HMMT 2022

Positive integers $a, b$, and $c$ are all powers of $k$ for some positive integer $k$. It is known that the equation $a x^{2}-b x+c=0$ has exactly one real solution $r$, and this value $r$ is less than 100 . Compute the maximum possible value of $r$.

Compute the number of positive integers that divide at least two of the integers in the set $\left\{1^{1}, 2^{2}, 3^{3}, 4^{4}, 5^{5}, 6^{6}, 7^{7}, 8^{8}, 9^{9}, 10^{10}\right\}$.

Let $x_{1}, x_{2}, \ldots, x_{2022}$ be nonzero real numbers. Suppose that $x_{k}+\frac{1}{x_{k+1}}<0$ for each $1 \leq k \leq 2022$, where $x_{2023}=x_{1}$. Compute the maximum possible number of integers $1 \leq n \leq 2022$ such that $x_{n}>0$.

Compute the sum of all 2-digit prime numbers $p$ such that there exists a prime number $q$ for which $100 q+p$ is a perfect square.

Given a positive integer $k$, let $\|k\|$ denote the absolute difference between $k$ and the nearest perfect square. For example, $\|13\|=3$ since the nearest perfect square to 13 is 16 . Compute the smallest positive integer $n$ such that

$$\frac{\|1\|+\|2\|+\cdots+\|n\|}{n}=100 .$$

Let $f$ be a function from $\{1,2, \ldots, 22\}$ to the positive integers such that $m n \mid f(m)+f(n)$ for all $m, n \in\{1,2, \ldots, 22\}$. If $d$ is the number of positive divisors of $f(20)$, compute the minimum possible value of $d$.

Let $\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right),\left(x_{3}, y_{3}\right),\left(x_{4}, y_{4}\right)$, and $\left(x_{5}, y_{5}\right)$ be the vertices of a regular pentagon centered at $(0,0)$. Compute the product of all positive integers $k$ such that the equality

$$x_{1}^{k}+x_{2}^{k}+x_{3}^{k}+x_{4}^{k}+x_{5}^{k}=y_{1}^{k}+y_{2}^{k}+y_{3}^{k}+y_{4}^{k}+y_{5}^{k}$$

must hold for all possible choices of the pentagon.

Positive integers $a_{1}, a_{2}, \ldots, a_{7}, b_{1}, b_{2}, \ldots, b_{7}$ satisfy $2 \leq a_{i} \leq 166$ and $a_{i}^{b_{i}} \equiv a_{i+1}^{2}(\bmod 167)$ for each $1 \leq i \leq 7$ (where $a_{8}=a_{1}$ ). Compute the minimum possible value of $b_{1} b_{2} \cdots b_{7}\left(b_{1}+b_{2}+\cdots+b_{7}\right)$.

Suppose $P(x)$ is a monic polynomial of degree 2023 such that

$$P(k)=k^{2023} P\left(1-\frac{1}{k}\right)$$

for every positive integer $1 \leq k \leq 2023$. Then $P(-1)=\frac{a}{b}$, where $a$ and $b$ relatively prime integers. Compute the unique integer $0 \leq n<2027$ such that $b n-a$ is divisible by the prime 2027 .

Compute the smallest positive integer $n$ for which there are at least two odd primes $p$ such that

$$\sum_{k=1}^{n}(-1)^{\nu_{p}(k!)}<0$$

Note: for a prime $p$ and a positive integer $m, \nu_{p}(m)$ is the exponent of the largest power of $p$ that divides $m$; for example, $\nu_{3}(18)=2$.

Let $A B C$ be a triangle with $\angle A=60^{\circ}$. Line $\ell$ intersects segments $A B$ and $A C$ and splits triangle $A B C$ into an equilateral triangle and a quadrilateral. Let $X$ and $Y$ be on $\ell$ such that lines $B X$ and $C Y$ are perpendicular to $\ell$. Given that $A B=20$ and $A C=22$, compute $X Y$.

Rectangle $R_{0}$ has sides of lengths 3 and 4. Rectangles $R_{1}, R_{2}$, and $R_{3}$ are formed such that:

• all four rectangles share a common vertex $P$,
• for each $n=1,2,3$, one side of $R_{n}$ is a diagonal of $R_{n-1}$,
• for each $n=1,2,3$, the opposite side of $R_{n}$ passes through a vertex of $R_{n-1}$ such that the center of $R_{n}$ is located counterclockwise of the center of $R_{n-1}$ with respect to $P$.

Compute the total area covered by the union of the four rectangles.

Let $A B C D$ and $A E F G$ be unit squares such that the area of their intersection is $\frac{20}{21}$. Given that $\angle B A E<45^{\circ}, \tan \angle B A E$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $100 a+b$.

Parallel lines $\ell_{1}, \ell_{2}, \ell_{3}, \ell_{4}$ are evenly spaced in the plane, in that order. Square $A B C D$ has the property that $A$ lies on $\ell_{1}$ and $C$ lies on $\ell_{4}$. Let $P$ be a uniformly random point in the interior of $A B C D$ and let $Q$ be a uniformly random point on the perimeter of $A B C D$. Given that the probability that $P$ lies between $\ell_{2}$ and $\ell_{3}$ is $\frac{53}{100}$, the probability that $Q$ lies between $\ell_{2}$ and $\ell_{3}$ can be expressed as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $100 a+b$.

Let triangle $A B C$ be such that $A B=A C=22$ and $B C=11$. Point $D$ is chosen in the interior of the triangle such that $A D=19$ and $\angle A B D+\angle A C D=90^{\circ}$. The value of $B D^{2}+C D^{2}$ can be expressed as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $100 a+b$.

Let $A B C D$ be a rectangle inscribed in circle $\Gamma$, and let $P$ be a point on minor arc $A B$ of $\Gamma$. Suppose that $P A \cdot P B=2, P C \cdot P D=18$, and $P B \cdot P C=9$. The area of rectangle $A B C D$ can be expressed as $\frac{a \sqrt{b}}{c}$, where $a$ and $c$ are relatively prime positive integers and $b$ is a squarefree positive integer. Compute $100 a+10 b+c$.

Point $P$ is located inside a square $A B C D$ of side length 10. Let $O_{1}, O_{2}, O_{3}, O_{4}$ be the circumcenters of $P A B, P B C, P C D$, and $P D A$, respectively. Given that $P A+P B+P C+P D=23 \sqrt{2}$ and the area of $O_{1} O_{2} O_{3} O_{4}$ is 50 , the second largest of the lengths $O_{1} O_{2}, O_{2} O_{3}, O_{3} O_{4}, O_{4} O_{1}$ can be written as $\sqrt{\frac{a}{b}}$, where $a$ and $b$ are relatively prime positive integers. Compute $100 a+b$.

Let $\mathcal{E}$ be an ellipse with foci $A$ and $B$. Suppose there exists a parabola $\mathcal{P}$ such that

• $\mathcal{P}$ passes through $A$ and $B$,
• the focus $F$ of $\mathcal{P}$ lies on $\mathcal{E}$,
• the orthocenter $H$ of $\triangle F A B$ lies on the directrix of $\mathcal{P}$.

If the major and minor axes of $\mathcal{E}$ have lengths 50 and 14 , respectively, compute $A H^{2}+B H^{2}$.

Let $A_{1} B_{1} C_{1}, A_{2} B_{2} C_{2}$, and $A_{3} B_{3} C_{3}$ be three triangles in the plane. For $1 \leq i \leq 3$, let $D_{i}, E_{i}$, and $F_{i}$ be the midpoints of $B_{i} C_{i}, A_{i} C_{i}$, and $A_{i} B_{i}$, respectively. Furthermore, for $1 \leq i \leq 3$ let $G_{i}$ be the centroid of $A_{i} B_{i} C_{i}$. Suppose that the areas of the triangles $A_{1} A_{2} A_{3}, B_{1} B_{2} B_{3}, C_{1} C_{2} C_{3}, D_{1} D_{2} D_{3}, E_{1} E_{2} E_{3}$, and $F_{1} F_{2} F_{3}$ are $2,3,4,20,21$, and 2020, respectively. Compute the largest possible area of $G_{1} G_{2} G_{3}$.

Suppose $\omega$ is a circle centered at $O$ with radius 8 . Let $A C$ and $B D$ be perpendicular chords of $\omega$. Let $P$ be a point inside quadrilateral $A B C D$ such that the circumcircles of triangles $A B P$ and $C D P$ are tangent, and the circumcircles of triangles $A D P$ and $B C P$ are tangent. If $A C=2 \sqrt{61}$ and $B D=6 \sqrt{7}$, then $O P$ can be expressed as $\sqrt{a}-\sqrt{b}$ for positive integers $a$ and $b$. Compute $100 a+b$.

Sets $A, B$, and $C$ satisfy $|A|=92,|B|=35,|C|=63,|A \cap B|=16,|A \cap C|=51,|B \cap C|=19$. Compute the number of possible values of $|A \cap B \cap C|$.

Compute the number of ways to color 3 cells in a $3 \times 3$ grid so that no two colored cells share an edge.

Michel starts with the string $H M M T$. An operation consists of either replacing an occurrence of $H$ with $H M$, replacing an occurrence of $M M$ with $M O M$, or replacing an occurrence of $T$ with $M T$. For example, the two strings that can be reached after one operation are $H M M M T$ and $H M O M T$. Compute the number of distinct strings Michel can obtain after exactly 10 operations.

Compute the number of nonempty subsets $S \subseteq\{-10,-9,-8, \ldots, 8,9,10\}$ that satisfy $|S|+\min (S)$. $\max (S)=0$.

Five cards labeled $1,3,5,7,9$ are laid in a row in that order, forming the five-digit number 13579 when read from left to right. A swap consists of picking two distinct cards, and then swapping them. After three swaps, the cards form a new five-digit number $n$ when read from left to right. Compute the expected value of $n$.

The numbers $1,2, \ldots, 10$ are randomly arranged in a circle. Let $p$ be the probability that for every positive integer $k<10$, there exists an integer $k^{\prime}>k$ such that there is at most one number between $k$ and $k^{\prime}$ in the circle. If $p$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$, compute $100 a+b$.

Let $S=\left\{(x, y) \in \mathbb{Z}^{2} \mid 0 \leq x \leq 11,0 \leq y \leq 9\right\}$. Compute the number of sequences $\left(s_{0}, s_{1}, \ldots, s_{n}\right)$ of elements in $S$ (for any positive integer $n \geq 2$ ) that satisfy the following conditions:

• $s_{0}=(0,0)$ and $s_{1}=(1,0)$,
• $s_{0}, s_{1}, \ldots, s_{n}$ are distinct,
• for all integers $2 \leq i \leq n, s_{i}$ is obtained by rotating $s_{i-2}$ about $s_{i-1}$ by either $90^{\circ}$ or $180^{\circ}$ in the clockwise direction.

Random sequences $a_{1}, a_{2}, \ldots$ and $b_{1}, b_{2}, \ldots$ are chosen so that every element in each sequence is chosen independently and uniformly from the set $\{0,1,2,3, \ldots, 100\}$. Compute the expected value of the smallest nonnegative integer $s$ such that there exist positive integers $m$ and $n$ with

$$s=\sum_{i=1}^{m} a_{i}=\sum_{j=1}^{n} b_{j} .$$

Consider permutations $\left(a_{0}, a_{1}, \ldots, a_{2022}\right)$ of $(0,1, \ldots, 2022)$ such that

• $a_{2022}=625$,
• for each $0 \leq i \leq 2022, a_{i} \geq \frac{625 i}{2022}$,
• for each $0 \leq i \leq 2022,\left\{a_{i}, \ldots, a_{2022}\right\}$ is a set of consecutive integers (in some order).

The number of such permutations can be written as $\frac{a!}{b!c!}$ for positive integers $a, b, c$, where $b>c$ and $a$ is minimal. Compute $100 a+10 b+c$.

Let $S$ be a set of size 11. A random 12 -tuple $\left(s_{1}, s_{2}, \ldots, s_{12}\right)$ of elements of $S$ is chosen uniformly at random. Moreover, let $\pi: S \rightarrow S$ be a permutation of $S$ chosen uniformly at random. The probability that $s_{i+1} \neq \pi\left(s_{i}\right)$ for all $1 \leq i \leq 12$ (where $s_{13}=s_{1}$ ) can be written as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Compute $a$.

Let $\left(a_{1}, a_{2}, \ldots, a_{8}\right)$ be a permutation of $(1,2, \ldots, 8)$. Find, with proof, the maximum possible number of elements of the set

$$\left\{a_{1}, a_{1}+a_{2}, \ldots, a_{1}+a_{2}+\cdots+a_{8}\right\}$$

that can be perfect squares.

Find, with proof, the maximum positive integer $k$ for which it is possible to color $6 k$ cells of $6 \times 6$ grid such that, for any choice of three distinct rows $R_{1}, R_{2}, R_{3}$ and three distinct columns $C_{1}, C_{2}, C_{3}$, there exists an uncolored cell $c$ and integers $1 \leq i, j \leq 3$ so that $c$ lies in $R_{i}$ and $C_{j}$.

Let triangle $A B C$ be an acute triangle with circumcircle $\Gamma$. Let $X$ and $Y$ be the midpoints of minor arcs $\widehat{A B}$ and $\widehat{A C}$ of $\Gamma$, respectively. If line $X Y$ is tangent to the incircle of triangle $A B C$ and the radius of $\Gamma$ is $R$, find, with proof, the value of $X Y$ in terms of $R$.

Suppose $n \geq 3$ is a positive integer. Let $a_{1}<a_{2}<\cdots<a_{n}$ be an increasing sequence of positive real numbers, and let $a_{n+1}=a_{1}$. Prove that

$$\sum_{k=1}^{n} \frac{a_{k}}{a_{k+1}}>\sum_{k=1}^{n} \frac{a_{k+1}}{a_{k}} .$$

Let $A B C$ be a triangle with centroid $G$, and let $E$ and $F$ be points on side $B C$ such that $B E=E F=F C$. Points $X$ and $Y$ lie on lines $A B$ and $A C$, respectively, so that $X, Y$, and $G$ are not collinear. If the line through $E$ parallel to $X G$ and the line through $F$ parallel to $Y G$ intersect at $P \neq G$, prove that $G P$ passes through the midpoint of $X Y$.

Let $P(x)=x^{4}+a x^{3}+b x^{2}+x$ be a polynomial with four distinct roots that lie on a circle in the complex plane. Prove that $a b \neq 9$.

Find, with proof, all functions $f: \mathbb{R} \backslash\{0\} \rightarrow \mathbb{R}$ such that

$$f(x)^{2}-f(y) f(z)=x(x+y+z)(f(x)+f(y)+f(z))$$

for all real $x, y, z$ such that $x y z=1$.

Let $P_{1} P_{2} \cdots P_{n}$ be a regular $n$-gon in the plane and $a_{1}, \ldots, a_{n}$ be nonnegative integers. It is possible to draw $m$ circles so that for each $1 \leq i \leq n$, there are exactly $a_{i}$ circles that contain $P_{i}$ on their interior. Find, with proof, the minimum possible value of $m$ in terms of the $a_{i}$.

Let $\Gamma_{1}$ and $\Gamma_{2}$ be two circles externally tangent to each other at $N$ that are both internally tangent to $\Gamma$ at points $U$ and $V$, respectively. A common external tangent of $\Gamma_{1}$ and $\Gamma_{2}$ is tangent to $\Gamma_{1}$ and $\Gamma_{2}$ at $P$ and $Q$, respectively, and intersects $\Gamma$ at points $X$ and $Y$. Let $M$ be the midpoint of the arc $\widehat{X Y}$ that does not contain $U$ and $V$. Let $Z$ be on $\Gamma$ such $M Z \perp N Z$, and suppose the circumcircles of $Q V Z$ and $P U Z$ intersect at $T \neq Z$. Find, with proof, the value of $T U+T V$, in terms of $R, r_{1}$, and $r_{2}$, the radii of $\Gamma, \Gamma_{1}$, and $\Gamma_{2}$, respectively.

On a board the following six vectors are written:

$$(1,0,0), (-1,0,0), (0,1,0), (0,-1,0), (0,0,1), (0,0,-1)$$

Given two vectors $v$ and $w$ on the board, a move consists of erasing $v$ and $w$ and replacing them with $\frac{1}{\sqrt{2}}(v+w)$ and $\frac{1}{\sqrt{2}}(v-w)$. After some number of moves, the sum of the six vectors on the board is $u$. Find, with proof, the maximum possible length of $u$.

A regular 2022-gon has perimeter 6.28. To the nearest positive integer, compute the area of the 2022-gon.

Three distinct vertices are randomly selected among the five vertices of a regular pentagon. Let $p$ be the probability that the triangle formed by the chosen vertices is acute. Compute $10 p$.

Herbert rolls 6 fair standard dice and computes the product of all of his rolls. If the probability that the product is prime can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$, compute $100 a+b$.

For a real number $x$, let $[x]$ be $x$ rounded to the nearest integer and $\langle x\rangle$ be $x$ rounded to the nearest tenth. Real numbers $a$ and $b$ satisfy $\langle a\rangle+[b]=98.6$ and $[a]+\langle b\rangle=99.3$. Compute the minimum possible value of $[10(a+b)]$. (Here, any number equally between two integers or tenths of integers, respectively, is rounded up. For example, $[-4.5]=-4$ and $\langle 4.35\rangle=4.4$.)

Compute the remainder when

10002000400080016003200640128025605121024204840968192 is divided by 100020004000800160032 .

Regular polygons $I C A O, V E N T I$, and $A L B E D O$ lie on a plane. Given that $I N=1$, compute the number of possible values of $O N$.

A jar contains 8 red balls and 2 blue balls. Every minute, a ball is randomly removed. The probability that there exists a time during this process where there are more blue balls than red balls in the jar can be expressed as $\frac{a}{b}$ for relatively prime integers $a$ and $b$. Compute $100 a+b$.

For any positive integer $n$, let $\tau(n)$ denote the number of positive divisors of $n$. If $n$ is a positive integer such that $\frac{\tau\left(n^{2}\right)}{\tau(n)}=3$, compute $\frac{\tau\left(n^{7}\right)}{\tau(n)}$.

An $E$-shape is a geometric figure in the two-dimensional plane consisting of three rays pointing in the same direction, along with a line segment such that

• the endpoints of the rays all lie on the segment,
• the segment is perpendicular to all three rays,
• both endpoints of the segment are endpoints of rays.

Suppose two $E$-shapes intersect each other $N$ times in the plane for some positive integer $N$. Compute the maximum possible value of $N$.

A positive integer $n$ is loose it has six positive divisors and satisfies the property that any two positive divisors $a<b$ of $n$ satisfy $b \geq 2 a$. Compute the sum of all loose positive integers less than 100 .

A regular dodecagon $P_{1} P_{2} \cdots P_{12}$ is inscribed in a unit circle with center $O$. Let $X$ be the intersection of $P_{1} P_{5}$ and $O P_{2}$, and let $Y$ be the intersection of $P_{1} P_{5}$ and $O P_{4}$. Let $A$ be the area of the region bounded by $X Y, X P_{2}, Y P_{4}$, and minor arc $\widehat{P_{2} P_{4}}$. Compute $\lfloor 120 A\rfloor$.

A unit square $A B C D$ and a circle $\Gamma$ have the following property: if $P$ is a point in the plane not contained in the interior of $\Gamma$, then $\min (\angle A P B, \angle B P C, \angle C P D, \angle D P A) \leq 60^{\circ}$. The minimum possible area of $\Gamma$ can be expressed as $\frac{a \pi}{b}$ for relatively prime positive integers $a$ and $b$. Compute $100 a+b$.

Let $z_{1}, z_{2}, z_{3}, z_{4}$ be the solutions to the equation $x^{4}+3 x^{3}+3 x^{2}+3 x+1=0$. Then $\left|z_{1}\right|+\left|z_{2}\right|+\left|z_{3}\right|+\left|z_{4}\right|$ can be written as $\frac{a+b \sqrt{c}}{d}$, where $c$ is a square-free positive integer, and $a, b, d$ are positive integers with $\operatorname{gcd}(a, b, d)=1$. Compute $1000 a+100 b+10 c+d$.

The area of the largest regular hexagon that can fit inside of a rectangle with side lengths 20 and 22 can be expressed as $a \sqrt{b}-c$, for positive integers $a, b$, and $c$, where $b$ is squarefree. Compute $100 a+10 b+c$.

Let $N$ be the number of triples of positive integers $(a, b, c)$ satisfying

$$a \leq b \leq c, \operatorname{gcd}(a, b, c)=1, a b c=6^{2020}$$

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

Let $A B C$ be an acute triangle with $A$-excircle $\Gamma$. Let the line through $A$ perpendicular to $B C$ intersect $B C$ at $D$ and intersect $\Gamma$ at $E$ and $F$. Suppose that $A D=D E=E F$. If the maximum value of $\sin B$ can be expressed as $\frac{\sqrt{a}+\sqrt{b}}{c}$ for positive integers $a, b$, and $c$, compute the minimum possible value of $a+b+c$.

Compute the number of positive real numbers $x$ that satisfy

$$\left(3 \cdot 2^{\left\lfloor\log _{2} x\right\rfloor}-x\right)^{16}=2022 x^{13}$$

Compute the number of permutations $\pi$ of the set $\{1,2, \ldots, 10\}$ so that for all (not necessarily distinct) $m, n \in\{1,2, \ldots, 10\}$ where $m+n$ is prime, $\pi(m)+\pi(n)$ is prime.

In right triangle $A B C$, a point $D$ is on hypotenuse $A C$ such that $B D \perp A C$. Let $\omega$ be a circle with center $O$, passing through $C$ and $D$ and tangent to line $A B$ at a point other than $B$. Point $X$ is chosen on $B C$ such that $A X \perp B O$. If $A B=2$ and $B C=5$, then $B X$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $100 a+b$.

Let $\pi$ be a uniformly random permutation of the set $\{1,2, \ldots, 100\}$. The probability that $\pi^{20}(20)=$ 20 and $\pi^{21}(21)=21$ can be expressed as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $100 a+b$. (Here, $\pi^{k}$ means $\pi$ iterated $k$ times.)

In the Cartesian plane, let $A=(0,0), B=(200,100)$, and $C=(30,330)$. Compute the number of ordered pairs $(x, y)$ of integers so that $\left(x+\frac{1}{2}, y+\frac{1}{2}\right)$ is in the interior of triangle $A B C$.

The function $f(x)$ is of the form $a x^{2}+b x+c$ for some integers $a, b$, and $c$. Given that

$$\begin{aligned} \{f(177883), f(348710), & f(796921), f(858522)\} \\ = & \{1324754875645,1782225466694,1984194627862,4388794883485\} \end{aligned}$$

compute $a$.

Let $A B C D$ be an isosceles trapezoid such that $A B=17, B C=D A=25$, and $C D=31$. Points $P$ and $Q$ are selected on sides $A D$ and $B C$, respectively, such that $A P=C Q$ and $P Q=25$. Suppose that the circle with diameter $P Q$ intersects the sides $A B$ and $C D$ at four points which are vertices of a convex quadrilateral. Compute the area of this quadrilateral.

Let $S_{0}$ be a unit square in the Cartesian plane with horizontal and vertical sides. For any $n>0$, the shape $S_{n}$ is formed by adjoining 9 copies of $S_{n-1}$ in a $3 \times 3$ grid, and then removing the center copy. For example, $S_{3}$ is shown below:

Let $a_{n}$ be the expected value of $\left|x-x^{\prime}\right|+\left|y-y^{\prime}\right|$, where $(x, y)$ and $\left(x^{\prime}, y^{\prime}\right)$ are two points chosen randomly within $S_{n}$. There exist relatively prime positive integers $a$ and $b$ such that

$$\lim _{n \rightarrow \infty} \frac{a_{n}}{3^{n}}=\frac{a}{b} .$$

Compute $100 a+b$.

Let $A B C$ be an acute scalene triangle with circumcenter $O$ and centroid $G$. Given that $A G O$ is a right triangle, $A O=9$, and $B C=15$, let $S$ be the sum of all possible values for the area of triangle $A G O$. Compute $S^{2}$.

Diana is playing a card game against a computer. She starts with a deck consisting of a single card labeled 0.9. Each turn, Diana draws a random card from her deck, while the computer generates a card with a random real number drawn uniformly from the interval [ 0,1$]$. If the number on Diana's card is larger, she keeps her current card and also adds the computer's card to her deck. Otherwise, the computer takes Diana's card. After $k$ turns, Diana's deck is empty. Compute the expected value of $k$.

In three-dimensional space, let $S$ be the region of points $(x, y, z)$ satisfying $-1 \leq z \leq 1$. Let $S_{1}, S_{2}, \ldots, S_{2022}$ be 2022 independent random rotations of $S$ about the origin ( $0,0,0$ ). The expected volume of the region $S_{1} \cap S_{2} \cap \cdots \cap S_{2022}$ can be expressed as $\frac{a \pi}{b}$, for relatively prime positive integers $a$ and $b$. Compute $100 a+b$.

Compute the nearest integer to

$$100 \sum_{n=1}^{\infty} 3^{n} \sin ^{3}\left(\frac{\pi}{3^{n}}\right)$$

Let $a \neq b$ be positive real numbers and $m, n$ be positive integers. An $m+n$-gon $P$ has the property that $m$ sides have length $a$ and $n$ sides have length $b$. Further suppose that $P$ can be inscribed in a circle of radius $a+b$. Compute the number of ordered pairs $(m, n)$, with $m, n \leq 100$, for which such a polygon $P$ exists for some distinct values of $a$ and $b$.

Let $\left(x_{1}, y_{1}\right), \ldots,\left(x_{k}, y_{k}\right)$ be the distinct real solutions to the equation

$$\left(x^{2}+y^{2}\right)^{6}=\left(x^{2}-y^{2}\right)^{4}=\left(2 x^{3}-6 x y^{2}\right)^{3}$$

Then $\sum_{i=1}^{k}\left(x_{i}+y_{i}\right)$ can be expressed as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $100 a+b$.

For a point $P=(x, y)$ in the Cartesian plane, let $f(P)=\left(x^{2}-y^{2}, 2 x y-y^{2}\right)$. If $S$ is the set of all $P$ so that the sequence $P, f(P), f(f(P)), f(f(f(P))), \ldots$ approaches $(0,0)$, then the area of $S$ can be expressed as $\pi \sqrt{r}$ for some positive real number $r$. Compute $\lfloor 100 r\rfloor$.

An ant starts at the point $(0,0)$ in the Cartesian plane. In the first minute, the ant faces towards $(1,0)$ and walks one unit. Each subsequent minute, the ant chooses an angle $\theta$ uniformly at random in the interval $\left[-90^{\circ}, 90^{\circ}\right]$, and then turns an angle of $\theta$ clockwise (negative values of $\theta$ correspond to counterclockwise rotations). Then, the ant walks one unit. After $n$ minutes, the ant's distance from $(0,0)$ is $d_{n}$. Let the expected value of $d_{n}^{2}$ be $a_{n}$. Compute the closest integer to

$$10 \lim _{n \rightarrow \infty} \frac{a_{n}}{n} .$$

In last year's HMMT Spring competition, 557 students submitted at least one answer to each of the three individual tests. Let $S$ be the set of these students, and let $P$ be the set containing the 30 problems on the individual tests. Estimate $A$, the number of subsets $R \subseteq P$ for which some student in $S$ answered the questions in $R$ correctly but no others. An estimate of $E$ earns $\max \left(0,\left\lfloor 20-\frac{2}{3}|A-E|\right\rfloor\right)$ points.

Estimate $A$, the number of unordered triples of integers $(a, b, c)$ so that there exists a nondegenerate triangle with side lengths $a, b$, and $c$ fitting inside a $100 \times 100$ square. An estimate of $E$ earns $\max (0,\lfloor 20-|A-E| / 1000\rfloor)$ points.

A random permutation of $\{1,2, \ldots, 100\}$ is given. It is then sorted to obtain the sequence $(1,2, \ldots, 100)$ as follows: at each step, two of the numbers which are not in their correct positions are selected at random, and the two numbers are swapped. If $s$ is the expected number of steps (i.e. swaps) required to obtain the sequence $(1,2, \cdots, 100)$, then estimate $A=\lfloor s\rfloor$. An estimate of $E$ earns $\max \left(0,\left\lfloor 20-\frac{1}{2}|A-E|\right\rfloor\right)$ points.

For a cubic polynomial $P(x)$ with complex roots $z_{1}, z_{2}, z_{3}$, let

$$M(P)=\frac{\max \left(\left|z_{1}-z_{2}\right|,\left|z_{1}-z_{3}\right|,\left|z_{2}-z_{3}\right|\right)}{\min \left(\left|z_{1}-z_{2}\right|,\left|z_{1}-z_{3}\right|,\left|z_{2}-z_{3}\right|\right)}$$

Over all polynomials $P(x)=x^{3}+a x^{2}+b x+c$, where $a, b, c$ are nonnegative integers at most 100 and $P(x)$ has no repeated roots, the twentieth largest possible value of $M(P)$ is $m$. Estimate $A=\lfloor m\rfloor$. An estimate of $E$ earns $\max \left(0,\left\lfloor 20-20|3 \ln (A / E)|^{1 / 2}\right\rfloor\right)$ points.