $N$ balls, each of which is red, white, or blue. If Almondine picks three balls from the bag without replacement, the probability that she picks one ball of each color is larger than 23 percent. Compute the largest possible value of $\left\lfloor\frac{N}{3}\right\rfloor$.

Let $f(x)=x^{3}-3 x$. Compute the number of positive divisors of

$$\left\lfloor f\left(f\left(f\left(f\left(f\left(f\left(f\left(f\left(\frac{5}{2}\right)\right)\right)\right)\right)\right)\right)\right)\right\rfloor$$

where $f$ is applied 8 times.

Bob knows that Alice has 2021 secret positive integers $x_{1}, \ldots, x_{2021}$ that are pairwise relatively prime. Bob would like to figure out Alice's integers. He is allowed to choose a set $S \subseteq\{1,2, \ldots, 2021\}$ and ask her for the product of $x_{i}$ over $i \in S$. Alice must answer each of Bob's queries truthfully, and Bob may use Alice's previous answers to decide his next query. Compute the minimum number of queries Bob needs to guarantee that he can figure out each of Alice's integers.

Let $E$ be a three-dimensional ellipsoid. For a plane $p$, let $E(p)$ be the projection of $E$ onto the plane $p$. The minimum and maximum areas of $E(p)$ are $9 \pi$ and $25 \pi$, and there exists a $p$ where $E(p)$ is a circle of area $16 \pi$. If $V$ is the volume of $E$, compute $V / \pi$.

Let $f: \mathbb{N} \rightarrow \mathbb{N}$ be a strictly increasing function such that $f(1)=1$ and $f(2 n) f(2 n+1)=$ $9 f(n)^{2}+3 f(n)$ for all $n \in \mathbb{N}$. Compute $f(137)$.

Let $P$ be a point selected uniformly at random in the cube $[0,1]^{3}$. The plane parallel to $x+y+z=0$ passing through $P$ intersects the cube in a two-dimensional region $\mathcal{R}$. Let $t$ be the expected value of the perimeter of $\mathcal{R}$. If $t^{2}$ can be written as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers, compute $100 a+b$.

Let $n$ be a positive integer. Claudio has $n$ cards, each labeled with a different number from 1 to n. He takes a subset of these cards, and multiplies together the numbers on the cards. He remarks that, given any positive integer $m$, it is possible to select some subset of the cards so that the difference between their product and $m$ is divisible by 100. Compute the smallest possible value of $n$.

Let triangle $A B C$ have incircle $\omega$, which touches $B C, C A$, and $A B$ at $D, E$, and $F$, respectively. Then, let $\omega_{1}$ and $\omega_{2}$ be circles tangent to $A D$ and internally tangent to $\omega$ at $E$ and $F$, respectively. Let $P$ be the intersection of line $E F$ and the line passing through the centers of $\omega_{1}$ and $\omega_{2}$. If $\omega_{1}$ and $\omega_{2}$ have radii 5 and 6 , respectively, compute $P E \cdot P F$.

Let $P$ be the set of points

$$\{(x, y) \mid 0 \leq x, y \leq 25, x, y \in \mathbb{Z}\}$$

and let $T$ be the set of triangles formed by picking three distinct points in $P$ (rotations, reflections, and translations count as distinct triangles). Compute the number of triangles in $T$ that have area larger than 300.

Caroline starts with the number 1, and every second she flips a fair coin; if it lands heads, she adds 1 to her number, and if it lands tails she multiplies her number by 2 . Compute the expected number of seconds it takes for her number to become a multiple of 2021.

Compute the number of complex numbers $z$ with $|z|=1$ that satisfy

$$1+z^{5}+z^{10}+z^{15}+z^{18}+z^{21}+z^{24}+z^{27}=0$$

Let $f(n)$ be the largest prime factor of $n^{2}+1$. Compute the least positive integer $n$ such that $f(f(n))=n$.

Roger initially has 20 socks in a drawer, each of which is either white or black. He chooses a sock uniformly at random from the drawer and throws it away. He repeats this action until there are equal numbers of white and black socks remaining. Suppose that the probability he stops before all socks are gone is $p$. If the sum of all distinct possible values of $p$ over all initial combinations of socks is $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$, compute $100 a+b$.

Let acute triangle $A B C$ have circumcenter $O$, and let $M$ be the midpoint of $B C$. Let $P$ be the unique point such that $\angle B A P=\angle C A M, \angle C A P=\angle B A M$, and $\angle A P O=90^{\circ}$. If $A O=53, O M=28$, and $A M=75$, compute the perimeter of $\triangle B P C$.

After the Guts round ends, HMMT organizers will collect all answers submitted to all 66 questions (including this one) during the individual rounds and the guts round. Estimate $N$, the smallest positive integer that no one will have submitted at any point during the tournament. An estimate of $E$ will receive $\max (0,24-4|E-N|)$ points.

Let $f(n)$ be the largest prime factor of $n$. Estimate

$$N=\left\lfloor 10^{4} \cdot \frac{\sum_{n=2}^{10^{6}} f\left(n^{2}-1\right)}{\sum_{n=2}^{10^{6}} f(n)}\right\rfloor .$$

An estimate of $E$ will receive $\max \left(0,\left\lfloor 20-20\left(\frac{|E-N|}{10^{3}}\right)^{1 / 3}\right\rfloor\right)$ points.

Geoff walks on the number line for 40 minutes, starting at the point 0 . On the $n$th minute, he flips a fair coin. If it comes up heads he walks $\frac{1}{n}$ in the positive direction and if it comes up tails he walks $\frac{1}{n}$ in the negative direction. Let $p$ be the probability that he never leaves the interval $[-2,2]$. Estimate $N=\left\lfloor 10^{4} p\right\rfloor$. An estimate of $E$ will receive $\max \left(0,\left\lfloor 20-20\left(\frac{|E-N|}{160}\right)^{1 / 3}\right\rfloor\right)$ points.

A set of 6 distinct lattice points is chosen uniformly at random from the set $\{1,2,3,4,5,6\}^{2}$. Let $A$ be the expected area of the convex hull of these 6 points. Estimate $N=\left\lfloor 10^{4} A\right\rfloor$. An estimate of $E$ will receive $\max \left(0,\left\lfloor 20-20\left(\frac{|E-N|}{10^{4}}\right)^{1 / 3}\right\rfloor\right)$ points.