Školjka

%V0 Let $n \geq 2$ be a fixed integer. Find the least constant $C$ such the inequality $$\sum_{i<j} x_{i}x_{j} \left(x^{2}_{i}+x^{2}_{j} \right) \leq C \left(\sum_{i}x_{i} \right)^4$$ holds for any $x_{1}, \ldots ,x_{n} \geq 0$ (the sum on the left consists of $\binom{n}{2}$ summands). For this constant $C$, characterize the instances of equality.

%V0 Find all the functions $f: \mathbb{R} \mapsto \mathbb{R}$ such that $$f(x-f(y))=f(f(y))+xf(y)+f(x)-1$$ for all $x,y \in \mathbb{R}$.

%V0 Let $b$ be an even positive integer. We say that two different cells of a $n \times n$ board are neighboring if they have a common side. Find the minimal number of cells on he $n \times n$ board that must be marked so that any cell marked or not marked) has a marked neighboring cell.

%V0 A set $S$ of points from the space will be called completely symmetric if it has at least three elements and fulfills the condition that for every two distinct points $A$ and $B$ from $S$, the perpendicular bisector plane of the segment $AB$ is a plane of symmetry for $S$. Prove that if a completely symmetric set is finite, then it consists of the vertices of either a regular polygon, or a regular tetrahedron or a regular octahedron.

%V0 Two circles $\Omega_{1}$ and $\Omega_{2}$ touch internally the circle $\Omega$ in M and N and the center of $\Omega_{2}$ is on $\Omega_{1}$. The common chord of the circles $\Omega_{1}$ and $\Omega_{2}$ intersects $\Omega$ in $A$ and $B$. $MA$ and $MB$ intersects $\Omega_{1}$ in $C$ and $D$. Prove that $\Omega_{2}$ is tangent to $CD$.

%V0 Find all the pairs of positive integers $(x,p)$ such that p is a prime, $x \leq 2p$ and $x^{p-1}$ is a divisor of $(p-1)^{x}+1$.