Let $ABC$ be a triangle inscribed in the parabola $y=x^2$ such that the line $AB \parallel$ the axis $Ox$. Also point $C$ is closer to the axis $Ox$ than the line $AB$. Given that the length of the segment $AB$ is 1 shorter than the length of the altitude $CH$ (of the triangle $ABC$). Determine the angle $\angle{ACB}$ .

Pairwise distinct prime numbers $p, q, r$ satisfy the equality $$rp^3 + p^2 + p = 2rq^2 +q^2 + q.$$Determine all possible values of the product $pqr$.

The angles at the vertices $A$ and $C$ in the convex quadrilateral $ABCD$ are not acute. Points $K, L, M$ and $N$ are marked on the sides $AB, BC, CD$ and $DA$ respectively. Prove that the perimeter of $KLMN$ is not less than the double length of the diagonal $AC$.

There are $N$ cities in a country, some of which are connected by two-way flights. No city is directly connected with every other city. For each pair $(A, B)$ of cities there is exactly one route using at most two flights between them. Prove that $N - 1$ is a square of an integer.

Prove that $\frac{1}{x+y+1}-\frac{1}{(x+1)(y+1)}<\frac{1}{11}$ for all positive $x$ and $y$.

Points $C_1, A_1$ and $B_1$ are marked on the sides $AB, BC$ and $CA$ of a triangle $ABC$ so that the segments $AA_1, BB_1$, and $CC_1$ are concurrent (see the fig.). It is known that the area of the white part of the triangle $ABC$ is equal to the area of its black part. Prove that at least one of the segments $AA_1, BB_1, CC_1$ is a median of the triangle $ABC$.

a) $n$ $2\times2$ squares are drawn on the Cartesian plane. The sides of these squares are parallel to the coordinate axes. It is known that the center of any square is not an inner point of any other square. Let $\Pi$ be a rectangle such that it contains all these $n$ squares and its sides are parallel to the coordinate axes. Prove that the perimeter of $\Pi$ is greater than or equal to $4(\sqrt{n}+1)$. b) Prove the sharp estimate: the perimeter of $\Pi$ is greater than or equal to $2\lceil \sqrt{n}+1) \rceil$ (here $\lceil a\rceil$ stands for the smallest integer which is greater than or equal to $a$).

An $n\times n$ square is divided into $n^2$ unit cells. Is it possible to cover this square with some layers of 4-cell figures of the following shape (i.e. each cell of the square must be covered with the same number of these figures) if a) $n=6$? b) $n=7$? (The sides of each figure must coincide with the sides of the cells; the figures may be rotated and turned over, but none of them can go beyond the bounds of the square.)