Find three different polynomials $P(x)$ with real coefficients such that $P\left(x^2 + 1\right) = P(x)^2 + 1$ for all real $x$.

Two perpendicular chords are drawn through a given interior point $P$ of a circle with radius $R.$ Determine, with proof, the maximum and the minimum of the sum of the lengths of these two chords if the distance from $P$ to the center of the circle is $kR.$

There are 15 people at a party; each person has 10 friends. To greet each other each person hugs all their friends. How many hugs are exchanged at this party?

Let $ABC$ be an acute-angled, nonisosceles triangle. Point $A_1, A_2$ are symmetric to the feet of the internal and the external bisectors of angle $A$ wrt the midpoint of $BC$. Segment $A_1A_2$ is a diameter of a circle $\alpha$. Circles $\beta$ and $\gamma$ are defined similarly. Prove that these three circles have two common points.

Determine all triples $(p, q, r)$ of prime numbers for which $p^q + p^r$ is a perfect square.

Starting with the triple $(1007\sqrt{2},2014\sqrt{2},1007\sqrt{14})$, define a sequence of triples $(x_{n},y_{n},z_{n})$ by $x_{n+1}=\sqrt{x_{n}(y_{n}+z_{n}-x_{n})}$ $y_{n+1}=\sqrt{y_{n}(z_{n}+x_{n}-y_{n})}$ $ z_{n+1}=\sqrt{z_{n}(x_{n}+y_{n}-z_{n})}$ for $n\geq 0$.Show that each of the sequences $\langle x_n\rangle _{n\geq 0},\langle y_n\rangle_{n\geq 0},\langle z_n\rangle_{n\geq 0}$ converges to a limit and find these limits.

In a convex quadrilateral $ABCD$, consider quadrilateral $KLMN$ formed by the centers of mass of triangles $ABC, BCD, DBA, CDA$. Prove that the straight lines connecting the midpoints of the opposite sides of quadrilateral $ABCD$ meet at the same point as the straight lines connecting the midpoints of the opposite sides of $KLMN$.

Two points on the distance 1 are given in a plane. It is allowed to draw a line through two marked points, as well as a circle centered in a marked point with radius equal to the distance between some two marked points. By marked points we mean the two initial points and intersection points of two lines, two circles, or a line and a circle constructed so far. Let $C(n)$ be the minimum number of circles needed to construct two points on the distance $n$ if only a compass is used, and let $LC(n)$ be the minimum total number of circles and lines needed to do so if a ruler and a compass are used, where $n$ is a natural number. Prove that the sequence $C(n)/LC(n)$ is not bounded. [i]A. Belov[/i]

The number $328$ is written on the board. Two players alternate writing positive divisors of $328$ on the board, subject to the following rules: $\bullet$ No divisor of a previously written number may be written. $\bullet$ The player who writes 328 loses. Who has a winning strategy, the first player or the second player?

Evaluate $50(\cos 39^{\circ}\cos21^{\circ}+\cos129^{\circ}\cos69^{\circ})$

Are there positive real numbers $x$, $y$ and $z$ such that $ x^4 + y^4 + z^4 = 13\text{,} $ $ x^3y^3z + y^3z^3x + z^3x^3y = 6\sqrt{3} \text{,} $ $ x^3yz + y^3zx + z^3xy = 5\sqrt{3} \text{?} $ [i]Proposed by Matko Ljulj.[/i]

None of the positive integers $k,m,n$ are divisible by $5$. Prove that at least one of the numbers $k^2-m^2,m^2-n^2,n^2-k^2$ is divisible by $5$.

Let $a_1, a_2, ..., a_n $ be a sequence such that the arithmetic mean of the $n$ terms is $n$. Consider $n = 2009$. Determine the sum of the $2009$ terms of the sequence.

Anna and Boris move simultaneously towards each other, from points A and B respectively. Their speeds are constant, but not necessarily equal. Had Anna started 30 minutes earlier, they would have met 2 kilometers nearer to B. Had Boris started 30 minutes earlier instead, they would have met some distance nearer to A. Can this distance be uniquely determined? [i](3 points)[/i]

For all positive integers $n$ define $a_n=2 \underbrace{33...3}_{n \, times}$, where digit $3$ occurs $n$ times. Show that the number $a_{2009}$ has infinitely many multiples in the set $\{a_n | n \in N*\}$.

Prove that there exist infinitely many integers $n$ such that $n$, $n+1$, $n+2$ are each the sum of the squares of two integers. [Example: $0=0^2+0^2$, $1=0^2+1^2$, $2=1^2+1^2$.]

A baker uses $6\tfrac{2}{3}$ cups of flour when she prepares $\tfrac{5}{3}$ recipes of rolls. She will use $9\tfrac{3}{4}$ cups of flour when she prepares $\tfrac{m}{n}$ recipes of rolls where m and n are relatively prime positive integers. Find $m + n.$

(a) Is there a natural number $n$ such that the number $2^n$ has last digit $6$ and the sum of the other digits is $2$? b) Are there natural numbers $a$ and $m\ge 3$ such that the number $a^m$ has last digit $6$ and the sum of the other digits is 3?

Let $O$ be the circumcircle of $\triangle ABC$, with $AC=BC$ end let $D=AO\cap BC$. If $BD$ and $CD$ are integer numbers and $AO-CD$ is prime, determine such three numbers.

Circle $ \Gamma_{1},$ with radius $ r,$ is internally tangent to circle $ \Gamma_{2}$ at $ S.$ Chord $ AB$ of $ \Gamma_{2}$ is tangent to $ \Gamma_{1}$ at $ C.$ Let $ M$ be the midpoint of arc $ AB$ (not containing $ S$), and let $ N$ be the foot of the perpendicular from $ M$ to line $ AB.$ Prove that $ AC\cdot CB\equal{}2r\cdot MN.$

Given an integer $n \geqslant 2$. Suppose there is a point $P$ inside a convex cyclic $2n$-gon $A_1 \ldots A_{2n}$ satisfying $$\angle PA_1A_2 = \angle PA_2A_3 = \ldots = \angle PA_{2n}A_1,$$prove that $$ \prod_{i=1}^{n} \left|A_{2i - 1}A_{2i} \right| = \prod_{i=1}^{n} \left|A_{2i}A_{2i+1} \right|,$$where $A_{2n + 1} = A_1$.

A massless elastic cord (that obeys Hooke's Law) will break if the tension in the cord exceeds $T_{max}$. One end of the cord is attached to a fixed point, the other is attached to an object of mass $3m$. If a second, smaller object of mass m moving at an initial speed $v_0$ strikes the larger mass and the two stick together, the cord will stretch and break, but the final kinetic energy of the two masses will be zero. If instead the two collide with a perfectly elastic one-dimensional collision, the cord will still break, and the larger mass will move off with a final speed of $v_f$. All motion occurs on a horizontal, frictionless surface. Find $v_f/v_0$. $ \textbf{(A)}\ 1/\sqrt{12}\qquad\textbf{(B)}\ 1/\sqrt{2}\qquad\textbf{(C)}\ 1/\sqrt{6} \qquad\textbf{(D)}\ 1/\sqrt{3}\qquad\textbf{(E)}\ \text{none of the above} $

Let $S_1$ and $S_2$ be sets of points on the coordinate plane $\mathbb{R}^2$ defined as follows \[S_1={(x,y)\in \mathbb{R}^2:|x+|x||+|y+|y||\le 2}\] \[S_2={(x,y)\in \mathbb{R}^2:|x-|x||+|y-|y||\le 2}\] Find the area of the intersection of $S_1$ and $S_2$

Let $ABC$ be a equilateral triangle and let $P$ be any point on the minor arc $AC$ of the circumcircle of $ABC$.Prove that $PB=PA+PC$

There are 100 apples on the table with total weight of 10 kg. Each apple weighs no less than 25 grams. The apples need to be cut for 100 children so that each of the children gets 100 grams. Prove that you can do it in such a way that each piece weighs no less than 25 grams.