HMMT 2010

Suppose that $x$ and $y$ are positive reals such that

$$x-y^{2}=3, x^{2}+y^{4}=13$$

Find $x$.

The rank of a rational number $q$ is the unique $k$ for which $q=\frac{1}{a_{1}}+\cdots+\frac{1}{a_{k}}$, where each $a_{i}$ is the smallest positive integer such that $q \geq \frac{1}{a_{1}}+\cdots+\frac{1}{a_{i}}$. Let $q$ be the largest rational number less than $\frac{1}{4}$ with rank 3 , and suppose the expression for $q$ is $\frac{1}{a_{1}}+\frac{1}{a_{2}}+\frac{1}{a_{3}}$. Find the ordered triple $\left(a_{1}, a_{2}, a_{3}\right)$.

Let $S_{0}=0$ and let $S_{k}$ equal $a_{1}+2 a_{2}+\ldots+k a_{k}$ for $k \geq 1$. Define $a_{i}$ to be 1 if $S_{i-1}<i$ and -1 if $S_{i-1} \geq i$. What is the largest $k \leq 2010$ such that $S_{k}=0$ ?

Suppose that there exist nonzero complex numbers $a, b, c$, and $d$ such that $k$ is a root of both the equations $a x^{3}+b x^{2}+c x+d=0$ and $b x^{3}+c x^{2}+d x+a=0$. Find all possible values of $k$ (including complex values).

Suppose that $x$ and $y$ are complex numbers such that $x+y=1$ and that $x^{20}+y^{20}=20$. Find the sum of all possible values of $x^{2}+y^{2}$.

Suppose that a polynomial of the form $p(x)=x^{2010} \pm x^{2009} \pm \cdots \pm x \pm 1$ has no real roots. What is the maximum possible number of coefficients of -1 in $p$ ?

Let $a, b, c, x, y$, and $z$ be complex numbers such that

$$a=\frac{b+c}{x-2}, b=\frac{c+a}{y-2}, c=\frac{a+b}{z-2} .$$

If $x y+y z+z x=67$ and $x+y+z=2010$, find the value of $x y z$.

How many polynomials of degree exactly 5 with real coefficients send the set $\{1,2,3,4,5,6\}$ to a permutation of itself?

Let $f(x)=c x(x-1)$, where $c$ is a positive real number. We use $f^{n}(x)$ to denote the polynomial obtained by composing $f$ with itself $n$ times. For every positive integer $n$, all the roots of $f^{n}(x)$ are real. What is the smallest possible value of $c$ ?

Let $p(x)$ and $q(x)$ be two cubic polynomials such that $p(0)=-24, q(0)=30$, and $$p(q(x))=q(p(x))$$ for all real numbers $x$. Find the ordered pair $(p(3), q(6))$.

Below is pictured a regular seven-pointed star. Find the measure of angle $a$ in radians.

A rectangular piece of paper is folded along its diagonal (as depicted below) to form a non-convex pentagon that has an area of $\frac{7}{10}$ of the area of the original rectangle. Find the ratio of the longer side of the rectangle to the shorter side of the rectangle.

For $0 \leq y \leq 2$, let $D_{y}$ be the half-disk of diameter 2 with one vertex at $(0, y)$, the other vertex on the positive $x$-axis, and the curved boundary further from the origin than the straight boundary. Find the area of the union of $D_{y}$ for all $0 \leq y \leq 2$.

Let $O$ be the point $(0,0)$. Let $A, B, C$ be three points in the plane such that $A O=15, B O=15$, and $C O=7$, and such that the area of triangle $A B C$ is maximal. What is the length of the shortest side of $A B C$ ?

Let $A B C D$ be a quadrilateral with an inscribed circle centered at $I$. Let $C I$ intersect $A B$ at $E$. If $\angle I D E=35^{\circ}, \angle A B C=70^{\circ}$, and $\angle B C D=60^{\circ}$, then what are all possible measures of $\angle C D A$ ?

Circles $\omega_{1}$ and $\omega_{2}$ intersect at points $A$ and $B$. Segment $P Q$ is tangent to $\omega_{1}$ at $P$ and to $\omega_{2}$ at $Q$, and $A$ is closer to $P Q$ than $B$. Point $X$ is on $\omega_{1}$ such that $P X \| Q B$, and point $Y$ is on $\omega_{2}$ such that $Q Y \| P B$. Given that $\angle A P Q=30^{\circ}$ and $\angle P Q A=15^{\circ}$, find the ratio $A X / A Y$.

Let $S=\{1,2,3,4,5,6,7,8,9,10\}$. How many (potentially empty) subsets $T$ of $S$ are there such that, for all $x$, if $x$ is in $T$ and $2 x$ is in $S$ then $2 x$ is also in $T$ ?

How many positive integers less than or equal to 240 can be expressed as a sum of distinct factorials? Consider 0! and 1! to be distinct.

How many ways are there to choose 2010 functions $f_{1}, \ldots, f_{2010}$ from $\{0,1\}$ to $\{0,1\}$ such that $f_{2010} \circ f_{2009} \circ \cdots \circ f_{1}$ is constant? Note: a function $g$ is constant if $g(a)=g(b)$ for all $a, b$ in the domain of $g$.

Manya has a stack of $85=1+4+16+64$ blocks comprised of 4 layers (the $k$ th layer from the top has $4^{k-1}$ blocks; see the diagram below). Each block rests on 4 smaller blocks, each with dimensions half those of the larger block. Laura removes blocks one at a time from this stack, removing only blocks that currently have no blocks on top of them. Find the number of ways Laura can remove precisely 5 blocks from Manya's stack (the order in which they are removed matters).

John needs to pay 2010 dollars for his dinner. He has an unlimited supply of 2, 5, and 10 dollar notes. In how many ways can he pay?

An ant starts out at $(0,0)$. Each second, if it is currently at the square $(x, y)$, it can move to $(x-1, y-1),(x-1, y+1),(x+1, y-1)$, or $(x+1, y+1)$. In how many ways can it end up at $(2010,2010)$ after 4020 seconds?

For each integer $x$ with $1 \leq x \leq 10$, a point is randomly placed at either $(x, 1)$ or $(x,-1)$ with equal probability. What is the expected area of the convex hull of these points? Note: the convex hull of a finite set is the smallest convex polygon containing it.

How many functions $f$ from $\{-1005, \ldots, 1005\}$ to $\{-2010, \ldots, 2010\}$ are there such that the following two conditions are satisfied?

• If $a<b$ then $f(a)<f(b)$.
• There is no $n$ in $\{-1005, \ldots, 1005\}$ such that $|f(n)|=|n|$.

Rosencrantz and Guildenstern are playing a game where they repeatedly flip coins. Rosencrantz wins if 1 heads followed by 2009 tails appears. Guildenstern wins if 2010 heads come in a row. They will flip coins until someone wins. What is the probability that Rosencrantz wins?

In a $16 \times 16$ table of integers, each row and column contains at most 4 distinct integers. What is the maximum number of distinct integers that there can be in the whole table?

Suppose that $p(x)$ is a polynomial and that $p(x)-p^{\prime}(x)=x^{2}+2 x+1$. Compute $p(5)$.

Let $f$ be a function such that $f(0)=1, f^{\prime}(0)=2$, and

$$f^{\prime \prime}(t)=4 f^{\prime}(t)-3 f(t)+1$$

for all $t$. Compute the 4th derivative of $f$, evaluated at 0 .

Let $p$ be a monic cubic polynomial such that $p(0)=1$ and such that all the zeros of $p^{\prime}(x)$ are also zeros of $p(x)$. Find $p$. Note: monic means that the leading coefficient is 1 .

Compute $\lim _{n \rightarrow \infty} \frac{\sum_{k=1}^{n}|\cos (k)|}{n}$.

Let the functions $f(\alpha, x)$ and $g(\alpha)$ be defined as

$$f(\alpha, x)=\frac{\left(\frac{x}{2}\right)^{\alpha}}{x-1} g(\alpha)=\left.\frac{d^{4} f}{d x^{4}}\right|_{x=2}$$

Then $g(\alpha)$ is a polynomial in $\alpha$. Find the leading coefficient of $g(\alpha)$.

Let $f(x)=x^{3}-x^{2}$. For a given value of $c$, the graph of $f(x)$, together with the graph of the line $c+x$, split the plane up into regions. Suppose that $c$ is such that exactly two of these regions have finite area. Find the value of $c$ that minimizes the sum of the areas of these two regions.

Let $a_{1}, a_{2}$, and $a_{3}$ be nonzero complex numbers with non-negative real and imaginary parts. Find the minimum possible value of

$$\frac{\left|a_{1}+a_{2}+a_{3}\right|}{\sqrt[3]{\left|a_{1} a_{2} a_{3}\right|}}$$

Let $f(n)=\sum_{k=2}^{\infty} \frac{1}{k^{n} \cdot k!}$. Calculate $\sum_{n=2}^{\infty} f(n)$.

Let $x(t)$ be a solution to the differential equation

$$\left(x+x^{\prime}\right)^{2}+x \cdot x^{\prime \prime}=\cos t$$

with $x(0)=x^{\prime}(0)=\sqrt{\frac{2}{5}}$. Compute $x\left(\frac{\pi}{4}\right)$.

Let $f(n)=\sum_{k=1}^{n} \frac{1}{k}$. Then there exists constants $\gamma, c$, and $d$ such that

$$f(n)=\ln (n)+\gamma+\frac{c}{n}+\frac{d}{n^{2}}+O\left(\frac{1}{n^{3}}\right)$$

where the $O\left(\frac{1}{n^{3}}\right)$ means terms of order $\frac{1}{n^{3}}$ or lower. Compute the ordered pair $(c, d)$.

A rectangular piece of paper is folded along its diagonal (as depicted below) to form a non-convex pentagon that has an area of $\frac{7}{10}$ of the area of the original rectangle. Find the ratio of the longer side of the rectangle to the shorter side of the rectangle.

Manya has a stack of $85=1+4+16+64$ blocks comprised of 4 layers (the $k$ th layer from the top has $4^{k-1}$ blocks; see the diagram below). Each block rests on 4 smaller blocks, each with dimensions half those of the larger block. Laura removes blocks one at a time from this stack, removing only blocks that currently have no blocks on top of them. Find the number of ways Laura can remove precisely 5 blocks from Manya's stack (the order in which they are removed matters).

A sphere is the set of points at a fixed positive distance $r$ from its center. Let $\mathcal{S}$ be a set of 2010dimensional spheres. Suppose that the number of points lying on every element of $\mathcal{S}$ is a finite number $n$. Find the maximum possible value of $n$.

Three unit circles $\omega_{1}, \omega_{2}$, and $\omega_{3}$ in the plane have the property that each circle passes through the centers of the other two. A square $S$ surrounds the three circles in such a way that each of its four sides is tangent to at least one of $\omega_{1}, \omega_{2}$ and $\omega_{3}$. Find the side length of the square $S$.

Below is pictured a regular seven-pointed star. Find the measure of angle $a$ in radians.

For $0 \leq y \leq 2$, let $D_{y}$ be the half-disk of diameter 2 with one vertex at $(0, y)$, the other vertex on the positive $x$-axis, and the curved boundary further from the origin than the straight boundary. Find the area of the union of $D_{y}$ for all $0 \leq y \leq 2$.

Suppose that there exist nonzero complex numbers $a, b, c$, and $d$ such that $k$ is a root of both the equations $a x^{3}+b x^{2}+c x+d=0$ and $b x^{3}+c x^{2}+d x+a=0$. Find all possible values of $k$ (including complex values).

Let $A B C D$ be an isosceles trapezoid such that $A B=10, B C=15, C D=28$, and $D A=15$. There is a point $E$ such that $\triangle A E D$ and $\triangle A E B$ have the same area and such that $E C$ is minimal. Find $E C$.

You are standing in an infinitely long hallway with sides given by the lines $x=0$ and $x=6$. You start at $(3,0)$ and want to get to $(3,6)$. Furthermore, at each instant you want your distance to $(3,6)$ to either decrease or stay the same. What is the area of the set of points that you could pass through on your journey from $(3,0)$ to $(3,6)$ ?

You are trying to sink a submarine. Every second, you launch a missile at a point of your choosing on the $x$-axis. If the submarine is at that point at that time, you sink it. A firing sequence is a sequence of real numbers that specify where you will fire at each second. For example, the firing sequence $2,3,5,6, \ldots$ means that you will fire at 2 after one second, 3 after two seconds, 5 after three seconds, 6 after four seconds, and so on. (a) [5] Suppose that the submarine starts at the origin and travels along the positive $x$-axis with an (unknown) positive integer velocity. Show that there is a firing sequence that is guaranteed to hit the submarine eventually.

Consider the following two-player game. Player 1 starts with a number, $N$. He then subtracts a proper divisor of $N$ from $N$ and gives the result to player 2 (a proper divisor of $N$ is a positive divisor of $N$ that is not equal to 1 or $N)$. Player 2 does the same thing with the number she gets from player 1 , and gives the result back to player 1. The two players continue until a player is given a prime number or 1 , at which point that player loses. For which values of $N$ does player 1 have a winning strategy?

Call a positive integer in base $10 k$-good if we can split it into two integers y and z , such that y is all digits on the left and z is all digits on the right, and such that $y=k \cdot z$. For example, 2010 is 2 -good because we can split it into 20 and 10 and $20=2 \cdot 10.20010$ is also 2 -good, because we can split it into 20 and 010 . In addition, it is 20 -good, because we can split it into 200 and 10 . Show that there exists a 48 -good perfect square.

Let

$$\begin{gathered} e^{x}+e^{y}=A \\ x e^{x}+y e^{y}=B \\ x^{2} e^{x}+y^{2} e^{y}=C \\ x^{3} e^{x}+y^{3} e^{y}=D \\ x^{4} e^{x}+y^{4} e^{y}=E . \end{gathered}$$

Prove that if $A, B, C$, and $D$ are all rational, then so is $E$.

Show that, for every positive integer $n$, there exists a monic polynomial of degree $n$ with integer coefficients such that the coefficients are decreasing and the roots of the polynomial are all integers.

Let $S$ be a convex set in the plane with a finite area $a$. Prove that either $a=0$ or $S$ is bounded. Note: a set is bounded if it is contained in a circle of finite radius. Note: a set is convex if, whenever two points $A$ and $B$ are in the set, the line segment between them is also in the set.

Point $P$ lies inside a convex pentagon $A F Q D C$ such that $F P D Q$ is a parallelogram. Given that $\angle F A Q=\angle P A C=10^{\circ}$, and $\angle P F A=\angle P D C=15^{\circ}$. What is $\angle A Q C ?$

A knight moves on a two-dimensional grid. From any square, it can move 2 units in one axisparallel direction, then move 1 unit in an orthogonal direction, the way a regular knight moves in a game of chess. The knight starts at the origin. As it moves, it keeps track of a number $t$, which is initially 0 . When the knight lands at the point $(a, b)$, the number is changed from $x$ to $a x+b$.

Show that, for any integers $a$ and $b$, it is possible for the knight to land at the points $(1, a)$ and $(-1, a)$ with $t$ equal to $b$.

Let $p(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\ldots+a_{0}$ be a polynomial with complex coefficients such that $a_{i} \neq 0$ for all $i$. Prove that $|r| \leq 2 \max _{i=0}^{n-1}\left|\frac{a_{i-1}}{a_{i}}\right|$ for all roots $r$ of all such polynomials $p$. Here we let $|z|$ denote the absolute value of the complex number $z$.

Call an $2 n$-digit base-10 number special if we can split its digits into two sets of size $n$ such that the sum of the numbers in the two sets is the same. Let $p_{n}$ be the probability that a randomly-chosen $2 n$-digit number is special. (We allow leading zeros in $2 n$-digit numbers). (a) [20] The sequence $p_{n}$ converges to a constant $c$. Find $c$.

How many ways are there to place pawns on an $8 \times 8$ chessboard, so that there is at most 1 pawn in each horizontal row? Express your answer in the form $p_{1}^{e_{1}} \cdot p_{2}^{e_{2}} \cdots$, where the $p_{i}$ are distinct primes and the $e_{i}$ are positive integers.

In the following figure, a regular hexagon of side length 1 is attached to a semicircle of diameter 1. What is the longest distance between any two points in the figure?

Let $p(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\ldots+a_{0}$, where each $a_{i}$ is either 1 or -1 . Let $r$ be a root of $p$. If $|r|>\frac{15}{8}$, what is the minimum possible value of $n$ ?

Find all 4-digit integers of the form $a a b b$ (when written in base 10) that are perfect squares.

Compute

$$\sum_{n=1}^{98} \frac{2}{\sqrt{n}+\sqrt{n+2}}+\frac{1}{\sqrt{n+1}+\sqrt{n+2}}$$

Into how many regions can a circle be cut by 10 parabolas?

Evaluate $$\sum_{k=1}^{2010} \cos ^{2}(k)$$

Consider the following two-player game. Player 1 starts with a number, $N$. He then subtracts a proper divisor of $N$ from $N$ and gives the result to player 2 (a proper divisor of $N$ is a positive divisor of $N$ that is not equal to 1 or $N)$. Player 2 does the same thing with the number she gets from player 1 , and gives the result back to player 1. The two players continue until a player is given a prime number, at which point that player loses. For how many values of $N$ between 2 and 100 inclusive does player 1 have a winning strategy?

Let $S$ be the set of ordered pairs of integers $(x, y)$ with $1 \leq x \leq 5$ and $1 \leq y \leq 3$. How many subsets $R$ of $S$ have the property that all the points of $R$ lie on the graph of a single cubic? A cubic is a polynomial of the form $y=a x^{3}+b x^{2}+c x+d$, where $a, b, c$, and $d$ are real numbers (meaning that $a$ is allowed to be 0 ).

Call an $2 n$-digit number special if we can split its digits into two sets of size $n$ such that the sum of the numbers in the two sets is the same. Let $p_{n}$ be the probability that a randomly-chosen $2 n$-digit number is special (we will allow leading zeros in the number). (a) [25] The sequence $p_{n}$ converges to a constant $c$. Find $c$.

If $A=10^{9}-987654321$ and $B=\frac{123456789+1}{10}$, what is the value of $\sqrt{A B}$ ?

Suppose that $x, y$, and $z$ are non-negative real numbers such that $x+y+z=1$. What is the maximum possible value of $x+y^{2}+z^{3}$ ?

In a group of people, there are 13 who like apples, 9 who like blueberries, 15 who like cantaloupe, and 6 who like dates. (A person can like more than 1 kind of fruit.) Each person who likes blueberries also likes exactly one of apples and cantaloupe. Each person who likes cantaloupe also likes exactly one of blueberries and dates. Find the minimum possible number of people in the group.

To set up for a Fourth of July party, David is making a string of red, white, and blue balloons. He places them according to the following rules:

• No red balloon is adjacent to another red balloon.
• White balloons appear in groups of exactly two, and groups of white balloons are separated by at least two non-white balloons.
• Blue balloons appear in groups of exactly three, and groups of blue balloons are separated by at least three non-blue balloons.

If David uses over 600 balloons, determine the smallest number of red balloons that he can use.

You have a length of string and 7 beads in the 7 colors of the rainbow. You place the beads on the string as follows - you randomly pick a bead that you haven't used yet, then randomly add it to either the left end or the right end of the string. What is the probability that, at the end, the colors of the beads are the colors of the rainbow in order? (The string cannot be flipped, so the red bead must appear on the left side and the violet bead on the right side.)

How many different numbers are obtainable from five 5 s by first concatenating some of the 5 s, then multiplying them together? For example, we could do $5 \cdot 55 \cdot 55,555 \cdot 55$, or 55555 , but not $5 \cdot 5$ or 2525 .

What are the last 8 digits of

$$11 \times 101 \times 1001 \times 10001 \times 100001 \times 1000001 \times 111 ?$$

Each square in the following hexomino has side length 1. Find the minimum area of any rectangle that contains the entire hexomino.

Indecisive Andy starts out at the midpoint of the 1-unit-long segment $\overline{H T}$. He flips 2010 coins. On each flip, if the coin is heads, he moves halfway towards endpoint $H$, and if the coin is tails, he moves halfway towards endpoint $T$. After his 2010 moves, what is the expected distance between Andy and the midpoint of $\overline{H T}$ ?

Let $A B C$ be a triangle with $A B=8, B C=15$, and $A C=17$. Point $X$ is chosen at random on line segment $A B$. Point $Y$ is chosen at random on line segment $B C$. Point $Z$ is chosen at random on line segment $C A$. What is the expected area of triangle $X Y Z$ ?

From the point $(x, y)$, a legal move is a move to $\left(\frac{x}{3}+u, \frac{y}{3}+v\right)$, where $u$ and $v$ are real numbers such that $u^{2}+v^{2} \leq 1$. What is the area of the set of points that can be reached from $(0,0)$ in a finite number of legal moves?

How many different collections of 9 letters are there? A letter can appear multiple times in a collection. Two collections are equal if each letter appears the same number of times in both collections.

A triangle in the $x y$-plane is such that when projected onto the $x$-axis, $y$-axis, and the line $y=x$, the results are line segments whose endpoints are $(1,0)$ and $(5,0),(0,8)$ and $(0,13)$, and $(5,5)$ and $(7.5,7.5)$, respectively. What is the triangle's area?

In how many ways can you fill a $3 \times 3$ table with the numbers 1 through 9 (each used once) such that all pairs of adjacent numbers (sharing one side) are relatively prime?

Pick a random integer between 0 and 4095, inclusive. Write it in base 2 (without any leading zeroes). What is the expected number of consecutive digits that are not the same (that is, the expected number of occurrences of either 01 or 10 in the base 2 representation)?

Jessica has three marbles colored red, green, and blue. She randomly selects a non-empty subset of them (such that each subset is equally likely) and puts them in a bag. You then draw three marbles from the bag with replacement. The colors you see are red, blue, red. What is the probability that the only marbles in the bag are red and blue?

An ant starts at the origin, facing in the positive $x$-direction. Each second, it moves 1 unit forward, then turns counterclockwise by $\sin ^{-1}\left(\frac{3}{5}\right)$ degrees. What is the least upper bound on the distance between the ant and the origin? (The least upper bound is the smallest real number $r$ that is at least as big as every distance that the ant ever is from the origin.)

Find two lines of symmetry of the graph of the function $y=x+\frac{1}{x}$. Express your answer as two equations of the form $y=a x+b$.

A 5-dimensional ant starts at one vertex of a 5 -dimensional hypercube of side length 1 . A move is when the ant travels from one vertex to another vertex at a distance of $\sqrt{2}$ away. How many ways can the ant make 5 moves and end up on the same vertex it started at?

Find the volume of the set of points $(x, y, z)$ satisfying

$$\begin{aligned} x, y, z & \geq 0 \\ x+y & \leq 1 \\ y+z & \leq 1 \\ z+x & \leq 1 \end{aligned}$$

Let $\triangle A B C$ be a scalene triangle. Let $h_{a}$ be the locus of points $P$ such that $|P B-P C|=|A B-A C|$. Let $h_{b}$ be the locus of points $P$ such that $|P C-P A|=|B C-B A|$. Let $h_{c}$ be the locus of points $P$ such that $|P A-P B|=|C A-C B|$. In how many points do all of $h_{a}, h_{b}$, and $h_{c}$ concur?

You are the general of an army. You and the opposing general both have an equal number of troops to distribute among three battlefields. Whoever has more troops on a battlefield always wins (you win ties). An order is an ordered triple of non-negative real numbers $(x, y, z)$ such that $x+y+z=1$, and corresponds to sending a fraction $x$ of the troops to the first field, $y$ to the second, and $z$ to the third. Suppose that you give the order $\left(\frac{1}{4}, \frac{1}{4}, \frac{1}{2}\right)$ and that the other general issues an order chosen uniformly at random from all possible orders. What is the probability that you win two out of the three battles?

In the country of Francisca, there are 2010 cities, some of which are connected by roads. Between any two cities, there is a unique path which runs along the roads and which does not pass through any city twice. What is the maximum possible number of cities in Francisca which have at least 3 roads running out of them?

Define a sequence of polynomials as follows: let $a_{1}=3 x^{2}-x$, let $a_{2}=3 x^{2}-7 x+3$, and for $n \geq 1$, let $a_{n+2}=\frac{5}{2} a_{n+1}-a_{n}$. As $n$ tends to infinity, what is the limit of the sum of the roots of $a_{n}$ ?

How many functions $f:\{1,2,3,4,5\} \rightarrow\{1,2,3,4,5\}$ have the property that $f(\{1,2,3\})$ and $f(f(\{1,2,3\}))$ are disjoint?

Suppose that there are real numbers $a, b, c \geq 1$ and that there are positive reals $x, y, z$ such that

$$\begin{aligned} a^{x}+b^{y}+c^{z} & =4 \\ x a^{x}+y b^{y}+z c^{z} & =6 \\ x^{2} a^{x}+y^{2} b^{y}+z^{2} c^{z} & =9 \end{aligned}$$

What is the maximum possible value of $c$ ?

Danielle Bellatrix Robinson is organizing a poker tournament with 9 people. The tournament will have 4 rounds, and in each round the 9 players are split into 3 groups of 3 . During the tournament, each player plays every other player exactly once. How many different ways can Danielle divide the 9 people into three groups in each round to satisfy these requirements?

Compute the remainder when

$$\sum_{k=1}^{30303} k^{k}$$

is divided by 101 .

A monomial term $x_{i_{1}} x_{i_{2}} \ldots x_{i_{k}}$ in the variables $x_{1}, x_{2}, \ldots x_{8}$ is square-free if $i_{1}, i_{2}, \ldots i_{k}$ are distinct. (A constant term such as 1 is considered square-free.) What is the sum of the coefficients of the squarefree terms in the following product?

$$\prod_{1 \leq i<j \leq 8}\left(1+x_{i} x_{j}\right)$$

In the Democratic Republic of Irun, 5 people are voting in an election among 5 candidates. If each person votes for a single candidate at random, what is the expected number of candidates that will be voted for?

There are 101 people participating in a Secret Santa gift exchange. As usual each person is randomly assigned another person for whom (s)he has to get a gift, such that each person gives and receives exactly one gift and no one gives a gift to themself. What is the probability that the first person neither gives gifts to or receives gifts from the second or third person? Express your answer as a decimal rounded to five decimal places.

Let $a_{1}=3$, and for $n>1$, let $a_{n}$ be the largest real number such that

$$4\left(a_{n-1}^{2}+a_{n}^{2}\right)=10 a_{n-1} a_{n}-9$$

What is the largest positive integer less than $a_{8}$ ?

3000 people each go into one of three rooms randomly. What is the most likely value for the maximum number of people in any of the rooms? Your score for this problem will be 0 if you write down a number less than or equal to 1000 . Otherwise, it will be $25-27 \frac{|A-C|}{\min (A, C)-1000}$.

Call an positive integer almost-square if it can be written as $a \cdot b$, where $a$ and $b$ are integers and $a \leq b \leq \frac{4}{3} a$. How many almost-square positive integers are less than or equal to 1000000 ? Your score will be equal to $25-65 \frac{|A-C|}{\min (A, C)}$.

Consider an infinite grid of unit squares. An $n$-omino is a subset of $n$ squares that is connected. Below are depicted examples of 8 -ominoes. Two $n$-ominoes are considered equivalent if one can be obtained from the other by translations and rotations. What is the number of distinct 15 -ominoes? Your score will be equal to $25-13|\ln (A)-\ln (C)|$.