HMMT 2005

How many real numbers $x$ are solutions to the following equation?

$$|x-1|=|x-2|+|x-3|$$

How many real numbers $x$ are solutions to the following equation?

$$2003^{x}+2004^{x}=2005^{x}$$

Let $x, y$, and $z$ be distinct real numbers that sum to 0 . Find the maximum possible value of

$$\frac{x y+y z+z x}{x^{2}+y^{2}+z^{2}} .$$

If $a, b, c>0$, what is the smallest possible value of $\left\lfloor\frac{a+b}{c}\right\rfloor+\left\lfloor\frac{b+c}{a}\right\rfloor+\left\lfloor\frac{c+a}{b}\right\rfloor$ ? (Note that $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$.)

Ten positive integers are arranged around a circle. Each number is one more than the greatest common divisor of its two neighbors. What is the sum of the ten numbers?

Find the sum of the $x$-coordinates of the distinct points of intersection of the plane curves given by $x^{2}=x+y+4$ and $y^{2}=y-15 x+36$.

Let $x$ be a positive real number. Find the maximum possible value of

$$\frac{x^{2}+2-\sqrt{x^{4}+4}}{x} .$$

Compute

$$\sum_{n=0}^{\infty} \frac{n}{n^{4}+n^{2}+1}$$

The number $27,000,001$ has exactly four prime factors. Find their sum.

Find the sum of the absolute values of the roots of $x^{4}-4 x^{3}-4 x^{2}+16 x-8=0$.

Let $A B C D$ be a rectangle with area 1 , and let $E$ lie on side $C D$. What is the area of the triangle formed by the centroids of triangles $A B E, B C E$, and $A D E$ ?

Let $X Y Z$ be a triangle with $\angle X=60^{\circ}$ and $\angle Y=45^{\circ}$. A circle with center $P$ passes through points $A$ and $B$ on side $X Y, C$ and $D$ on side $Y Z$, and $E$ and $F$ on side $Z X$. Suppose $A B=C D=E F$. Find $\angle X P Y$ in degrees.

Let $A B C D$ be a tetrahedron such that edges $A B, A C$, and $A D$ are mutually perpendicular. Let the areas of triangles $A B C, A C D$, and $A D B$ be denoted by $x, y$, and $z$, respectively. In terms of $x, y$, and $z$, find the area of triangle $B C D$.

Let $T$ be a triangle with side lengths 26,51 , and 73 . Let $S$ be the set of points inside $T$ which do not lie within a distance of 5 of any side of $T$. Find the area of $S$.

Let $A C$ be a diameter of a circle $\omega$ of radius 1 , and let $D$ be the point on $A C$ such that $C D=1 / 5$. Let $B$ be the point on $\omega$ such that $D B$ is perpendicular to $A C$, and let $E$ be the midpoint of $D B$. The line tangent to $\omega$ at $B$ intersects line $C E$ at the point $X$. Compute $A X$.

Let $A B$ be the diameter of a semicircle $\Gamma$. Two circles, $\omega_{1}$ and $\omega_{2}$, externally tangent to each other and internally tangent to $\Gamma$, are tangent to the line $A B$ at $P$ and $Q$, respectively, and to semicircular arc $A B$ at $C$ and $D$, respectively, with $A P<A Q$. Suppose $F$ lies on $\Gamma$ such that $\angle F Q B=\angle C Q A$ and that $\angle A B F=80^{\circ}$. Find $\angle P D Q$ in degrees.

A true-false test has ten questions. If you answer five questions "true" and five "false," your score is guaranteed to be at least four. How many answer keys are there for which this is true?

How many nonempty subsets of $\{1,2,3, \ldots, 12\}$ have the property that the sum of the largest element and the smallest element is 13 ?

The Red Sox play the Yankees in a best-of-seven series that ends as soon as one team wins four games. Suppose that the probability that the Red Sox win Game $n$ is $\frac{n-1}{6}$. What is the probability that the Red Sox will win the series?

In how many ways can 4 purple balls and 4 green balls be placed into a $4 \times 4$ grid such that every row and column contains one purple ball and one green ball? Only one ball may be placed in each box, and rotations and reflections of a single configuration are considered different.

Doug and Ryan are competing in the 2005 Wiffle Ball Home Run Derby. In each round, each player takes a series of swings. Each swing results in either a home run or an out, and an out ends the series. When Doug swings, the probability that he will hit a home run is $1 / 3$. When Ryan swings, the probability that he will hit a home run is $1 / 2$. In one round, what is the probability that Doug will hit more home runs than Ryan hits?

Three fair six-sided dice, each numbered 1 through 6 , are rolled. What is the probability that the three numbers that come up can form the sides of a triangle?

What is the maximum number of bishops that can be placed on an $8 \times 8$ chessboard such that at most three bishops lie on any diagonal?

Every second, Andrea writes down a random digit uniformly chosen from the set $\{1,2,3,4\}$. She stops when the last two numbers she has written sum to a prime number. What is the probability that the last number she writes down is 1 ?

Eight coins are arranged in a circle heads up. A move consists of flipping over two adjacent coins. How many different sequences of six moves leave the coins alternating heads up and tails up?

You start out with a big pile of $3^{2004}$ cards, with the numbers $1,2,3, \ldots, 3^{2004}$ written on them. You arrange the cards into groups of three any way you like; from each group, you keep the card with the largest number and discard the other two. You now again arrange these $3^{2003}$ remaining cards into groups of three any way you like, and in each group, keep the card with the smallest number and discard the other two. You now have $3^{2002}$ cards, and you again arrange these into groups of three and keep the largest number in each group. You proceed in this manner, alternating between keeping the largest number and keeping the smallest number in each group, until you have just one card left.

How many different values are possible for the number on this final card?

Let $f(x)=x^{3}+a x+b$, with $a \neq b$, and suppose the tangent lines to the graph of $f$ at $x=a$ and $x=b$ are parallel. Find $f(1)$.

A plane curve is parameterized by $x(t)=\int_{t}^{\infty} \frac{\cos u}{u} d u$ and $y(t)=\int_{t}^{\infty} \frac{\sin u}{u} d u$ for $1 \leq t \leq$ 2. What is the length of the curve?

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a continuous function with $\int_{0}^{1} f(x) f^{\prime}(x) d x=0$ and $\int_{0}^{1} f(x)^{2} f^{\prime}(x) d x=$ 18. What is $\int_{0}^{1} f(x)^{4} f^{\prime}(x) d x$ ?

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a smooth function such that $f^{\prime}(x)^{2}=f(x) f^{\prime \prime}(x)$ for all $x$. Suppose $f(0)=1$ and $f^{(4)}(0)=9$. Find all possible values of $f^{\prime}(0)$.

Calculate

$$\lim _{x \rightarrow 0^{+}}\left(x^{x^{x}}-x^{x}\right) .$$

The graph of $r=2+\cos 2 \theta$ and its reflection over the line $y=x$ bound five regions in the plane. Find the area of the region containing the origin.

Two ants, one starting at $(-1,1)$, the other at $(1,1)$, walk to the right along the parabola $y=x^{2}$ such that their midpoint moves along the line $y=1$ with constant speed 1. When the left ant first hits the line $y=\frac{1}{2}$, what is its speed?

If $f$ is a continuous real function such that $f(x-1)+f(x+1) \geq x+f(x)$ for all $x$, what is the minimum possible value of $\int_{1}^{2005} f(x) d x$ ?

Compute

$$\sum_{k=0}^{\infty} \frac{4}{(4 k)!}$$

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a smooth function such that $f^{\prime}(x)=f(1-x)$ for all $x$ and $f(0)=1$. Find $f(1)$.

Let $A B C D$ be a regular tetrahedron with side length 2 . The plane parallel to edges $A B$ and $C D$ and lying halfway between them cuts $A B C D$ into two pieces. Find the surface area of one of these pieces.

Find all real solutions to $x^{3}+(x+1)^{3}+(x+2)^{3}=(x+3)^{3}$.

In an election, there are two candidates, $A$ and $B$, who each have 5 supporters. Each supporter, independent of other supporters, has a $\frac{1}{2}$ probability of voting for his or her candidate and a $\frac{1}{2}$ probability of being lazy and not voting. What is the probability of a tie (which includes the case in which no one votes)?

A triangular piece of paper of area 1 is folded along a line parallel to one of the sides and pressed flat. What is the minimum possible area of the resulting figure?

What is the smallest integer $x$ larger than 1 such that $x^{2}$ ends in the same three digits as $x$ does?

The volume of a cube (in cubic inches) plus three times the total length of its edges (in inches) is equal to twice its surface area (in square inches). How many inches long is its long diagonal?

Find three real numbers $a<b<c$ satisfying:

$$\begin{aligned} a+b+c & =21 / 4 \\ 1 / a+1 / b+1 / c & =21 / 4 \\ a b c & =1 \end{aligned}$$

Working together, Jack and Jill can paint a house in 3 days; Jill and Joe can paint the same house in 4 days; or Joe and Jack can paint the house in 6 days. If Jill, Joe, and Jack all work together, how many days will it take them?

In how many ways can 8 people be arranged in a line if Alice and Bob must be next to each other, and Carol must be somewhere behind Dan?

You and I play the following game on an $8 \times 8$ square grid of boxes: Initially, every box is empty. On your turn, you choose an empty box and draw an $X$ in it; if any of the four adjacent boxes are empty, you mark them with an $X$ as well. (Two boxes are adjacent if they share an edge.) We alternate turns, with you moving first, and whoever draws the last $X$ wins. How many choices do you have for a first move that will enable you to guarantee a win no matter how I play?

A cube with side length 2 is inscribed in a sphere. A second cube, with faces parallel to the first, is inscribed between the sphere and one face of the first cube. What is the length of a side of the smaller cube?

Three distinct lines are drawn in the plane. Suppose there exist exactly $n$ circles in the plane tangent to all three lines. Find all possible values of $n$.

In how many ways can the cells of a $4 \times 4$ table be filled in with the digits $1,2, \ldots, 9$ so that each of the 4 -digit numbers formed by the columns is divisible by each of the 4-digit numbers formed by the rows?

Let $\lfloor x\rfloor$ denote the greatest integer less than or equal to $x$. How many positive integers less than 2005 can be expressed in the form $\lfloor x\lfloor x\rfloor\rfloor$ for some positive real $x$ ?

Prove that for any two types of dominoes, there exists a rectangle that can be tiled by dominoes of either type.

Suppose $0<a \leq b$ and $4 \nmid m n$. Prove that the number of ways in which an $m \times n$ rectangle can be partitioned into dominoes of type ( $a, b$ ) is even.

Show that no rectangle of the form $1 \times k$ or $2 \times n$, where $4 \nmid n$, is $(1,2)$-tileable.

Show that all other rectangles of even area are (1,2)-tileable.

Show that for $b$ even, there exists some $M$ such that for every $n>M$, a $2 b \times n$ rectangle is $(1, b)$-tileable.

Show that for $b$ even, there exists some $M$ such that for every $m, n>M$ with $m n$ even, an $m \times n$ rectangle is $(1, b)$-tileable.

Prove that neither of the previous two problems holds if $b$ is odd.

Suppose $x$ is a fifth root of unity. Find, in radical form, all possible values of

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

Let $A_{1} A_{2} \ldots A_{k}$ be a regular $k$-gon inscribed in a circle of radius 1 , and let $P$ be a point lying on or inside the circumcircle. Find the maximum possible value of $\left(P A_{1}\right)\left(P A_{2}\right) \cdots\left(P A_{k}\right)$.

Let $P$ be a regular $k$-gon inscribed in a circle of radius 1 . Find the sum of the squares of the lengths of all the sides and diagonals of $P$.

Let $P(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{0}$ be a polynomial with real coefficients, $a_{n} \neq 0$. Suppose every root of $P$ is a root of unity, but $P(1) \neq 0$. Show that the coefficients of $P$ are symmetric; that is, show that $a_{n}=a_{0}, a_{n-1}=a_{1}, \ldots$

Suppose the elements of $A$ are either bounded below or bounded above. Show that if $S$ tiles $A$, then it does so uniquely, i.e., there is a unique tiling of $A$ by $S$.

Let $B$ be a set of integers either bounded below or bounded above. Then show that if $S$ tiles all other integers $\mathbf{Z} \backslash B$, then $S$ tiles all integers $\mathbf{Z}$.

Suppose $S$ tiles the natural numbers $\mathbf{N}$. Show that $S$ tiles the set $\{1,2, \ldots, k\}$ for some positive integer $k$.

Suppose $S$ tiles N. Show that $S$ is symmetric; that is, if $-S=\left\{-s_{n}, \ldots,-s_{0}\right\}$, show that $S \sim-S$.

Let $0<m \leq n$ be integers. How many different (i.e., noncongruent) dominoes can be formed by choosing two squares of an $m \times n$ array?

What are the dimensions of the rectangle of smallest area that is $(a, b)$-tileable?

Prove that every $(a, b)$-tileable rectangle contains a rectangle of these dimensions.

Prove that an $m \times n$ rectangle is $(b, b)$-tileable if and only if $2 b \mid m$ and $2 b \mid n$.

Prove that an $m \times n$ rectangle is ( $0, b$ )-tileable if and only if $2 b \mid m$ or $2 b \mid n$.

Let $k$ be an integer such that $k \mid a$ and $k \mid b$. Prove that if an $m \times n$ rectangle is ( $a, b$ )-tileable, then $2 k \mid m$ or $2 k \mid n$.

Find a real, irreducible quartic polynomial with leading coefficient 1 whose roots are all twelfth roots of unity.

Let $x$ and $y$ be two $k$ th roots of unity. Prove that $(x+y)^{k}$ is real.

Let $x$ and $y$ be two distinct roots of unity. Prove that $x+y$ is also a root of unity if and only if $\frac{y}{x}$ is a cube root of unity.

Let $x, y$, and $z$ be three roots of unity. Prove that $x+y+z$ is also a root of unity if and only if $x+y=0, y+z=0$, or $z+x=0$.

Find all sets $S$ with minimum element 1 that tile $A=\{1, \ldots, 12\}$.

Let $A$ be a finite set with more than one element. Prove that the number of nonequivalent sets $S$ which tile $A$ is always even.

Exhibit a set $S$ which tiles the integers $\mathbf{Z}$ but not the natural numbers $\mathbf{N}$.

Suppose that $S$ tiles the set of all integer cubes. Prove that $S$ has only one element.

Suppose that $S$ tiles the set of odd prime numbers. Prove that $S$ has only one element.

Find the largest positive integer $n$ such that $1+2+3+\cdots+n^{2}$ is divisible by $1+2+$ $3+\cdots+n$.

Let $x, y$, and $z$ be positive real numbers such that $(x \cdot y)+z=(x+z) \cdot(y+z)$. What is the maximum possible value of $x y z$ ?

Find the sum

$$\frac{2^{1}}{4^{1}-1}+\frac{2^{2}}{4^{2}-1}+\frac{2^{4}}{4^{4}-1}+\frac{2^{8}}{4^{8}-1}+\cdots .$$

What is the probability that in a randomly chosen arrangement of the numbers and letters in "HMMT2005," one can read either "HMMT" or " 2005 " from left to right? (For example, in "5HM0M20T," one can read "HMMT.")

For how many integers $n$ between 1 and 2005 , inclusive, is $2 \cdot 6 \cdot 10 \cdots(4 n-2)$ divisible by $n!$ ?

Let $m \circ n=(m+n) /(m n+4)$. Compute $((\cdots((2005 \circ 2004) \circ 2003) \circ \cdots \circ 1) \circ 0)$.

Five people of different heights are standing in line from shortest to tallest. As it happens, the tops of their heads are all collinear; also, for any two successive people, the horizontal distance between them equals the height of the shorter person. If the shortest person is 3 feet tall and the tallest person is 7 feet tall, how tall is the middle person, in feet?

Let $A B C D$ be a convex quadrilateral inscribed in a circle with shortest side $A B$. The ratio $[B C D] /[A B D]$ is an integer (where $[X Y Z]$ denotes the area of triangle $X Y Z$.) If the lengths of $A B, B C, C D$, and $D A$ are distinct integers no greater than 10 , find the largest possible value of $A B$.

Farmer Bill's 1000 animals - ducks, cows, and rabbits - are standing in a circle. In order to feel safe, every duck must either be standing next to at least one cow or between two rabbits. If there are 600 ducks, what is the least number of cows there can be for this to be possible?

You are given a set of cards labeled from 1 to 100 . You wish to make piles of three cards such that in any pile, the number on one of the cards is the product of the numbers on the other two cards. However, no card can be in more than one pile. What is the maximum number of piles you can form at once?

The Dingoberry Farm is a 10 mile by 10 mile square, broken up into 1 mile by 1 mile patches. Each patch is farmed either by Farmer Keith or by Farmer Ann. Whenever Ann farms a patch, she also farms all the patches due west of it and all the patches due south of it. Ann puts up a scarecrow on each of her patches that is adjacent to exactly two of Keith's patches (and nowhere else). If Ann farms a total of 30 patches, what is the largest number of scarecrows she could put up?

Two vertices of a cube are given in space. The locus of points that could be a third vertex of the cube is the union of $n$ circles. Find $n$.

Triangle $A B C$ has $A B=1, B C=\sqrt{7}$, and $C A=\sqrt{3}$. Let $\ell_{1}$ be the line through $A$ perpendicular to $A B, \ell_{2}$ the line through $B$ perpendicular to $A C$, and $P$ the point of intersection of $\ell_{1}$ and $\ell_{2}$. Find $P C$.

Three noncollinear points and a line $\ell$ are given in the plane. Suppose no two of the points lie on a line parallel to $\ell$ (or $\ell$ itself). There are exactly $n$ lines perpendicular to $\ell$ with the following property: the three circles with centers at the given points and tangent to the line all concur at some point. Find all possible values of $n$.

Let $S$ be the set of lattice points inside the circle $x^{2}+y^{2}=11$. Let $M$ be the greatest area of any triangle with vertices in $S$. How many triangles with vertices in $S$ have area $M$ ?

A regular octahedron has a side length of 1 . What is the distance between two opposite faces?

Compute

$$\sqrt[{2 \sqrt{2 \sqrt[3]{2 \sqrt[4]{2 \sqrt[5]{2 \cdots}}}}}]{ }$$

If $a, b$, and $c$ are random real numbers from 0 to 1 , independently and uniformly chosen, what is the average (expected) value of the smallest of $a, b$, and $c$ ?

Regular tetrahedron $A B C D$ is projected onto a plane sending $A, B, C$, and $D$ to $A^{\prime}, B^{\prime}, C^{\prime}$, and $D^{\prime}$ respectively. Suppose $A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ is a convex quadrilateral with $A^{\prime} B^{\prime}=A^{\prime} D^{\prime}$ and $C^{\prime} B^{\prime}=C^{\prime} D^{\prime}$, and suppose that the area of $A^{\prime} B^{\prime} C^{\prime} D^{\prime}=4$. Given these conditions, the set of possible lengths of $A B$ consists of all real numbers in the interval $[a, b)$. Compute $b$.

If $n$ is a positive integer, let $s(n)$ denote the sum of the digits of $n$. We say that $n$ is zesty if there exist positive integers $x$ and $y$ greater than 1 such that $x y=n$ and $s(x) s(y)=s(n)$. How many zesty two-digit numbers are there?

In triangle $A B C$ with altitude $A D, \angle B A C=45^{\circ}, D B=3$, and $C D=2$. Find the area of triangle $A B C$.

Find

$$\{\ln (1+e)\}+\left\{\ln \left(1+e^{2}\right)\right\}+\left\{\ln \left(1+e^{4}\right)\right\}+\left\{\ln \left(1+e^{8}\right)\right\}+\cdots,$$

where $\{x\}=x-\lfloor x\rfloor$ denotes the fractional part of $x$.

The sides of a regular hexagon are trisected, resulting in 18 points, including vertices. These points, starting with a vertex, are numbered clockwise as $A_{1}, A_{2}, \ldots, A_{18}$. The line segment $A_{k} A_{k+4}$ is drawn for $k=1,4,7,10,13,16$, where indices are taken modulo 18. These segments define a region containing the center of the hexagon. Find the ratio of the area of this region to the area of the large hexagon.

In the base 10 arithmetic problem $H M M T+G U T S=R O U N D$, each distinct letter represents a different digit, and leading zeroes are not allowed. What is the maximum possible value of $R O U N D$ ?

An ant starts at one vertex of a tetrahedron. Each minute it walks along a random edge to an adjacent vertex. What is the probability that after one hour the ant winds up at the same vertex it started at?

In triangle $A B C, A C=3 A B$. Let $A D$ bisect angle $A$ with $D$ lying on $B C$, and let $E$ be the foot of the perpendicular from $C$ to $A D$. Find $[A B D] /[C D E]$. (Here, $[X Y Z]$ denotes the area of triangle $X Y Z$ ).

In a chess-playing club, some of the players take lessons from other players. It is possible (but not necessary) for two players both to take lessons from each other. It so happens that for any three distinct members of the club, $A, B$, and $C$, exactly one of the following three statements is true: $A$ takes lessons from $B ; B$ takes lessons from $C ; C$ takes lessons from $A$. What is the largest number of players there can be?

There are three pairs of real numbers $\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)$, and $\left(x_{3}, y_{3}\right)$ that satisfy both $x^{3}-3 x y^{2}=2005$ and $y^{3}-3 x^{2} y=2004$. Compute $\left(1-\frac{x_{1}}{y_{1}}\right)\left(1-\frac{x_{2}}{y_{2}}\right)\left(1-\frac{x_{3}}{y_{3}}\right)$.

Let $n>0$ be an integer. Each face of a regular tetrahedron is painted in one of $n$ colors (the faces are not necessarily painted different colors.) Suppose there are $n^{3}$ possible colorings, where rotations, but not reflections, of the same coloring are considered the same. Find all possible values of $n$.

A cuboctahedron is a polyhedron whose faces are squares and equilateral triangles such that two squares and two triangles alternate around each vertex, as shown.

What is the volume of a cuboctahedron of side length 1 ?

The L shape made by adjoining three congruent squares can be subdivided into four smaller L shapes.

Each of these can in turn be subdivided, and so forth. If we perform 2005 successive subdivisions, how many of the $4^{2005} L$ 's left at the end will be in the same orientation as the original one?

Let $a_{1}=3$, and for $n \geq 1$, let $a_{n+1}=(n+1) a_{n}-n$. Find the smallest $m \geq 2005$ such that $a_{m+1}-1 \mid a_{m}^{2}-1$.

Triangle $A B C$ has incircle $\omega$ which touches $A B$ at $C_{1}, B C$ at $A_{1}$, and $C A$ at $B_{1}$. Let $A_{2}$ be the reflection of $A_{1}$ over the midpoint of $B C$, and define $B_{2}$ and $C_{2}$ similarly. Let $A_{3}$ be the intersection of $A A_{2}$ with $\omega$ that is closer to $A$, and define $B_{3}$ and $C_{3}$ similarly. If $A B=9, B C=10$, and $C A=13$, find $\left[A_{3} B_{3} C_{3}\right] /[A B C]$. (Here $[X Y Z]$ denotes the area of triangle $X Y Z$.)

A regular octahedron $A B C D E F$ is given such that $A D, B E$, and $C F$ are perpendicular. Let $G, H$, and $I$ lie on edges $A B, B C$, and $C A$ respectively such that $\frac{A G}{G B}=\frac{B H}{H C}=\frac{C I}{I A}=\rho$. For some choice of $\rho>1, G H, H I$, and $I G$ are three edges of a regular icosahedron, eight of whose faces are inscribed in the faces of $A B C D E F$. Find $\rho$.

Let $p=2^{24036583}-1$, the largest prime currently known. For how many positive integers $c$ do the quadratics $\pm x^{2} \pm p x \pm c$ all have rational roots?

One hundred people are in line to see a movie. Each person wants to sit in the front row, which contains one hundred seats, and each has a favorite seat, chosen randomly and independently. They enter the row one at a time from the far right. As they walk, if they reach their favorite seat, they sit, but to avoid stepping over people, if they encounter a person already seated, they sit to that person's right. If the seat furthest to the right is already taken, they sit in a different row. What is the most likely number of people that will get to sit in the first row?

Let $a_{1}, a_{2}, \ldots, a_{2005}$ be real numbers such that

$$\begin{array}{ccccccccccc} a_{1} \cdot 1 & + & a_{2} \cdot 2 & + & a_{3} \cdot 3 & + & \cdots & + & a_{2005} \cdot 2005 & = & 0 \\ a_{1} \cdot 1^{2} & + & a_{2} \cdot 2^{2} & + & a_{3} \cdot 3^{2} & + & \cdots & + & a_{2005} \cdot 2005^{2} & = & 0 \\ a_{1} \cdot 1^{3} & + & a_{2} \cdot 2^{3} & + & a_{3} \cdot 3^{3} & + & \cdots & + & a_{2005} \cdot 2005^{3} & = & 0 \\ \vdots & & \vdots & & \vdots & & & & \vdots & & \vdots \\ a_{1} \cdot 1^{2004} & + & a_{2} \cdot 2^{2004} & + & a_{3} \cdot 3^{2004} & + & \cdots & + & a_{2005} \cdot 2005^{2004} & = & 0 \end{array}$$

and

$$a_{1} \cdot 1^{2005}+a_{2} \cdot 2^{2005}+a_{3} \cdot 3^{2005}+\cdots+a_{2005} \cdot 2005^{2005}=1 .$$

What is the value of $a_{1}$ ?

In how many ways can the set of ordered pairs of integers be colored red and blue such that for all $a$ and $b$, the points $(a, b),(-1-b, a+1)$, and $(1-b, a-1)$ are all the same color?

How many regions of the plane are bounded by the graph of

$$x^{6}-x^{5}+3 x^{4} y^{2}+10 x^{3} y^{2}+3 x^{2} y^{4}-5 x y^{4}+y^{6}=0 ?$$

In a town of $n$ people, a governing council is elected as follows: each person casts one vote for some person in the town, and anyone that receives at least five votes is elected to council. Let $c(n)$ denote the average number of people elected to council if everyone votes randomly. Find $\lim _{n \rightarrow \infty} c(n) / n$.

There are 42 stepping stones in a pond, arranged along a circle. You are standing on one of the stones. You would like to jump among the stones so that you move counterclockwise by either 1 stone or 7 stones at each jump. Moreover, you would like to do this in such a way that you visit each stone (except for the starting spot) exactly once before returning to your initial stone for the first time. In how many ways can you do this?

In how many ways can 6 purple balls and 6 green balls be placed into a $4 \times 4$ grid of boxes such that every row and column contains two balls of one color and one ball of the other color? Only one ball may be placed in each box, and rotations and reflections of a single configuration are considered different.