Find all finite sets of positive integers with at least two elements such that for any two numbers $ a$, $ b$ ($ a > b$) belonging to the set, the number $ \frac {b^2}{a \minus{} b}$ belongs to the set, too.

Frank mistakenly believes that the number $1011$ is prime and for some integer $x$ writes down $(x+1)^{1011}\equiv x^{1011}+1\pmod{1011}$. However, it turns out that for Frank's choice of $x$, this statement is actually true. If $x$ is positive and less than $1011$, what is the sum of the possible values of $x$?

For a point $P = (a,a^2)$ in the coordinate plane, let $l(P)$ denote the line passing through $P$ with slope $2a$. Consider the set of triangles with vertices of the form $P_1 = (a_1, a_1^2), P_2 = (a_2, a_2^2), P_3 = (a_3, a_3^2)$, such that the intersection of the lines $l(P_1), l(P_2), l(P_3)$ form an equilateral triangle $\triangle$. Find the locus of the center of $\triangle$ as $P_1P_2P_3$ ranges over all such triangles.

Let $A,B,C,D,E$ be five points on a circle; some segments are drawn between the points so that each of the $5C2 = 10$ pairs of points is connected by either zero or one segments. Determine the number of sets of segments that can be drawn such that: • It is possible to travel from any of the five points to any other of the five points along drawn segments. • It is possible to divide the five points into two nonempty sets $S$ and $T$ such that each segment has one endpoint in $S$ and the other endpoint in $T$.

For function $ f: \mathbb{R} \to \mathbb{R}$ given that $ f(x^2 +x +3) +2 \cdot f(x^2 - 3x + 5) = 6x^2 - 10x +17$, calculate $ f(2009)$.

In a triangle $ABC$ with $AB = AC$, let $M$ be the midpoint of $CB$ and let $D$ be a point in $BC$ such that $\angle BAD = \frac{\angle BAC}{6}$. The perpendicular line to $AD$ by $C$ intersects $AD$ in $N$ where $DN = DM$. Find the angles of the triangle $BAC$.

Prove that any real number $0<x<1$ can be written as a difference of two positive and less than $1$ irrational numbers.

A coin is flipped until there is a head followed by two tails. What is the probability that this will take exactly $12$ flips?

Consider an alphabetized list of all the arrangements of the letters in the word BETWEEN. Then BEEENTW would be in position $1$ in the list, BEEENWT would be in position $2$ in the list, and so forth. Find the position that BETWEEN would be in the list.

Suppose that $p(x),q(x)$ are monic polynomials with nonnegative integer coefficients such that \[\frac{1}{5x}\ge\frac{1}{q(x)}-\frac{1}{p(x)}\ge\frac{1}{3x^2}\] for all integers $x\ge2$. Compute the minimum possible value of $p(1)\cdot q(1)$.

Let $AB$ be a chord of circle $\Gamma$. Let $O$ be the centre of a circle which is tangent to $AB$ at $C$ and internally tangent to $\Gamma$ at $P$. Point $C$ lies between $A$ and $B$. Let the circumcircle of triangle $POC$ intersect $\Gamma$ at distinct points $P$ and $Q$. Prove that $\angle{AQP}=\angle{CQB}$.

Compute $\sum_{k=0}^{\infty}\int_{0}^{\frac{\pi}{3}}\sin^{2k} x \, dx$.

An alien race has three genders: male, female and emale. A married triple consists of three persons, one from each gender who all like each other. Any person is allowed to belong to at most one married triple. The feelings are always mutual ( if $x$ likes $y$ then $y$ likes $x$). The race wants to colonize a planet and sends $n$ males, $n$ females and $n$ emales. Every expedition member likes at least $k$ persons of each of the two other genders. The problem is to create as many married triples so that the colony could grow. a) Prove that if $n$ is even and $k\geq 1/2$ then there might be no married triple. b) Prove that if $k \geq 3n/4$ then there can be formed $n$ married triple ( i.e. everybody is in a triple).

Given an $n \times n$ board which is divided into $n^2$ squares of size $1 \times 1$, all of which are white. Then, Aqua selects several squares from this board and colors them black. Ruby then places exactly one $1\times 2$ domino on the board, so that the domino covers exactly two squares on the board. Ruby can rotate the domino into a $2\times 1$ domino. After Aqua colors, it turns out there are exactly $2024$ ways for Ruby to place a domino on the board so that it covers exactly $1$ black square and $1$ white square. Determine the smallest possible value of $n$ so that Aqua and Ruby can do this. [i]Proposed by Muhammad Afifurrahman, Indonesia [/i]

Given a circle ${\omega }_{1}$ , and a circle ${\omega }_{2}$ inside it. An arbitrary circle ${\omega }_{3}$ is chosen which is tangent to the two latter circles and both tangencies are internal. The tangency points are linked by a segment. A tangent line to ${\omega }_{2}$ is drawn through the meet point of this segment and the circle ${\omega }_{2}$ . Thus a chord of the circle ${\omega }_{3}$ is obtained. Prove that the ends of all such chords (obtained by all possible choices of ${\omega }_{3}$ ) belong to a fixed circle. Pavel Kozhevnikov

An 8-by-8 square is divided into 64 unit squares in the usual way. Each unit square is colored black or white. The number of black unit squares is even. We can take two adjacent unit squares (forming a 1-by-2 or 2-by-1 rectangle), and flip their colors: black becomes white and white becomes black. We call this operation a [i]step[/i]. If $C$ is the original coloring, let $S(C)$ be the least number of steps required to make all the unit squares black. Find with proof the greatest possible value of $S(C)$.

Given an acute-angled $\vartriangle AB$C with altitude $AH$ ( $\angle BAC > 45^o > \angle AB$C). The perpendicular bisector of $AB$ intersects $BC$ at point $D$. Let $K$ be the midpoint of $BF$, where $F$ is the foot of the perpendicular from $C$ on $AD$. Point $H'$ is the symmetric to $H$ wrt $K$. Point $P$ lies on the line $AD$, such that $H'P \perp AB$. Prove that $AK = KP$.