HMMT 2008

Positive real numbers $x, y$ satisfy the equations $x^{2}+y^{2}=1$ and $x^{4}+y^{4}=\frac{17}{18}$. Find $x y$.

Let $f(n)$ be the number of times you have to hit the $\sqrt{ }$ key on a calculator to get a number less than 2 starting from $n$. For instance, $f(2)=1, f(5)=2$. For how many $1<m<2008$ is $f(m)$ odd?

Determine all real numbers $a$ such that the inequality $\left|x^{2}+2 a x+3 a\right| \leq 2$ has exactly one solution in $x$.

The function $f$ satisfies

$$f(x)+f(2 x+y)+5 x y=f(3 x-y)+2 x^{2}+1$$

for all real numbers $x, y$. Determine the value of $f(10)$.

Let $f(x)=x^{3}+x+1$. Suppose $g$ is a cubic polynomial such that $g(0)=-1$, and the roots of $g$ are the squares of the roots of $f$. Find $g(9)$.

A root of unity is a complex number that is a solution to $z^{n}=1$ for some positive integer $n$. Determine the number of roots of unity that are also roots of $z^{2}+a z+b=0$ for some integers $a$ and $b$.

Compute $\sum_{n=1}^{\infty} \sum_{k=1}^{n-1} \frac{k}{2^{n+k}}$.

Compute $\arctan \left(\tan 65^{\circ}-2 \tan 40^{\circ}\right)$. (Express your answer in degrees as an angle between $0^{\circ}$ and $180^{\circ}$.)

Let $S$ be the set of points $(a, b)$ with $0 \leq a, b \leq 1$ such that the equation

$$x^{4}+a x^{3}-b x^{2}+a x+1=0$$

has at least one real root. Determine the area of the graph of $S$.

Evaluate the infinite sum

$$\sum_{n=0}^{\infty}\binom{2 n}{n} \frac{1}{5^{n}}$$

How many different values can $\angle A B C$ take, where $A, B, C$ are distinct vertices of a cube?

Let $A B C$ be an equilateral triangle. Let $\Omega$ be its incircle (circle inscribed in the triangle) and let $\omega$ be a circle tangent externally to $\Omega$ as well as to sides $A B$ and $A C$. Determine the ratio of the radius of $\Omega$ to the radius of $\omega$.

Let $A B C$ be a triangle with $\angle B A C=90^{\circ}$. A circle is tangent to the sides $A B$ and $A C$ at $X$ and $Y$ respectively, such that the points on the circle diametrically opposite $X$ and $Y$ both lie on the side $B C$. Given that $A B=6$, find the area of the portion of the circle that lies outside the triangle.

A piece of paper is folded in half. A second fold is made such that the angle marked below has measure $\phi\left(0^{\circ}<\phi<90^{\circ}\right)$, and a cut is made as shown below.

When the piece of paper is unfolded, the resulting hole is a polygon. Let $O$ be one of its vertices. Suppose that all the other vertices of the hole lie on a circle centered at $O$, and also that $\angle X O Y=144^{\circ}$, where $X$ and $Y$ are the the vertices of the hole adjacent to $O$. Find the value(s) of $\phi$ (in degrees).

Let $A B C$ be a triangle with $\angle A=45^{\circ}$. Let $P$ be a point on side $B C$ with $P B=3$ and $P C=5$. Let $O$ be the circumcenter of $A B C$. Determine the length $O P$.

Let $C_{1}$ and $C_{2}$ be externally tangent circles with radius 2 and 3 , respectively. Let $C_{3}$ be a circle internally tangent to both $C_{1}$ and $C_{2}$ at points $A$ and $B$, respectively. The tangents to $C_{3}$ at $A$ and $B$ meet at $T$, and $T A=4$. Determine the radius of $C_{3}$.

Let $A B C$ be a triangle, and $I$ its incenter. Let the incircle of $A B C$ touch side $B C$ at $D$, and let lines $B I$ and $C I$ meet the circle with diameter $A I$ at points $P$ and $Q$, respectively. Given $B I=$ $6, C I=5, D I=3$, determine the value of $(D P / D Q)^{2}$.

Let $A B C$ be a triangle with $B C=2007, C A=2008, A B=2009$. Let $\omega$ be an excircle of $A B C$ that touches the line segment $B C$ at $D$, and touches extensions of lines $A C$ and $A B$ at $E$ and $F$, respectively (so that $C$ lies on segment $A E$ and $B$ lies on segment $A F$ ). Let $O$ be the center of $\omega$. Let $\ell$ be the line through $O$ perpendicular to $A D$. Let $\ell$ meet line $E F$ at $G$. Compute the length $D G$.

A $3 \times 3 \times 3$ cube composed of 27 unit cubes rests on a horizontal plane. Determine the number of ways of selecting two distinct unit cubes from a $3 \times 3 \times 1$ block (the order is irrelevant) with the property that the line joining the centers of the two cubes makes a $45^{\circ}$ angle with the horizontal plane.

Let $S=\{1,2, \ldots, 2008\}$. For any nonempty subset $A \subset S$, define $m(A)$ to be the median of $A$ (when $A$ has an even number of elements, $m(A)$ is the average of the middle two elements). Determine the average of $m(A)$, when $A$ is taken over all nonempty subsets of $S$.

Farmer John has 5 cows, 4 pigs, and 7 horses. How many ways can he pair up the animals so that every pair consists of animals of different species? Assume that all animals are distinguishable from each other. (Please write your answer as an integer, without any incomplete computations.)

Kermit the frog enjoys hopping around the infinite square grid in his backyard. It takes him 1 Joule of energy to hop one step north or one step south, and 1 Joule of energy to hop one step east or one step west. He wakes up one morning on the grid with 100 Joules of energy, and hops till he falls asleep with 0 energy. How many different places could he have gone to sleep?

Let $S$ be the smallest subset of the integers with the property that $0 \in S$ and for any $x \in S$, we have $3 x \in S$ and $3 x+1 \in S$. Determine the number of non-negative integers in $S$ less than 2008.

A Sudoku matrix is defined as a $9 \times 9$ array with entries from $\{1,2, \ldots, 9\}$ and with the constraint that each row, each column, and each of the nine $3 \times 3$ boxes that tile the array contains each digit from 1 to 9 exactly once. A Sudoku matrix is chosen at random (so that every Sudoku matrix has equal probability of being chosen). We know two of squares in this matrix, as shown. What is the probability that the square marked by? contains the digit 3 ?

Let $P_{1}, P_{2}, \ldots, P_{8}$ be 8 distinct points on a circle. Determine the number of possible configurations made by drawing a set of line segments connecting pairs of these 8 points, such that: (1) each $P_{i}$ is the endpoint of at most one segment and (2) two no segments intersect. (The configuration with no edges drawn is allowed. An example of a valid configuration is shown below.)

Determine the number of ways to select a sequence of 8 sets $A_{1}, A_{2}, \ldots, A_{8}$, such that each is a subset (possibly empty) of $\{1,2\}$, and $A_{m}$ contains $A_{n}$ if $m$ divides $n$.

On an infinite chessboard (whose squares are labeled by $(x, y)$, where $x$ and $y$ range over all integers), a king is placed at $(0,0)$. On each turn, it has probability of 0.1 of moving to each of the four edgeneighboring squares, and a probability of 0.05 of moving to each of the four diagonally-neighboring squares, and a probability of 0.4 of not moving. After 2008 turns, determine the probability that the king is on a square with both coordinates even. An exact answer is required.

Determine the number of 8-tuples of nonnegative integers $\left(a_{1}, a_{2}, a_{3}, a_{4}, b_{1}, b_{2}, b_{3}, b_{4}\right)$ satisfying $0 \leq$ $a_{k} \leq k$, for each $k=1,2,3,4$, and $a_{1}+a_{2}+a_{3}+a_{4}+2 b_{1}+3 b_{2}+4 b_{3}+5 b_{4}=19$.

Let $f(x)=1+x+x^{2}+\cdots+x^{100}$. Find $f^{\prime}(1)$.

Let $\ell$ be the line through $(0,0)$ and tangent to the curve $y=x^{3}+x+16$. Find the slope of $\ell$.

Find all $y>1$ satisfying $\int_{1}^{y} x \ln x d x=\frac{1}{4}$.

Let $a, b$ be constants such that $\lim _{x \rightarrow 1} \frac{(\ln (2-x))^{2}}{x^{2}+a x+b}=1$. Determine the pair $(a, b)$.

Let $f(x)=\sin ^{6}\left(\frac{x}{4}\right)+\cos ^{6}\left(\frac{x}{4}\right)$ for all real numbers $x$. Determine $f^{(2008)}(0)$ (i.e., $f$ differentiated 2008 times and then evaluated at $x=0$ ).

Determine the value of $\lim _{n \rightarrow \infty} \sum_{k=0}^{n}\binom{n}{k}^{-1}$.

Find $p$ so that $\lim _{x \rightarrow \infty} x^{p}(\sqrt[3]{x+1}+\sqrt[3]{x-1}-2 \sqrt[3]{x})$ is some non-zero real number.

Let $T=\int_{0}^{\ln 2} \frac{2 e^{3 x}+e^{2 x}-1}{e^{3 x}+e^{2 x}-e^{x}+1} d x$. Evaluate $e^{T}$.

Evaluate the limit $\lim _{n \rightarrow \infty} n^{-\frac{1}{2}\left(1+\frac{1}{n}\right)}\left(1^{1} \cdot 2^{2} \cdots \cdots n^{n}\right)^{\frac{1}{n^{2}}}$.

Evaluate the integral $\int_{0}^{1} \ln x \ln (1-x) d x$.

Let $A B C D$ be a unit square (that is, the labels $A, B, C, D$ appear in that order around the square). Let $X$ be a point outside of the square such that the distance from $X$ to $A C$ is equal to the distance from $X$ to $B D$, and also that $A X=\frac{\sqrt{2}}{2}$. Determine the value of $C X^{2}$.

Find the smallest positive integer $n$ such that $107 n$ has the same last two digits as $n$.

There are 5 dogs, 4 cats, and 7 bowls of milk at an animal gathering. Dogs and cats are distinguishable, but all bowls of milk are the same. In how many ways can every dog and cat be paired with either a member of the other species or a bowl of milk such that all the bowls of milk are taken?

In a triangle $A B C$, take point $D$ on $B C$ such that $D B=14, D A=13, D C=4$, and the circumcircle of $A D B$ is congruent to the circumcircle of $A D C$. What is the area of triangle $A B C$ ?

The equation $x^{3}-9 x^{2}+8 x+2=0$ has three real roots $p, q, r$. Find $\frac{1}{p^{2}}+\frac{1}{q^{2}}+\frac{1}{r^{2}}$.

Let $S$ be the smallest subset of the integers with the property that $0 \in S$ and for any $x \in S$, we have $3 x \in S$ and $3 x+1 \in S$. Determine the number of positive integers in $S$ less than 2008.

A Sudoku matrix is defined as a $9 \times 9$ array with entries from $\{1,2, \ldots, 9\}$ and with the constraint that each row, each column, and each of the nine $3 \times 3$ boxes that tile the array contains each digit from 1 to 9 exactly once. A Sudoku matrix is chosen at random (so that every Sudoku matrix has equal probability of being chosen). We know two of squares in this matrix, as shown. What is the probability that the square marked by ? contains the digit 3 ?

Let $A B C$ be an equilateral triangle with side length 2 , and let $\Gamma$ be a circle with radius $\frac{1}{2}$ centered at the center of the equilateral triangle. Determine the length of the shortest path that starts somewhere on $\Gamma$, visits all three sides of $A B C$, and ends somewhere on $\Gamma$ (not necessarily at the starting point). Express your answer in the form of $\sqrt{p}-q$, where $p$ and $q$ are rational numbers written as reduced fractions.

Four students from Harvard, one of them named Jack, and five students from MIT, one of them named Jill, are going to see a Boston Celtics game. However, they found out that only 5 tickets remain, so 4 of them must go back. Suppose that at least one student from each school must go see the game, and at least one of Jack and Jill must go see the game, how many ways are there of choosing which 5 people can see the game?

Let $A B C$ be an equilateral triangle. Let $\Omega$ be a circle inscribed in $A B C$ and let $\omega$ be a circle tangent externally to $\Omega$ as well as to sides $A B$ and $A C$. Determine the ratio of the radius of $\Omega$ to the radius of $\omega$.

A $3 \times 3 \times 3$ cube composed of 27 unit cubes rests on a horizontal plane. Determine the number of ways of selecting two distinct unit cubes (order is irrelevant) from a $3 \times 3 \times 1$ block with the property that the line joining the centers of the two cubes makes a $45^{\circ}$ angle with the horizontal plane.

Suppose that $a, b, c, d$ are real numbers satisfying $a \geq b \geq c \geq d \geq 0, a^{2}+d^{2}=1, b^{2}+c^{2}=1$, and $a c+b d=1 / 3$. Find the value of $a b-c d$.

A root of unity is a complex number that is a solution to $z^{n}=1$ for some positive integer $n$. Determine the number of roots of unity that are also roots of $z^{2}+a z+b=0$ for some integers $a$ and $b$.

A piece of paper is folded in half. A second fold is made such that the angle marked below has measure $\phi\left(0^{\circ}<\phi<90^{\circ}\right)$, and a cut is made as shown below.

When the piece of paper is unfolded, the resulting hole is a polygon. Let $O$ be one of its vertices. Suppose that all the other vertices of the hole lie on a circle centered at $O$, and also that $\angle X O Y=144^{\circ}$, where $X$ and $Y$ are the the vertices of the hole adjacent to $O$. Find the value(s) of $\phi$ (in degrees).

Let $A B C$ be a triangle, and $I$ its incenter. Let the incircle of $A B C$ touch side $B C$ at $D$, and let lines $B I$ and $C I$ meet the circle with diameter $A I$ at points $P$ and $Q$, respectively. Given $B I=$ $6, C I=5, D I=3$, determine the value of $(D P / D Q)^{2}$.

Determine the number of 8-tuples of nonnegative integers ( $a_{1}, a_{2}, a_{3}, a_{4}, b_{1}, b_{2}, b_{3}, b_{4}$ ) satisfying $0 \leq$ $a_{k} \leq k$, for each $k=1,2,3,4$, and $a_{1}+a_{2}+a_{3}+a_{4}+2 b_{1}+3 b_{2}+4 b_{3}+5 b_{4}=19$.

Determine the number of ways of walking from $(0,0)$ to $(5,5)$ using only up and right unit steps such that the path does not pass through any of the following points: $(1,1),(1,4),(4,1),(4,4)$.

Let $n>2$ be a positive integer. Prove that there are $\frac{1}{2}(n-2)(n+1)$ ways to walk from $(0,0)$ to $(n, 2)$ using only up and right unit steps such that the walk never visits the line $y=x$ after it leaves the origin.

Let $n>4$ be a positive integer. Determine the number of ways to walk from $(0,0)$ to $(n, 2)$ using only up and right unit steps such that the path does not meet the lines $y=x$ or $y=x-n+2$ except at the start and at the end.

Let $n>6$ be a positive integer. Determine the number of ways to walk from $(0,0)$ to $(n, 3)$ using only up and right unit steps such that the path does not meet the lines $y=x$ or $y=x-n+3$ except at the start and at the end.

Let $S$ be a set of 5 points in the 2 -dimensional lattice. Show that we can always choose a pair of points in $S$ whose midpoint is also a lattice point.

Construct a set of $2^{d} d$-dimensional lattice points so that for any two chosen points $A, B$, the line segment $A B$ does not pass through any other lattice point.

Show that for positive integers $n$ and $d$,

$$(n-1) 2^{d}+1 \leq f(n, d) \leq(n-1) n^{d}+1 .$$

Show that for positive integers $n_{1}, n_{2}$ and $d$,

$$f\left(n_{1} n_{2}, d\right) \leq f\left(n_{1}, d\right)+n_{1}\left(f\left(n_{2}, d\right)-1\right)$$

Determine, with proof, a simple closed-form expression for $f\left(2^{a}, d\right)$.

On the circumcircle of $A B C$, let $A^{\prime}$ be the midpoint of arc $B C$ (not containing $A$ ). (a) $[\mathbf{1 0}]$ Show that $A, I, A^{\prime}$ are collinear. (b) $[\mathbf{2 0}]$ Show that $A^{\prime}$ is the circumcenter of BIC.

Let lines $B I$ and $E F$ meet at $K$. Show that $I, K, E, C, D$ are concyclic.

Let $K$ be as in the previous problem. Let $M$ be the midpoint of $B C$ and $N$ the midpoint of $A C$. Show that $K$ lies on line $M N$.

Let $M$ be the midpoint of $B C$, and $T$ diametrically opposite to $D$ on the incircle of $A B C$. Show that $D T, A M, E F$ are concurrent.

Let $P$ be a point inside the incircle of $A B C$. Let lines $D P, E P, F P$ meet the incircle again at $D^{\prime}, E^{\prime}, F^{\prime}$. Show that $A D^{\prime}, B E^{\prime}, C F^{\prime}$ are concurrent.

(Distributive law) Prove that $(x \oplus y) \odot z=x \odot z \oplus y \odot z$ for all $x, y, z \in \mathbb{R} \cup\{\infty\}$.

(Freshman's Dream) Let $z^{n}$ denote $z \odot z \odot z \odot \cdots \odot z$ with $z$ appearing $n$ times. Prove that $(x \oplus y)^{n}=x^{n} \oplus y^{n}$ for all $x, y \in \mathbb{R} \cup\{\infty\}$ and positive integer $n$.

By a tropical polynomial we mean a function of the form

$$p(x)=a_{n} \odot x^{n} \oplus a_{n-1} \odot x^{n-1} \oplus \cdots \oplus a_{1} \odot x \oplus a_{0}$$

where exponentiation is as defined in the previous problem. Let $p$ be a tropical polynomial. Prove that

$$p\left(\frac{x+y}{2}\right) \geq \frac{p(x)+p(y)}{2}$$

for all $x, y \in \mathbb{R} \cup\{\infty\}$. (This means that all tropical polynomials are concave.)

(Fundamental Theorem of Algebra) Let $p$ be a tropical polynomial:

$$p(x)=a_{n} \odot x^{n} \oplus a_{n-1} \odot x^{n-1} \oplus \cdots \oplus a_{1} \odot x \oplus a_{0}, a_{n} \neq \infty$$

Prove that we can find $r_{1}, r_{2}, \ldots, r_{n} \in \mathbb{R} \cup\{\infty\}$ so that

$$p(x)=a_{n} \odot\left(x \oplus r_{1}\right) \odot\left(x \oplus r_{2}\right) \odot \cdots \odot\left(x \oplus r_{n}\right)$$

for all $x$.

Suppose that $j(0) j(1) \cdots j(n-1)$ is a valid juggling sequence. For $i=0,1, \ldots, n-1$, Let $a_{i}$ denote the remainder of $j(i)+i$ when divided by $n$. Prove that $\left(a_{0}, a_{1}, \ldots, a_{n-1}\right)$ is a permutation of $(0,1, \ldots, n-1)$.

Determine the number of juggling sequences of length $n$ with exactly 1 ball.

Prove that the number of balls $b$ in a juggling sequence $j(0) j(1) \cdots j(n-1)$ is simply the average

$$b=\frac{j(0)+j(1)+\cdots+j(n-1)}{n} .$$

Show that the converse of the previous statement is false by providing a non-juggling sequence $j(0) j(1) j(2)$ of length 3 where the average $\frac{1}{3}(j(0)+j(1)+j(2))$ is an integer. Show that your example works.

Let $a, b, c$ denote the side lengths of $B C, C A, A B$. Find the lengths of $A E, B F, C D$ in terms of $a, b, c$.

Show that lines $A D, B E, C F$ pass through a common point.

Show that the incenter of triangle $A E F$ lies on the incircle of $A B C$.

Let $A_{1}, B_{1}, C_{1}$ be the incenters of triangle $A E F, B D F, C D E$, respectively. Show that $A_{1} D, B_{1} E, C_{1} F$ all pass through the orthocenter of $A_{1} B_{1} C_{1}$.

Let $X$ be the point on side $B C$ such that $B X=C D$. Show that the excircle $A B C$ opposite of vertex $A$ touches segment $B C$ at $X$.

Let $X$ be as in the previous problem. Let $T$ be the point diametrically opposite to $D$ on on the incircle of $A B C$. Show that $A, T, X$ are collinear.

Determine all pairs $(a, b)$ of real numbers such that $10, a, b, a b$ is an arithmetic progression.

Given right triangle $A B C$, with $A B=4, B C=3$, and $C A=5$. Circle $\omega$ passes through $A$ and is tangent to $B C$ at $C$. What is the radius of $\omega$ ?

How many ways can you color the squares of a $2 \times 2008$ grid in 3 colors such that no two squares of the same color share an edge?

Find the real solution(s) to the equation $(x+y)^{2}=(x+1)(y-1)$.

A Vandal and a Moderator are editing a Wikipedia article. The article originally is error-free. Each day, the Vandal introduces one new error into the Wikipedia article. At the end of the day, the moderator checks the article and has a $2 / 3$ chance of catching each individual error still in the article. After 3 days, what is the probability that the article is error-free?

Determine the number of non-degenerate rectangles whose edges lie completely on the grid lines of the following figure.

Given that $x+\sin y=2008$ and $x+2008 \cos y=2007$, where $0 \leq y \leq \pi / 2$, find the value of $x+y$.

Trodgor the dragon is burning down a village consisting of 90 cottages. At time $t=0$ an angry peasant arises from each cottage, and every 8 minutes ( 480 seconds) thereafter another angry peasant spontaneously generates from each non-burned cottage. It takes Trodgor 5 seconds to either burn a peasant or to burn a cottage, but Trodgor cannot begin burning cottages until all the peasants around him have been burned. How many seconds does it take Trodgor to burn down the entire village?

Consider a circular cone with vertex $V$, and let $A B C$ be a triangle inscribed in the base of the cone, such that $A B$ is a diameter and $A C=B C$. Let $L$ be a point on $B V$ such that the volume of the cone is 4 times the volume of the tetrahedron $A B C L$. Find the value of $B L / L V$.

Find the number of subsets $S$ of $\{1,2, \ldots 63\}$ the sum of whose elements is 2008 .

Let $f(r)=\sum_{j=2}^{2008} \frac{1}{j^{r}}=\frac{1}{2^{r}}+\frac{1}{3^{r}}+\cdots+\frac{1}{2008^{r}}$. Find $\sum_{k=2}^{\infty} f(k)$.

Suppose we have an (infinite) cone $\mathcal{C}$ with apex $A$ and a plane $\pi$. The intersection of $\pi$ and $\mathcal{C}$ is an ellipse $\mathcal{E}$ with major axis $B C$, such that $B$ is closer to $A$ than $C$, and $B C=4, A C=5, A B=3$. Suppose we inscribe a sphere in each part of $\mathcal{C}$ cut up by $\mathcal{E}$ with both spheres tangent to $\mathcal{E}$. What is the ratio of the radii of the spheres (smaller to larger)?

Let $P(x)$ be a polynomial with degree 2008 and leading coefficient 1 such that

$$P(0)=2007, P(1)=2006, P(2)=2005, \ldots, P(2007)=0 .$$

Determine the value of $P(2008)$. You may use factorials in your answer.

Evaluate the infinite sum $\sum_{n=1}^{\infty} \frac{n}{n^{4}+4}$.

In a game show, Bob is faced with 7 doors, 2 of which hide prizes. After he chooses a door, the host opens three other doors, of which one is hiding a prize. Bob chooses to switch to another door. What is the probability that his new door is hiding a prize?

Point $A$ lies at $(0,4)$ and point $B$ lies at $(3,8)$. Find the $x$-coordinate of the point $X$ on the $x$-axis maximizing $\angle A X B$.

Solve the equation

$$\sqrt{x+\sqrt{4 x+\sqrt{16 x+\sqrt{\ldots+\sqrt{4^{2008} x+3}}}}}-\sqrt{x}=1$$

Express your answer as a reduced fraction with the numerator and denominator written in their prime factorization.

Let $A B C$ be a right triangle with $\angle A=90^{\circ}$. Let $D$ be the midpoint of $A B$ and let $E$ be a point on segment $A C$ such that $A D=A E$. Let $B E$ meet $C D$ at $F$. If $\angle B F C=135^{\circ}$, determine $B C / A B$.

Let $A B C D$ be a regular tetrahedron, and let $O$ be the centroid of triangle $B C D$. Consider the point $P$ on $A O$ such that $P$ minimizes $P A+2(P B+P C+P D)$. Find $\sin \angle P B O$.

For how many ordered triples $(a, b, c)$ of positive integers are the equations $a b c+9=a b+b c+c a$ and $a+b+c=10$ satisfied?

Let $A B C$ be a triangle with $A B=5, B C=4$ and $A C=3$. Let $\mathcal{P}$ and $\mathcal{Q}$ be squares inside $A B C$ with disjoint interiors such that they both have one side lying on $A B$. Also, the two squares each have an edge lying on a common line perpendicular to $A B$, and $\mathcal{P}$ has one vertex on $A C$ and $\mathcal{Q}$ has one vertex on $B C$. Determine the minimum value of the sum of the areas of the two squares.

For a positive integer $n$, let $\theta(n)$ denote the number of integers $0 \leq x<2010$ such that $x^{2}-n$ is divisible by 2010. Determine the remainder when $\sum_{n=0}^{2009} n \cdot \theta(n)$ is divided by 2010.

Two mathematicians, Kelly and Jason, play a cooperative game. The computer selects some secret positive integer $n<60$ (both Kelly and Jason know that $n<60$, but that they don't know what the value of $n$ is). The computer tells Kelly the unit digit of $n$, and it tells Jason the number of divisors of $n$. Then, Kelly and Jason have the following dialogue: Kelly: I don't know what $n$ is, and I'm sure that you don't know either. However, I know that $n$ is divisible by at least two different primes. Jason: Oh, then I know what the value of $n$ is. Kelly: Now I also know what $n$ is. Assuming that both Kelly and Jason speak truthfully and to the best of their knowledge, what are all the possible values of $n$ ?

Suppose that $A B C$ is an isosceles triangle with $A B=A C$. Let $P$ be the point on side $A C$ so that $A P=2 C P$. Given that $B P=1$, determine the maximum possible area of $A B C$.

Alice and the Cheshire Cat play a game. At each step, Alice either (1) gives the cat a penny, which causes the cat to change the number of (magic) beans that Alice has from $n$ to $5 n$ or (2) gives the cat a nickel, which causes the cat to give Alice another bean. Alice wins (and the cat disappears) as soon as the number of beans Alice has is greater than 2008 and has last two digits 42 . What is the minimum number of cents Alice can spend to win the game, assuming she starts with 0 beans?

Let $\mathcal{P}$ be a parabola, and let $V_{1}$ and $F_{1}$ be its vertex and focus, respectively. Let $A$ and $B$ be points on $\mathcal{P}$ so that $\angle A V_{1} B=90^{\circ}$. Let $\mathcal{Q}$ be the locus of the midpoint of $A B$. It turns out that $\mathcal{Q}$ is also a parabola, and let $V_{2}$ and $F_{2}$ denote its vertex and focus, respectively. Determine the ratio $F_{1} F_{2} / V_{1} V_{2}$.

Cyclic pentagon $A B C D E$ has a right angle $\angle A B C=90^{\circ}$ and side lengths $A B=15$ and $B C=20$. Supposing that $A B=D E=E A$, find $C D$.

Let $P$ be a polyhedron where every face is a regular polygon, and every edge has length 1. Each vertex of $P$ is incident to two regular hexagons and one square. Choose a vertex $V$ of the polyhedron. Find the volume of the set of all points contained in $P$ that are closer to $V$ than to any other vertex.

Let $(x, y)$ be a pair of real numbers satisfying

$$56 x+33 y=\frac{-y}{x^{2}+y^{2}}, \text { and } 33 x-56 y=\frac{x}{x^{2}+y^{2}}$$

Determine the value of $|x|+|y|$.

Triangle $A B C$ obeys $A B=2 A C$ and $\angle B A C=120^{\circ}$. Points $P$ and $Q$ lie on segment $B C$ such that

$$\begin{aligned} A B^{2}+B C \cdot C P & =B C^{2} \\ 3 A C^{2}+2 B C \cdot C Q & =B C^{2} \end{aligned}$$

Find $\angle P A Q$ in degrees.

Let $\mathcal{C}$ be the hyperbola $y^{2}-x^{2}=1$. Given a point $P_{0}$ on the $x$-axis, we construct a sequence of points $\left(P_{n}\right)$ on the $x$-axis in the following manner: let $\ell_{n}$ be the line with slope 1 passing passing through $P_{n}$, then $P_{n+1}$ is the orthogonal projection of the point of intersection of $\ell_{n}$ and $\mathcal{C}$ onto the $x$-axis. (If $P_{n}=0$, then the sequence simply terminates.) Let $N$ be the number of starting positions $P_{0}$ on the $x$-axis such that $P_{0}=P_{2008}$. Determine the remainder of $N$ when divided by 2008.

Cyclic pentagon $A B C D E$ has side lengths $A B=B C=5, C D=D E=12$, and $A E=14$. Determine the radius of its circumcircle.

Let $a, b, c$ be nonzero real numbers such that $a+b+c=0$ and $a^{3}+b^{3}+c^{3}=a^{5}+b^{5}+c^{5}$. Find the value of $a^{2}+b^{2}+c^{2}$.

Who Wants to Be a Millionaire. In 2000, the Clay Mathematics Institute named seven Millennium Prize Problems, with each carrying a prize of $\$ 1$ Million for its solution. Write down the name of ONE of the seven Clay Millennium Problems. If your submission is incorrect or misspelled, then your submission is disqualified. If another team wrote down the same Millennium Problem as you, then you get 0 points, otherwise you get 20 points.

NUMB3RS. The RSA Factoring Challenge, which ended in 2007, challenged computational mathematicians to factor extremely large numbers that were the product of two prime numbers. The largest number successfully factored in this challenge was RSA-640, which has 193 decimal digits and carried a prize of $\$ 20,000$. The next challenge number carried prize of $\$ 30,000$, and contains $N$ decimal digits. Your task is to submit a guess for $N$. Only the team(s) that have the closest guess(es) receives points. If $k$ teams all have the closest guesses, then each of them receives $\left\lceil\frac{20}{k}\right\rceil$ points.

The History Channel. Below is a list of famous mathematicians. Your task is to list a subset of them in the chronological order of their birth dates. Your submission should be a sequence of letters. If your sequence is not in the correct order, then you get 0 points. Otherwise your score will be $\min \{\max \{5(N-4), 0\}, 25\}$, where $N$ is the number of letters in your sequence. (A) Niels Abel (B) Arthur Cayley (C) Augustus De Morgan (D) Gustav Dirichlet (E) Leonhard Euler (F) Joseph Fourier (G) Évariste Galois (H) Carl Friedrich Gauss (I) Marie-Sophie Germain (J) Joseph Louis Lagrange (K) Pierre-Simon Laplace (L) Henri Poincaré (N) Bernhard Riemann