Find all functions $ f: \mathbb{R} \rightarrow \mathbb{R}$ for which the equality \[f(f(x)\minus{}y)\equal{}f(x)\plus{}f(f(y)\minus{}f(\minus{}x))\plus{}x\] holds for all real $x,y$.

Consider the complex numbers $a_1, a_2,\cdots , a_n$, $n \geq 2$. Which have the following properties: [list] [*] $|a_i|=1$ $\forall$ $i=1,2,\cdots , n$ [*] $\sum \limits_{k=1}^n arg(a_k)\leq \pi$ [/list] Show that the inequality$$\left(n^2\cot \left(\frac{\pi}{2n}\right)\right)^{-1}\left |\sum \limits_{k=0}^n(-1)^k\left[3n^2-(8k+5)n+4k(k+1)\sigma_k\right]\right |\geq \sqrt{\left(1+\frac{1}{n}\right)^2\cot^2 \left(\frac{\pi}{2n}\right)}+16\left |\sum \limits_{k=0}^n(-1)^k\sigma_k\right |$$where $\sigma_0=1$, $\sigma_k=\sum \limits_{1\leq i_1\leq i_2\leq \cdots \leq i_k\leq n}a_{i_1}a_{i_2}\cdots a_{i_k}$, $\forall$ $k=1,2,\cdots , n$

Prove that for every natural number $n > 1$ there exists a suquence $a_1$,$a_2$, $...$, $a_n$ of the numbers $1,2,...,n$ such that for each $k \in \{1,2,...,n-1\}$ the number $a_{k+1}$ divides $a_1+a_2+...+a_k$.

Let $ABC$ be a triangle, and let $D$ be a point on the internal angle bisector of $BAC$. Let $x$ be the ellipse with foci $B$ and $C$ passing through $D$, $y$ be the ellipse with foci $A$ and $C$ passing through $D$, and $z$ be the ellipse with foci $A$ and $B$ passing through $D$. Ellipses $x$ and $z$ intersect at distinct points $D$ and $E$, and ellipses $x$ and $y$ intersect at distinct points $D$ and $F$. Prove that $AD$ bisects angle $EAF$. [i]Andrew Carratu[/i]

We are given a triangle $ABC$ in a plane $P$. To any line $D$, not parallel to any side of the triangle, we associate the barycenter $G_D$ of the set of intersection points of $D$ with the sides of $\triangle ABC$. The object of this problem is determining the set $\mathfrak F$ of points $G_D$ when $D$ varies. (a) If $D$ goes over all lines parallel to a given line $\delta$, prove that $G_D$ describes a line $\Delta_\delta$. (b) Assume $\triangle ABC$ is equilateral. Prove that all lines $\Delta_\delta$ are tangent to the same circle as $\delta$ varies, and describe the set $\mathfrak F$. (c) If $ABC$ is an arbitrary triangle, prove that one can find a plane $P$ and an equilateral triangle $A'B'C'$ whose orthogonal projection onto $P$ is $\triangle ABC$, and describe the set $\mathfrak F$ in the general case.

To each vertex of a regular pentagon an integer is assigned, so that the sum of all five numbers is positive. If three consecutive vertices are assigned the numbers $x,y,z$ respectively, and $y<0$, then the following operation is allowed: $x,y,z$ are replaced by $x+y,-y,z+y$ respectively. Such an operation is performed repeatedly as long as at least one of the five numbers is negative. Determine whether this procedure necessarily comes to an end after a finite number of steps.

What’s the smallest integer $n>1$ such that $p \mid \left(n^{p-1}-1\right)$ for all integers $2 \leq p \leq 10?$ [i]Individual #6[/i]

Place points $A$, $B$, $C$, $D$, $E$, and $F$ evenly spaced on a unit circle. Compute the area of the shaded $12$-sided region, where the region is bounded by line segments $AD$, $DF$, $FB$, $BE$, $EC$, and $CA$. [center]<see attached>[/center]

For which pairs of positive integers $(m,n)$ there exists a set $A$ such that for all positive integers $x,y$, if $|x-y|=m$, then at least one of the numbers $x,y$ belongs to the set $A$, and if $|x-y|=n$, then at least one of the numbers $x,y$ does not belong to the set $A$? [i]Adapted by Dan Schwarz from A.M.M.[/i]

Find all the functions $ f :\mathbb{R}\mapsto\mathbb{R} $ with the following property: $ \forall x$ $f(x)= f(x/2) + (x/2)f'(x)$

Consider the equation $ax^{2}+by^{2}=1$, where $a,b$ are fixed rational numbers. Prove that either such an equation has no solutions in rational numbers, or it has infinitely many solutions.

A convex polyhedron has no diagonals (every pair of vertices are connected by an edge). Prove that it is a tetrahedron.

Consider an acute triangle $ABC$ of area $S$. Let $CD \perp AB$ ($D \in AB$), $DM \perp AC$ ($M \in AC$) and $DN \perp BC$ ($N \in BC$). Denote by $H_1$ and $H_2$ the orthocentres of the triangles $MNC$, respectively $MND$. Find the area of the quadrilateral $AH_1BH_2$ in terms of $S$.

Let $n$ be a given positive integer. Sisyphus performs a sequence of turns on a board consisting of $n + 1$ squares in a row, numbered $0$ to $n$ from left to right. Initially, $n$ stones are put into square $0$, and the other squares are empty. At every turn, Sisyphus chooses any nonempty square, say with $k$ stones, takes one of these stones and moves it to the right by at most $k$ squares (the stone should say within the board). Sisyphus' aim is to move all $n$ stones to square $n$. Prove that Sisyphus cannot reach the aim in less than \[ \left \lceil \frac{n}{1} \right \rceil + \left \lceil \frac{n}{2} \right \rceil + \left \lceil \frac{n}{3} \right \rceil + \dots + \left \lceil \frac{n}{n} \right \rceil \] turns. (As usual, $\lceil x \rceil$ stands for the least integer not smaller than $x$. )

How many integers $n \geq 2$ are there such that whenever $z_1, z_2, ..., z_n$ are complex numbers such that $$|z_1| = |z_2| = ... = |z_n| = 1 \text{ and } z_1 + z_2 + ... + z_n = 0,$$ then the numbers $z_1, z_2, ..., z_n$ are equally spaced on the unit circle in the complex plane? $\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 5$

Rui Xuen has a circle $\omega$ with center $O$, and a square $ABCJ$ with vertices on $\omega$. Let $M$ be the midpoint of $AB$, and let $\Gamma$ be the circle passing through the points $J$, $O$, $M$. Suppose $\Gamma$ intersect line $AJ$ at a point $P \neq J$, and suppose $\Gamma$ intersect $\omega$ at a point $Q \neq J$. A point $R$ lies on side $BC$ so that $RC = 3RB$. Help Rui Xuen prove that the points $P$, $Q$, $R$ are collinear.

For positive integers $n$, let $S_n$ be the set of integers $x$ such that $n$ distinct lines, no three concurrent, can divide a plane into $x$ regions (for example, $S_2=\{3,4\}$, because the plane is divided into 3 regions if the two lines are parallel, and 4 regions otherwise). What is the minimum $i$ such that $S_i$ contains at least 4 elements?

Consider an infinite white plane divided into square cells. For which $k$ it is possible to paint a positive finite number of cells black so that on each horizontal, vertical and diagonal line of cells there is either exactly $k$ black cells or none at all? A. Dinev, K. Garov, N Belukhov

Find the smallest positive integer $k$ such that there is exactly one prime number of the form $kx + 60$ for the integers $0 \le x \le 10$.

Let $ABC$ be an acute-angled triangle and variable $D \in [BC]$ . Let's denote by $E, F$ the feet of the perpendiculars from $D$ to $AB$, $AC$ respectively . a) Show that $$\frac{4S^2}{b^2+c^2}\le DE^2 + DF^2\le max \{h_B^2 + h_C^2 \}.$$ b) Proved that, if $D_0 \in [BC]$ is the point where the minimum of the sum $DE^2 + DF^2$ is achieved, then $D_0$ is the leg of the symmetrical median of $A$ facing the bisector of angle $A$. c) Specify the position, of $D \in [BC]$ for which the maximum of the sum $DE^2 + DF^2$ is achieved. (The area of the triangle $ABC$ was denoted by $S$ and $h_b, h_c$ are the lengths of the altitudes from $B$ and $C$ respectively)

Let $ABC$ be a non-right triangle with its altitudes intersecting in point $H$. Prove that $ABH$ is an acute triangle if and only if $\angle ACB$ is obtuse.

Find $n$ such that the mean of $\frac74$, $\frac65$, and $\frac1n$ is $1$.

Given cyclic quadrilateral $ABCD$. Four circles each touching its diagonals and the circumcircle internally are equal. Is $ABCD$ a square? (C.Pohoata, A.Zaslavsky)

The acute triangle $ABC$ with $AB\neq AC$ has circumcircle $\Gamma$, circumcenter $O$, and orthocenter $H$. The midpoint of $BC$ is $M$, and the extension of the median $AM$ intersects $\Gamma$ at $N$. The circle of diameter $AM$ intersects $\Gamma$ again at $A$ and $P$. Show that the lines $AP$, $BC$, and $OH$ are concurrent if and only if $AH = HN$.

In some cells of a rectangular table $m\times n (m, n> 1)$ is one checker. $Baby$ cut along the lines of the grid this table so that it is split into two equal parts, with the number of pieces on each side were the same. $Carlson$ changed the arrangement of checkers on the board (and on each side of the cage is still worth no more than one pieces). Prove that the $Baby$ may again cut the board into two equal parts containing an equal number of pieces