Find all functions $f : \mathbb{R} \to \mathbb{R} $ which verify the relation \[(x-2)f(y)+f(y+2f(x))= f(x+yf(x)), \qquad \forall x,y \in \mathbb R.\]

Let $AXYZB$ be a convex pentagon inscribed in a semicircle of diameter $AB$. Denote by $P$, $Q$, $R$, $S$ the feet of the perpendiculars from $Y$ onto lines $AX$, $BX$, $AZ$, $BZ$, respectively. Prove that the acute angle formed by lines $PQ$ and $RS$ is half the size of $\angle XOZ$, where $O$ is the midpoint of segment $AB$.

For which digits $a$ do exist integers $n \geq 4$ such that each digit of $\frac{n(n+1)}{2}$ equals $a \ ?$

An geometric progression starting at $a_0 = 3$ has an even number of terms. Suppose the difference between the odd indexed terms and even indexed terms is $39321$ and that the sum of the first and last term is $49155$. Find the common ratio of this geometric progression.

Let $A, B,C$ be colinear points in this order, $\omega$ an arbitrary circle passing through $B$ and $C$, and $l$ an arbitrary line different from $BC$, passing through A and intersecting $\omega$ at $M$ and $N$. The bisectors of the angles $\angle CMB$ and $\angle CNB$ intersect $BC$ at $P$ and $Q$. Prove that $AP\cdot AQ = AB \cdot AC$.

Consider a point $ M$ found in the same plane with the triangle $ ABC$, but not found on any of the lines $ AB,BC$ and $ CA$. Denote by $ S_1,S_2$ and $ S_3$ the areas of the triangles $ AMB,BMC$ and $ CMA$, respectively. Find the locus of $ M$ satisfying the relation: $ (MA^2\plus{}MB^2\plus{}MC^2)^2\equal{}16(S_1^2\plus{}S_2^2\plus{}S_3^2)$

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We call a permutation of the set of real numbers $\{a_1,\cdots,a_n\}$, $n\in\mathbb{N}$ [i]average increasing[/i] if the arithmetic mean of its first $k$ elements for $k=1,\cdots ,n$ form a strictly increasing sequence. 1) Depending on $n$, determine the smallest number that can be the last term of some average increasing permutation of the numbers $\{1,\cdots,n\}$; 2) Depending on $n$, determine the lowest position (in some general order) that the number $n$ can be achieved in some average increasing permutation of the numbers $\{1,\cdots,n\}.$ [i] Proposed by David Hruska, Czech Republic[/i]

A $50$-card deck consists of $4$ cards labeled " i" for $i = 1, 2,..., 12$ and $2$ cards labeled "$13$". If Bob randomly chooses $2$ cards from the deck without replacement, what is the probability that his $2$ cards have the same label?

For any number $n\in\mathbb{N},n\ge 2$, denote by $P(n)$ the number of pairs $(a,b)$ whose elements are of positive integers such that \[\frac{n}{a}\in (0,1),\quad \frac{a}{b}\in (1,2)\quad \text{and}\quad \frac{b}{n}\in (2,3). \] $a)$ Calculate $P(3)$. $b)$ Find $n$ such that $P(n)=2002$.

Prove that in a finite group G the number of subgroups with index n is at most $| G |^{2 \log_2 n}$.

Determine the value of the parameter $m$ such that the equation $(m-2)x^2+(m^2-4m+3)x-(6m^2-2)=0$ has real solutions, and the sum of the third powers of these solutions is equal to zero.

Given is a triangle $ABC$ with side lengths $a, b,c$. Three spheres touch each other in pairs and also touch the plane of the triangle at points $A,B$ and $C$, respectively. Determine the radii of these spheres.

Let $N$ denote the numbers of ordered triples of positive integers $(a, b, c)$ such that $a, b, c \le 3^6$ and $a^3 + b^3 + c^3$ is a multiple of $3^7$. Find the remainder when $N$ is divided by $1000$.

If $a, b, c$ are side lengths of a triangle, prove that \[(a + b)(b + c)(c + a) \geq 8(a + b - c)(b + c - a)(c + a - b).\]

For which $k$ the number $N = 101 ... 0101$ with $k$ ones is a prime?

In a triangle $ABC$, points $D,E$ lie on sides $AB,AC$ respectively. The lines $BE$ and $CD$ intersect at $F$. Prove that if $\color{white}\ .\quad\ \color{black}\ \quad BC^2=BD\cdot BA+CE\cdot CA,$ then the points $A,D,F,E$ lie on a circle.

In the triangle $ABC$, the side $BC$ is equal to $a$. Point $F$ is midpoint of $AB$, $I$ is the point of intersection of the bisectors of triangle $ABC$. It turned out that $\angle AIF = \angle ACB$. Find the perimeter of the triangle $ABC$. (Grigory Filippovsky)

Let $ABC$ be a triangle, $O$ its circumcenter and $R=1$ its circumradius. Let $G_1,G_2,G_3$ be the centroids of the triangles $OBC, OAC$ and $OAB.$ Prove that the triangle $ABC$ is equilateral if and only if $$AG_1+BG_2+CG_3=4$$

In a $2\times n$ array we have positive reals s.t. the sum of the numbers in each of the $n$ columns is $1$. Show that we can select a number in each column s.t. the sum of the selected numbers in each row is at most $\frac{n+1}4$.

Two courier companies offer services in the country of Old Aland. For any two towns, at least one of the companies offers a direct link in both directions between them. Additionally, each company is willing to chain together deliveries (so if they offer a direct link from $A$ to $B$, and $B$ to $C$, and $C$ to $D$ for instance, they will deliver from $A$ to $D$.) Show that at least one of the two companies must be able to deliver packages between any two towns in Old Aland.

For every positive integers $n,m$, show that there exist two sets $A,B$ which satisfy the following. [list] [*]$A$ is a set of $n$ successive positive integers, and $B$ is a set of $m$ successive positive integers. [*]$A\cup B = \phi$ [*]For every $a\in A$ and $b\in B$, $a$ and $b$ are not relatively prime. [/list]

For which real numbers $k > 1$ does there exist a bounded set of positive real numbers $S$ with at least $3$ elements such that $$k(a - b)\in S$$ for all $a,b\in S $ with $a > b?$ Remark: A set of positive real numbers $S$ is bounded if there exists a positive real number $M$ such that $x < M$ for all $x \in S.$

Given four points $ A_1, A_2, A_3, A_4$ in the plane, no three collinear, such that \[ A_1A_2 \cdot A_3 A_4 \equal{} A_1 A_3 \cdot A_2 A_4 \equal{} A_1 A_4 \cdot A_2 A_3, \] denote by $ O_i$ the circumcenter of $ \triangle A_j A_k A_l$ with $ \{i,j,k,l\} \equal{} \{1,2,3,4\}.$ Assuming $ \forall i A_i \neq O_i ,$ prove that the four lines $ A_iO_i$ are concurrent or parallel. [i]Nikolai Ivanov Beluhov, Bulgaria[/i]

Let $a$ and $b$ be positive integers and $c$ be a positive real number satisfying $$\frac{a + 1}{b + c}=\frac{b}{a}.$$ Prove that $c \ge 1$ holds. (Karl Czakler)