Let $x_1, x_2, ... , x_n$ and $y_1, y_2, ..., y_n$ be positive real numbers so that $$x_1 + x_2 + ...+ x_n \ge x_1y_1 + x_2y_2 + ... + x_ny_n.$$Show that for any non-negative integer $p$ the following inequality holds $$\frac{x_1}{y_1^p} +\frac{ x_2}{y_2^p} + ...+ \frac{x_n}{y_n^p} \ge x_1 + x_2 + ...+ x_n.$$

Let $\omega$ be a circle in the plane and $A,B$ two points lying on it. We denote by $M$ the midpoint of $AB$ and let $P \ne M$ be a new point on $AB$. Build circles $\gamma$ and $\delta$ tangent to $AB$ at $P$ and to $\omega$ at $C$, respectively $D$. Consider $E$ to be the point diametrically opposed to $D$ in $\omega$. Prove that the circumcenter of $\triangle BMC$ lies on the line $BE$.

Let $A,B,C$ be nodes of the lattice $Z\times Z$ such that inside the triangle $ABC$ lies a unique node $P$ of the lattice. Denote $E = AP \cap BC$. Determine max $\frac{AP}{PE}$ , over all such configurations.

Determine all non-constant polynomials $ f\in \mathbb{Z}[X]$ with the property that there exists $ k\in\mathbb{N}^*$ such that for any prime number $ p$, $ f(p)$ has at most $ k$ distinct prime divisors.

The cells of a $(n^2-n+1)\times(n^2-n+1)$ matrix are coloured using $n$ colours. A colour is called dominant on a row (or a column) if there are at least $n$ cells of this colour on that row (or column). A cell is called extremal if its colour is dominant both on its row, and its column. Find all $n \ge 2$ for which there is a colouring with no extremal cells. Iurie Boreico (Moldova)