HMMT 2006

Larry can swim from Harvard to MIT (with the current of the Charles River) in 40 minutes, or back (against the current) in 45 minutes. How long does it take him to row from Harvard to MIT, if he rows the return trip in 15 minutes? (Assume that the speed of the current and Larry's swimming and rowing speeds relative to the current are all constant.) Express your answer in the format mm:ss.

Find all real solutions $(x, y)$ of the system $x^{2}+y=12=y^{2}+x$.

The train schedule in Hummut is hopelessly unreliable. Train A will enter Intersection X from the west at a random time between 9:00 am and 2:30 pm; each moment in that interval is equally likely. Train B will enter the same intersection from the north at a random time between 9:30 am and 12:30 pm, independent of Train A; again, each moment in the interval is equally likely. If each train takes 45 minutes to clear the intersection, what is the probability of a collision today?

Let $a_{1}, a_{2}, \ldots$ be a sequence defined by $a_{1}=a_{2}=1$ and $a_{n+2}=a_{n+1}+a_{n}$ for $n \geq 1$. Find

$$\sum_{n=1}^{\infty} \frac{a_{n}}{4^{n+1}}$$

Tim has a working analog 12 -hour clock with two hands that run continuously (instead of, say, jumping on the minute). He also has a clock that runs really slow-at half the correct rate, to be exact. At noon one day, both clocks happen to show the exact time. At any given instant, the hands on each clock form an angle between $0^{\circ}$ and $180^{\circ}$ inclusive. At how many times during that day are the angles on the two clocks equal?

Let $a, b, c$ be the roots of $x^{3}-9 x^{2}+11 x-1=0$, and let $s=\sqrt{a}+\sqrt{b}+\sqrt{c}$. Find $s^{4}-18 s^{2}-8 s$.

Let

$$f(x)=x^{4}-6 x^{3}+26 x^{2}-46 x+65 .$$

Let the roots of $f(x)$ be $a_{k}+i b_{k}$ for $k=1,2,3,4$. Given that the $a_{k}, b_{k}$ are all integers, find $\left|b_{1}\right|+\left|b_{2}\right|+\left|b_{3}\right|+\left|b_{4}\right|$.

Solve for all complex numbers $z$ such that $z^{4}+4 z^{2}+6=z$.

Compute the value of the infinite series

$$\sum_{n=2}^{\infty} \frac{n^{4}+3 n^{2}+10 n+10}{2^{n} \cdot\left(n^{4}+4\right)}$$

Determine the maximum value attained by

$$\frac{x^{4}-x^{2}}{x^{6}+2 x^{3}-1}$$

over real numbers $x>1$.

Suppose $A B C$ is a scalene right triangle, and $P$ is the point on hypotenuse $\overline{A C}$ such that $\angle A B P=45^{\circ}$. Given that $A P=1$ and $C P=2$, compute the area of $A B C$.

Let $A, B, C$, and $D$ be points on a circle such that $A B=11$ and $C D=19$. Point $P$ is on segment $A B$ with $A P=6$, and $Q$ is on segment $C D$ with $C Q=7$. The line through $P$ and $Q$ intersects the circle at $X$ and $Y$. If $P Q=27$, find $X Y$.

Let $A B C$ be a triangle such that $A B=2, C A=3$, and $B C=4$. A semicircle with its diameter on $\overline{B C}$ is tangent to $\overline{A B}$ and $\overline{A C}$. Compute the area of the semicircle.

Triangle $A B C$ has side lengths $A B=2 \sqrt{5}, B C=1$, and $C A=5$. Point $D$ is on side $A C$ such that $C D=1$, and $F$ is a point such that $B F=2$ and $C F=3$. Let $E$ be the intersection of lines $A B$ and $D F$. Find the area of $C D E B$.

A circle of radius $t$ is tangent to the hypotenuse, the incircle, and one leg of an isosceles right triangle with inradius $r=1+\sin \frac{\pi}{8}$. Find $r t$.

Suppose $A B C D$ is an isosceles trapezoid in which $\overline{A B} \| \overline{C D}$. Two mutually externally tangent circles $\omega_{1}$ and $\omega_{2}$ are inscribed in $A B C D$ such that $\omega_{1}$ is tangent to $\overline{A B}, \overline{B C}$, and $\overline{C D}$ while $\omega_{2}$ is tangent to $\overline{A B}, \overline{D A}$, and $\overline{C D}$. Given that $A B=1, C D=6$, compute the radius of either circle.

Triangle $A B C$ has a right angle at $B$. Point $D$ lies on side $B C$ such that $3 \angle B A D=$ $\angle B A C$. Given $A C=2$ and $C D=1$, compute $B D$.

Four spheres, each of radius $r$, lie inside a regular tetrahedron with side length 1 such that each sphere is tangent to three faces of the tetrahedron and to the other three spheres. Find $r$.

Triangle $A B C$ has side lengths $A B=65, B C=33$, and $A C=56$. Find the radius of the circle tangent to sides $A C$ and $B C$ and to the circumcircle of triangle $A B C$.

Vernonia High School has 85 seniors, each of whom plays on at least one of the school's three varsity sports teams: football, baseball, and lacrosse. It so happens that 74 are on the football team; 26 are on the baseball team; 17 are on both the football and lacrosse teams; 18 are on both the baseball and football teams; and 13 are on both the baseball and lacrosse teams. Compute the number of seniors playing all three sports, given that twice this number are members of the lacrosse team.

Compute

$$\sum_{n_{60}=0}^{2} \sum_{n_{59}=0}^{n_{60}} \cdots \sum_{n_{2}=0}^{n_{3}} \sum_{n_{1}=0}^{n_{2}} \sum_{n_{0}=0}^{n_{1}} 1$$

A moth starts at vertex $A$ of a certain cube and is trying to get to vertex $B$, which is opposite $A$, in five or fewer "steps," where a step consists in traveling along an edge from one vertex to another. The moth will stop as soon as it reaches $B$. How many ways can the moth achieve its objective?

A dot is marked at each vertex of a triangle $A B C$. Then, 2,3 , and 7 more dots are marked on the sides $A B, B C$, and $C A$, respectively. How many triangles have their vertices at these dots?

Fifteen freshmen are sitting in a circle around a table, but the course assistant (who remains standing) has made only six copies of today's handout. No freshman should get more than one handout, and any freshman who does not get one should be able to read a neighbor's. If the freshmen are distinguishable but the handouts are not, how many ways are there to distribute the six handouts subject to the above conditions?

For how many ordered triplets $(a, b, c)$ of positive integers less than 10 is the product $a \times b \times c$ divisible by 20 ?

Let $n$ be a positive integer, and let Pushover be a game played by two players, standing squarely facing each other, pushing each other, where the first person to lose balance loses. At the HMPT, $2^{n+1}$ competitors, numbered 1 through $2^{n+1}$ clockwise, stand in a circle. They are equals in Pushover: whenever two of them face off, each has a $50 \%$ probability of victory. The tournament unfolds in $n+1$ rounds. In each round, the referee randomly chooses one of the surviving players, and the players pair off going clockwise, starting from the chosen one. Each pair faces off in Pushover, and the losers leave the circle. What is the probability that players 1 and $2^{n}$ face each other in the last round? Express your answer in terms of $n$.

In how many ways can we enter numbers from the set $\{1,2,3,4\}$ into a $4 \times 4$ array so that all of the following conditions hold? (a) Each row contains all four numbers. (b) Each column contains all four numbers. (c) Each "quadrant" contains all four numbers. (The quadrants are the four corner $2 \times 2$ squares.)

Eight celebrities meet at a party. It so happens that each celebrity shakes hands with exactly two others. A fan makes a list of all unordered pairs of celebrities who shook hands with each other. If order does not matter, how many different lists are possible?

Somewhere in the universe, $n$ students are taking a 10 -question math competition. Their collective performance is called laughable if, for some pair of questions, there exist 57 students such that either all of them answered both questions correctly or none of them answered both questions correctly. Compute the smallest $n$ such that the performance is necessarily laughable.

A nonzero polynomial $f(x)$ with real coefficients has the property that $f(x)=f^{\prime}(x) f^{\prime \prime}(x)$. What is the leading coefficient of $f(x)$ ?

Compute $\lim _{x \rightarrow 0} \frac{e^{x \cos x}-1-x}{\sin \left(x^{2}\right)}$.

At time 0 , an ant is at $(1,0)$ and a spider is at $(-1,0)$. The ant starts walking counterclockwise along the unit circle, and the spider starts creeping to the right along the $x$-axis. It so happens that the ant's horizontal speed is always half the spider's. What will the shortest distance ever between the ant and the spider be?

Compute $\sum_{k=1}^{\infty} \frac{k^{4}}{k!}$.

Compute

$$\int_{0}^{1} \frac{d x}{\sqrt{x}+\sqrt[3]{x}}$$

A triangle with vertices at $(1003,0),(1004,3)$, and $(1005,1)$ in the $x y$-plane is revolved all the way around the $y$-axis. Find the volume of the solid thus obtained.

Find all positive real numbers $c$ such that the graph of $f: \mathbb{R} \rightarrow \mathbb{R}$ given by $f(x)=$ $x^{3}-c x$ has the property that the circle of curvature at any local extremum is centered at a point on the $x$-axis.

Compute

$$\int_{0}^{\pi / 3} x \tan ^{2}(x) d x$$

Compute the sum of all real numbers $x$ such that

$$2 x^{6}-3 x^{5}+3 x^{4}+x^{3}-3 x^{2}+3 x-1=0$$

Suppose $f$ and $g$ are differentiable functions such that

$$x g(f(x)) f^{\prime}(g(x)) g^{\prime}(x)=f(g(x)) g^{\prime}(f(x)) f^{\prime}(x)$$

for all real $x$. Moreover, $f$ is nonnegative and $g$ is positive. Furthermore,

$$\int_{0}^{a} f(g(x)) d x=1-\frac{e^{-2 a}}{2}$$

for all reals $a$. Given that $g(f(0))=1$, compute the value of $g(f(4))$.

How many positive integers $x$ are there such that $3 x$ has 3 digits and $4 x$ has four digits?

What is the probability that two cards randomly selected (without replacement) from a standard 52 -card deck are neither of the same value nor the same suit?

A square and an equaliteral triangle together have the property that the area of each is the perimeter of the other. Find the square's area.

Find

$$\frac{\sqrt{31+\sqrt{31+\sqrt{31+\ldots}}}}{\sqrt{1+\sqrt{1+\sqrt{1+\ldots}}}}$$

In the plane, what is the length of the shortest path from $(-2,0)$ to $(2,0)$ that avoids the interior of the unit circle (i.e., circle of radius 1 ) centered at the origin?

Six celebrities meet at a party. It so happens that each celebrity shakes hands with exactly two others. A fan makes a list of all unordered pairs of celebrities who shook hands with each other. If order does not matter, how many different lists are possible?

A dot is marked at each vertex of a triangle $A B C$. Then, 2,3 , and 7 more dots are marked on the sides $A B, B C$, and $C A$, respectively. How many triangles have their vertices at these dots?

Take a unit sphere $S$, i.e., a sphere with radius 1 . Circumscribe a cube $C$ about $S$, and inscribe a cube $D$ in $S$, so that every edge of cube $C$ is parallel to some edge of cube $D$. What is the shortest possible distance from a point on a face of $C$ to a point on a face of $D$ ?

A positive integer $n$ is called "flippant" if $n$ does not end in 0 (when written in decimal notation) and, moreover, $n$ and the number obtained by reversing the digits of $n$ are both divisible by 7 . How many flippant integers are there between 10 and 1000 ?

Find

$$\frac{2^{2}}{2^{2}-1} \cdot \frac{3^{2}}{3^{2}-1} \cdot \frac{4^{2}}{4^{2}-1} \cdots \cdots \frac{2006^{2}}{2006^{2}-1}$$

Let $C$ be the unit circle. Four distinct, smaller congruent circles $C_{1}, C_{2}, C_{3}, C_{4}$ are internally tangent to $C$ such that $C_{i}$ is externally tangent to $C_{i-1}$ and $C_{i+1}$ for $i=$ $1, \ldots, 4$ where $C_{5}$ denotes $C_{1}$ and $C_{0}$ represents $C_{4}$. Compute the radius of $C_{1}$.

If $a, b$ are nonzero real numbers such that $a^{2}+b^{2}=8 a b$, find the value of $\left|\frac{a+b}{a-b}\right|$.

Octagon $A B C D E F G H$ is equiangular. Given that $A B=1, B C=2, C D=3$, $D E=4$, and $E F=F G=2$, compute the perimeter of the octagon.

What is the smallest positive integer $n$ such that $n^{2}$ and $(n+1)^{2}$ both contain the digit 7 but $(n+2)^{2}$ does not?

Six people, all of different weights, are trying to build a human pyramid: that is, they get into the formation

A B C D E F

We say that someone not in the bottom row is "supported by" each of the two closest people beneath her or him. How many different pyramids are possible, if nobody can be supported by anybody of lower weight?

For this problem, our lawn consists of a row of $n$ clumps of grass. This row runs in an east-west direction. In our formation, each mobot may be oriented toward the north, south, east, or west. One example of an allowable formation if $n=6$ is symbolized below:

$$\cdot \longleftarrow \uparrow^{\downarrow}$$

(The mobot on the third clump will move westward, mowing the first three clumps. Each of the last three clumps is mowed by a different mobot.) Here's another allowable formation for $n=6$, considered different from the first:

Compute the number of different allowable formations for any given $n$.

For this problem, our lawn is an $m \times n$ rectangular grid of clumps, that is, with $m$ rows running east-west and $n$ columns running north-south. To be even more explicit, we might say our clumps are at the lattice points

$$\left\{(x, y) \in \mathbb{Z}^{2} \mid 0 \leq x<n \text { and } 0 \leq y<m\right\}$$

However, mobots are now allowed to be oriented to go either north or east only. So one allowable formation for $m=2, n=3$ might be as follows:

$$\begin{array}{lll} \cdot & \rightarrow \\ \uparrow & \rightarrow & . \end{array}$$

Prove that the number of allowable formations for given $m$ and $n$ is $\frac{(m+n)!}{m!n!}$.

In this problem, we stipulate that $m \geq n$, and the lawn is shaped differently. The clumps are now at the lattice points in a trapezoid:

$$\left\{(x, y) \in \mathbb{Z}^{2} \mid 0 \leq x<n \text { and } 0 \leq y<m+1-n+x\right\}$$

As in problem 2, mobots can be set to move either north or east. For given $m$ and $n$, determine with proof the number of allowable formations.

In this problem and the next, the lawn consists of points in a triangular grid of size $n$, so that for $n=3$ the lawn looks like

Mobots are allowed to be oriented to the east, $30^{\circ}$ west of north, or $30^{\circ}$ west of south. Under these conditions, for any given $n$, what is the minimum number of mobots needed to now the lawn?

With the same lawn and the same allowable mobot orientations as in the previous problem, let us call a formation "happy" if it is invariant under $120^{\circ}$ rotations. (A rotation applies both to the positions of the mobots and to their orientations.) An example of a happy formation for $n=2$ might be

Find the number of happy formations for a given $n$.

Let $n$ be an integer at least 5. At most how many diagonals of a regular $n$-gon can be simultaneously drawn so that no two are parallel? Prove your answer.

Given a convex $n$-gon, $n \geq 4$, at most how many diagonals can be drawn such that each drawn diagonal intersects every other drawn diagonal either in the interior of the $n$-gon or at a vertex? Prove your answer.

Given a regular $n$-gon with sides of length 1 , what is the smallest radius $r$ such that there is a non-empty intersection of $n$ circles of radius $r$ centered at the vertices of the $n$-gon? Give $r$ as a formula in terms of $n$. Be sure to prove your answer.

Let $n \geq 3$ be a positive integer. Prove that given any $n$ angles $0<\theta_{1}, \theta_{2}, \ldots, \theta_{n}<$ $180^{\circ}$, such that their sum is $180(n-2)$ degrees, there exists a convex $n$-gon having exactly those angles, in that order.

Suppose we have an $n$-gon such that each interior angle, measured in degrees, is a positive integer. Suppose further that all angles are less than $180^{\circ}$, and that all angles are different sizes. What is the maximum possible value of $n$ ? Prove your answer.

The lottery cards of a certain lottery contain all nine-digit numbers that can be formed with the digits 1,2 and 3 . There is exactly one number on each lottery card. There are only red, yellow and blue lottery cards. Two lottery numbers that differ from each other in all nine digits always appear on cards of different color. Someone draws a red card and a yellow card. The red card has the number 122222222 and the yellow card has the number 222222 222. The first prize goes to the lottery card with the number 123123 123. What color(s) can it possibly have? Prove your answer.

A $3 \times 3 \times 3$ cube is built from 27 unit cubes. Suddenly five of those cubes mysteriously teleport away. What is the minimum possible surface area of the remaining solid? Prove your answer.

Having lost a game of checkers and my temper, I dash all the pieces to the ground but one. This last checker, which is perfectly circular in shape, remains completely on the board, and happens to cover equal areas of red and black squares. Prove that the center of this piece must lie on a boundary between two squares (or at a junction of four).

A number $n$ is called bummed out if there is exactly one ordered pair of positive integers $(x, y)$ such that

$$\left\lfloor x^{2} / y\right\rfloor+\left\lfloor y^{2} / x\right\rfloor=n .$$

Find all bummed out numbers.

Find, with proof, all positive integer palindromes whose square is also a palindrome.

Prove that the maximum number of mobots you need to mow your lawn is $m+n-1$.

Prove that the minimum number of mobots you need to mow your lawn is $\min \{m, n\}$.

Prove that, given any formation, each mobot may be colored in one of three colors - say, white, black, and blue - such that no two adjacent clumps of grass are mowed by different mobots of the same color. Two clumps of grass are adjacent if the distance between them is 1 . In your proof, you may use the Four-Color Theorem if you're familiar with it.

For $n=m=4$, find a formation with 6 mobots for which there are exactly 12 ways to color the mobots in three colors as in problem 3. (No proof is necessary.)

For $n, m \geq 3$, prove that a formation has exactly six possible colorings satisfying the conditions in problem 3 if and only if there is a mobot that starts at $(1,1)$.

Suppose we have a regular hexagon and draw all its sides and diagonals. Into how many regions do the segments divide the hexagon? (No proof is necessary.)

Suppose we have an octagon with all angles of $135^{\circ}$, and consecutive sides of alternating length 1 and $\sqrt{2}$. We draw all its sides and diagonals. Into how many regions do the segments divide the octagon? (No proof is necessary.)

A regular 12-sided polygon is inscribed in a circle of radius 1. How many chords of the circle that join two of the vertices of the 12-gon have lengths whose squares are rational? (No proof is necessary.)

Show a way to construct an equiangular hexagon with side lengths $1,2,3,4,5$, and 6 (not necessarily in that order).

Given a convex $n$-gon, $n \geq 4$, at most how many diagonals can be drawn such that each drawn diagonal intersects every other drawn diagonal strictly in the interior of the $n$-gon? Prove that your answer is correct.

Find the largest positive integer $n$ such that $1!+2!+3!+\cdots+n$ ! is a perfect square. Prove that your answer is correct.

Find all ordered triples $(x, y, z)$ of positive reals such that $x+y+z=27$ and $x^{2}+y^{2}+z^{2}-x y-y z-z x=0$. Prove that your answer is correct.

Four circles with radii $1,2,3$, and $r$ are externally tangent to one another. Compute $r$. (No proof is necessary.)

Find the prime factorization of

$$2006^{2} \cdot 2262-669^{2} \cdot 3599+1593^{2} \cdot 1337$$

(No proof is necessary.)

Let $a, b, c, d$ be real numbers so that $c, d$ are not both 0 . Define the function

$$m(x)=\frac{a x+b}{c x+d}$$

on all real numbers $x$ except possibly $-d / c$, in the event that $c \neq 0$. Suppose that the equation $x=m(m(x))$ has at least one solution that is not a solution of $x=m(x)$. Find all possible values of $a+d$. Prove that your answer is correct.

A bear walks one mile south, one mile east, and one mile north, only to find itself where it started. Another bear, more energetic than the first, walks two miles south, two miles east, and two miles north, only to find itself where it started. However, the bears are not white and did not start at the north pole. At most how many miles apart, to the nearest .001 mile, are the two bears' starting points?

Compute the positive integer less than 1000 which has exactly 29 positive proper divisors. (Here we refer to positive integer divisors other than the number itself.)

At a nursey, 2006 babies sit in a circle. Suddenly each baby pokes the baby immediately to either its left or its right, with equal probability. What is the expected number of unpoked babies?

Ann and Anne are in bumper cars starting 50 meters apart. Each one approaches the other at a constant ground speed of $10 \mathrm{~km} / \mathrm{hr}$. A fly starts at Ann, flies to Anne, then back to Ann, and so on, back and forth until it gets crushed when the two bumper cars collide. When going from Ann to Anne, the fly flies at $20 \mathrm{~km} / \mathrm{hr}$; when going in the opposite direction the fly flies at $30 \mathrm{~km} / \mathrm{hr}$ (thanks to a breeze). How many meters does the fly fly?

Find the number of solutions in positive integers $\left(k ; a_{1}, a_{2}, \ldots, a_{k} ; b_{1}, b_{2}, \ldots, b_{k}\right)$ to the equation

$$a_{1}\left(b_{1}\right)+a_{2}\left(b_{1}+b_{2}\right)+\cdots+a_{k}\left(b_{1}+b_{2}+\cdots+b_{k}\right)=7 .$$

Suppose $A B C$ is a triangle such that $A B=13, B C=15$, and $C A=14$. Say $D$ is the midpoint of $\overline{B C}, E$ is the midpoint of $\overline{A D}, F$ is the midpoint of $\overline{B E}$, and $G$ is the midpoint of $\overline{D F}$. Compute the area of triangle $E F G$.

Find all real numbers $x$ such that

$$x^{2}+\left\lfloor\frac{x}{2}\right\rfloor+\left\lfloor\frac{x}{3}\right\rfloor=10$$

How many ways are there to label the faces of a regular octahedron with the integers $1-$ 8 , using each exactly once, so that any two faces that share an edge have numbers that are relatively prime? Physically realizable rotations are considered indistinguishable, but physically unrealizable reflections are considered different.

Four unit circles are centered at the vertices of a unit square, one circle at each vertex. What is the area of the region common to all four circles?

Let $f(x)=x^{2}-2 x$. How many distinct real numbers $c$ satisfy $f(f(f(f(c))))=3$ ?

Find all positive integers $n>1$ for which $\frac{n^{2}+7 n+136}{n-1}$ is the square of a positive integer.

For each positive integer $n$ let $S_{n}$ denote the set $\{1,2,3, \ldots, n\}$. Compute the number of triples of subsets $A, B, C$ of $S_{2006}$ (not necessarily nonempty or proper) such that $A$ is a subset of $B$ and $S_{2006}-A$ is a subset of $C$.

Let $Z$ be as in problem 15. Let $X$ be the greatest integer such that $|X Z| \leq 5$. Find $X$.

Let $X$ be as in problem 13. Let $Y$ be the number of ways to order $X$ crimson flowers, $X$ scarlet flowers, and $X$ vermillion flowers in a row so that no two flowers of the same hue are adjacent. (Flowers of the same hue are mutually indistinguishable.) Find $Y$.

Let $Y$ be as in problem 14. Find the maximum $Z$ such that three circles of radius $\sqrt{Z}$ can simultaneously fit inside an equilateral triangle of area $Y$ without overlapping each other.

A sequence $a_{1}, a_{2}, a_{3}, \ldots$ of positive reals satisfies $a_{n+1}=\sqrt{\frac{1+a_{n}}{2}}$. Determine all $a_{1}$ such that $a_{i}=\frac{\sqrt{6}+\sqrt{2}}{4}$ for some positive integer $i$.

Begining at a vertex, an ant is crawls between the vertices of a regular octahedron. After reaching a vertex, it randomly picks a neighboring vertex (sharing an edge) and walks to that vertex along the adjoining edge (with all possibilities equally likely.) What is the probability that after walking along 2006 edges, the ant returns to the vertex where it began?

Cyclic quadrilateral $A B C D$ has side lengths $A B=1, B C=2, C D=3$ and $D A=4$. Points $P$ and $Q$ are the midpoints of $\overline{B C}$ and $\overline{D A}$. Compute $P Q^{2}$.

Let $A B C$ be a triangle with $A B=2, C A=3, B C=4$. Let $D$ be the point diametrically opposite $A$ on the circumcircle of $A B C$, and let $E$ lie on line $A D$ such that $D$ is the midpoint of $\overline{A E}$. Line $l$ passes through $E$ perpendicular to $\overline{A E}$, and $F$ and $G$ are the intersections of the extensions of $\overline{A B}$ and $\overline{A C}$ with $l$. Compute $F G$.

Compute the number of real solutions $(x, y, z, w)$ to the system of equations:

$$\begin{array}{rlrl} x & =z+w+z w x & z & =x+y+x y z \\ y & =w+x+w x y & w & =y+z+y z w \end{array}$$

Find the smallest positive integer $k$ such that $z^{10}+z^{9}+z^{6}+z^{5}+z^{4}+z+1$ divides $z^{k}-1$.

Let $f(x)$ be a degree 2006 polynomial with complex roots $c_{1}, c_{2}, \ldots, c_{2006}$, such that the set

$$\left\{\left|c_{1}\right|,\left|c_{2}\right|, \ldots,\left|c_{2006}\right|\right\}$$

consists of exactly 1006 distinct values. What is the minimum number of real roots of $f(x)$ ?

Let $a_{0}, a_{1}, a_{2}, \ldots$ be a sequence of real numbers defined by $a_{0}=21, a_{1}=35$, and $a_{n+2}=4 a_{n+1}-4 a_{n}+n^{2}$ for $n \geq 2$. Compute the remainder obtained when $a_{2006}$ is divided by 100 .

Two 18-24-30 triangles in the plane share the same circumcircle as well as the same incircle. What's the area of the region common to both the triangles?

Points $A, C$, and $B$ lie on a line in that order such that $A C=4$ and $B C=2$. Circles $\omega_{1}, \omega_{2}$, and $\omega_{3}$ have $\overline{B C}, \overline{A C}$, and $\overline{A B}$ as diameters. Circle $\Gamma$ is externally tangent to $\omega_{1}$ and $\omega_{2}$ at $D$ and $E$ respectively, and is internally tangent to $\omega_{3}$. Compute the circumradius of triangle $C D E$.

Let $a \geq b \geq c$ be real numbers such that

$$\begin{aligned} a^{2} b c+a b^{2} c+a b c^{2}+8 & =a+b+c \\ a^{2} b+a^{2} c+b^{2} c+b^{2} a+c^{2} a+c^{2} b+3 a b c & =-4 \\ a^{2} b^{2} c+a b^{2} c^{2}+a^{2} b c^{2} & =2+a b+b c+c a \end{aligned}$$

If $a+b+c>0$, then compute the integer nearest to $a^{5}$.

Let $N$ denote the number of subsets of $\{1,2,3, \ldots, 100\}$ that contain more prime numbers than multiples of 4 . Compute the largest integer $k$ such that $2^{k}$ divides $N$.

A pebble is shaped as the intersection of a cube of side length 1 with the solid sphere tangent to all of the cube's edges. What is the surface area of this pebble?

Find the area in the first quadrant bounded by the hyperbola $x^{2}-y^{2}=1$, the $x$-axis, and the line $3 x=4 y$.

$A B C$ is an acute triangle with incircle $\omega$. $\omega$ is tangent to sides $\overline{B C}, \overline{C A}$, and $\overline{A B}$ at $D, E$, and $F$ respectively. $P$ is a point on the altitude from $A$ such that $\Gamma$, the circle with diameter $\overline{A P}$, is tangent to $\omega$. $\Gamma$ intersects $\overline{A C}$ and $\overline{A B}$ at $X$ and $Y$ respectively. Given $X Y=8, A E=15$, and that the radius of $\Gamma$ is 5 , compute $B D \cdot D C$.

Let $A$ be as in problem 33. Let $W$ be the sum of all positive integers that divide $A$. Find $W$.

In the alphametic $W E \times E Y E=S C E N E$, each different letter stands for a different digit, and no word begins with a 0 . The $W$ in this problem has the same value as the $W$ in problem 31. Find $S$.

Let $W, S$ be as in problem 32. Let $A$ be the least positive integer such that an acute triangle with side lengths $S, A$, and $W$ exists. Find $A$.

In bridge, a standard 52 -card deck is dealt in the usual way to 4 players. By convention, each hand is assigned a number of "points" based on the formula

$$4 \times(\# \mathrm{~A} \text { 's })+3 \times(\# \mathrm{~K} \text { 's })+2 \times(\# \text { Q's })+1 \times(\# \mathrm{~J} \text { 's })$$

Given that a particular hand has exactly 4 cards that are A, K, Q, or J, find the probability that its point value is 13 or higher.

A sequence is defined by $A_{0}=0, A_{1}=1, A_{2}=2$, and, for integers $n \geq 3$,

$$A_{n}=\frac{A_{n-1}+A_{n-2}+A_{n-3}}{3}+\frac{1}{n^{4}-n^{2}}$$

Compute $\lim _{N \rightarrow \infty} A_{N}$.

Four points are independently chosen uniformly at random from the interior of a regular dodecahedron. What is the probability that they form a tetrahedron whose interior contains the dodecahedron's center?

Compute

$$\sum_{n=1}^{\infty} \frac{2 n+5}{2^{n} \cdot\left(n^{3}+7 n^{2}+14 n+8\right)}$$

Suppose $A B C$ is a triangle with incircle $\omega$, and $\omega$ is tangent to $\overline{B C}$ and $\overline{C A}$ at $D$ and $E$ respectively. The bisectors of $\angle A$ and $\angle B$ intersect line $D E$ at $F$ and $G$ respectively, such that $B F=1$ and $F G=G A=6$. Compute the radius of $\omega$.

A fat coin is one which, when tossed, has a $2 / 5$ probability of being heads, $2 / 5$ of being tails, and $1 / 5$ of landing on its edge. Mr. Fat starts at 0 on the real line. Every minute, he tosses a fat coin. If it's heads, he moves left, decreasing his coordinate by 1 ; if it's tails, he moves right, increasing his coordinate by 1 . If the coin lands on its edge, he moves back to 0 . If Mr. Fat does this ad infinitum, what fraction of his time will he spend at 0 ?

Compute

$$\sum_{k=1}^{\infty} \frac{3 k+1}{2 k^{3}+k^{2}} \cdot(-1)^{k+1}$$

Let $\Gamma$ denote the circumcircle of triangle $A B C$. Point $D$ is on $\overline{A B}$ such that $\overline{C D}$ bisects $\angle A C B$. Points $P$ and $Q$ are on $\Gamma$ such that $\overline{P Q}$ passes through $D$ and is perpendicular to $\overline{C D}$. Compute $P Q$, given that $B C=20, C A=80, A B=65$.

Suppose hypothetically that a certain, very corrupt political entity in a universe holds an election with two candidates, say $A$ and $B$. A total of $5,825,043$ votes are cast, but, in a sudden rainstorm, all the ballots get soaked. Undaunted, the election officials decide to guess what the ballots say. Each ballot has a $51 \%$ chance of being deemed a vote for $A$, and a $49 \%$ chance of being deemed a vote for $B$. The probability that $B$ will win is $10^{-X}$. What is $X$ rounded to the nearest 10 ?

Write down at least one, and up to ten, different 3-digit prime numbers. If you somehow fail to do this, we will ignore your submission for this problem. Otherwise, you're entered into a game with other teams. In this game, you start with 10 points, and each number you write down is like a bet: if no one else writes that number, you gain 1 point, but if anyone else writes that number, you lose 1 point. Thus, your score on this problem can be anything from 0 to 20 .

On the Euclidean plane are given 14 points:

$$\begin{array}{rlrr} A=(0,428) & B=(9,85) & C=(42,865) & D=(192,875) \\ E=(193,219) & F=(204,108) & G=(292,219) & H=(316,378) \\ I=(375,688) & J=(597,498) & K=(679,766) & L=(739,641) \\ & M=(772,307) & & N=(793,0) \\ & \end{array}$$

A fly starts at $A$, visits all the other points, and comes back to $A$ in such a way as to minimize the total distance covered. What path did the fly take? Give the names of the points it visits in order. Your score will be

$$20+\lfloor\text { the optimal distance }\rfloor-\lfloor\text { your distance }\rfloor$$

or 0 , whichever is greater.

On your answer sheet, clearly mark at least seven points, as long as (i) No three are collinear. (ii) No seven form a convex heptagon.

Please do not cross out any points; erase if you can do so neatly. If the graders deem that your paper is too messy, or if they determine that you violated one of those conditions, your submission for this problem will be disqualified. Otherwise, your score will be the number of points you marked minus 6 , even if you actually violated one of the conditions but were able to fool the graders.