Let $a$ be the largest root of the equation $x^3 - 3x^2 + 1 = 0$. Find the first $200$ decimal digits for the number $a^{2000}$.

Let $ABC$ be an isosceles acute triangle ($AB = BC$). On the side $BC$ we mark a point $P$, such that $\angle PAC = 45^o$, and $Q$ is the point of intersection of the perpendicular bisector of the segment $AP$ with the side $AB$. Prove that $PQ \perp BC$.

We create a sequence by setting $a_1 = 2010$ and requiring that $a_n-a_{n-1}\leq n$ and $a_n$ is also divisible by $n$. Show that $a_{100},a_{101},a_{102},\dots$ form an arithmetic sequence.

Determine all functions $f: \mathbb{Z}\to\mathbb{Z}$ satisfying the following property: For each pair of integers $m$ and $n$ (not necessarily distinct), $\mathrm{gcd}(m, n)$ divides $f(m) + f(n)$. Note: If $n\in\mathbb{Z}$, $\mathrm{gcd}(m, n)=\mathrm{gcd}(|m|, |n|)$ and $\mathrm{gcd}(n, 0)=n$.

What is the largest $n$ such that there exists a non-degenerate convex $n$-gon such that each of its angles are an integer number of degrees, and are all distinct?

Let $A$ and $B$ be rectangles in the plane and $f : A \rightarrow B$ be a mapping which is uniform on the interior of $A$, maps the boundary of $A$ homeomorphically to the boundary of $B$ by mapping the sides of $A$ to corresponding sides in $B$. Prove that $f$ is an affine transformation.

Let $f(x) = x^3 + 3x - 1$ have roots $ a, b, c$. Given that $\frac{1}{a^3 + b^3}+\frac{1}{b^3 + c^3}+\frac{1}{c^3 + a^3}$ can be written as $\frac{m}{n}$, where $m, n$ are positive integers and $gcd(m, n) = 1$, find $100m + n$.

Given prime numbers $p$ and $q$ with $p<q$, determine all pairs $(x,y)$ of positive integers such that $$\frac1x+\frac1y=\frac1p-\frac1q.$$

You are standing at a pole and a snail is moving directly away from the pole at $1$ cm/s. When the snail is $1$ meter away, you start "Round 1". In Round $n$ ($n\ge 1$), you move directly toward the snail at $n+1$ cm/s. When you reach the snail, you immediately turn around and move back to the starting pole at $n + 1$ cm/s. When you reach the pole, you immediately turn around and Round $n + 1$ begins. At the start of Round $100$, how many meters away is the snail?

Real numbers $x,y,a$ are such that $x+y=x^2+y^2=x^3+y^3=a$. Find all the possible values of $a$.

Integers not divisible by $2012$ are arranged on the arcs of an oriented graph. We call the [i]weight of a vertex [/i]the difference between the sum of the numbers on the arcs coming into it and the sum of the numbers on the arcs going away from it. It is known that the weight of each vertex is divisible by $2012$. Prove that non-zero integers with absolute values not exceeding $2012$ can be arranged on the arcs of this graph, so that the weight of each vertex is zero. [i]Proposed by W. Tutte[/i]

Let $y_1$ and $y_2$ be two linearly independent solutions of the equation $$y''+P(x)y'+Q(x)=0.$$ Find the differential equation satisfied by the product $z=y_1 y_2$.

The nine squares in the table shown are to be filled so that every row and every column contains each of the numbers $1,2,3$. Then $A+B=$ \[\begin{tabular}{|c|c|c|} \hline 1 & & \\ \hline & 2 & A \\ \hline & & B \\ \hline \end{tabular}\] $\text{(A)}\ 2 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 5 \qquad \text{(E)}\ 6$

There were $n{}$ positive integers. For each pair of those integers Boris wrote their arithmetic mean onto a blackboard and their geometric mean onto a whiteboard. It so happened that for each pair at least one of those means was integer. Prove that on at least one of the boards all the numbers are integer. [i]Boris Frenkin[/i]

Let $C_1$ and $C_2$ be two tangent unit circles inside a circle $C$ of radius $2$. Circle $C_3$ inside $C$ is tangent to the circles $C,C_1,C_2$, and circle $C_4$ inside $C$ is tangent to $C,C_1,C_3$. Prove that the centers of $C,C_1,C_3$ and $C_4$ are vertices of a rectangle.

Three mutually externally tangent circles are internally tangent to a circle with radius $1$. If two of the inner circles have radius $\frac{1}{3}$, the largest possible radius of the third inner circle can be expressed in the form $\frac{a+b\sqrt{c}}{d}$ where $c$ is squarefree and $\gcd(a,b,d)=1$. Find $a+b+c+d$.

The vertices of a triangle are lattice points (they have integer coordinates). There are no other lattice points on the boundary of the triangle, but there is exactly one lattice point inside the triangle. Show that it must be the centroid.

Let $k$ be any fixed positive integer. Let's look at integer pairs $(a, b)$, for which the quadratic equations $x^2 - 2ax + b = 0$ and $y^2 + 2ay + b = 0$ are real solutions (not necessarily different), which can be denoted by $x_1, x_2$ and $y_1, y_2$, respectively, in such an order that the equation $x_1 y_1 - x_2 y_2 = 4k$. a) Find the largest possible value of the second component $b$ of such a pair of numbers ($a, b)$. b) Find the sum of the other components of all such pairs of numbers.

All vertices of a hexagon, whose sides may intersect at points other than the vertices, lie on a circle. (a) Draw a hexagon such that it has the largest possible number of points of self-intersection. (b) Prove that this number is indeed maximum. (NB Vassiliev)

In triangle $ABC$, $M,N$ and $P$ are midpoints of sides $BC,CA$ and $AB$. Point $K$ lies on segment $NP$ so that $AK$ bisects $\angle BKC$. Lines $MN,BK$ intersects at $E$ and lines $MP,CK$ intersects at $F$. Suppose that $H$ be the foot of perpendicular line from $A$ to $BC$ and $L$ the second intersection of circumcircle of triangles $AKH, HEF$. Prove that $MK,EF$ and $HL$ are concurrent. [i]Proposed by Alireza Dadgarnia[/i]

Show that the four perpendiculars dropped from the midpoints of the sides of a cyclic quadrilateral to the respective opposite sides are concurrent. [b]Note by Darij:[/b] A [i]cyclic quadrilateral [/i]is a quadrilateral inscribed in a circle.

Let $S$ be the sum of all possible values of $a \cdot c$ such that $$a^3+3ab^2-72ab+432a=4c^3$$ if $a$, $b$, and $c$ are positive integers, $a+b > 11$, $a > b-13$, and $c \le 1000$. Find the sum of all distinct prime factors of $S$. [i]Proposed by Kevin Zhao[/i]

Hannah Kubik leads a local volunteer group of thirteen adults that takes turns holding classes for patients at the Children's Hospital. At the end of August, Hannah took a tour of the hospital and talked with some members of the staff. Dr. Yang told Hannah that it looked like there would be more girls than boys in the hospital during September. The next day Hannah brought the volunteers together and it was decided that three women and two men would volunteer to run the September classes at the Children's Hospital. If there are exactly six women in the volunteer group, how many combinations of three women and two men could Hannah choose from the volunteer group to run the classes?

Let $\omega$ be a unit circle with center $O$ and diameter $AB$. A point $C$ is chosen on $\omega$. Let $M$, $N$ be the midpoints of arc $AC$, $BC$, respectively, and let $AN,BM$ intersect at $I$. Suppose that $AM,BC,OI$ concur at a point. Find the area of $\triangle ABC$. [i]Proposed by Kevin You[/i]

Line $l$ tangents unit circle $S$ in point $P$. Point $A$ and circle $S$ are on the same side of $l$, and the distance from $A$ to $l$ is $h$ ($h > 2$). Two tangents of circle $S$ are drawn from $A$, and intersect line $l$ at points $B$ and $C$ respectively. Find the value of $PB \cdot PC$.