ÖMO 2010

Show that $\frac{(x - y)^7 + (y - z)^7 + (z - x)^7 - (x - y)(y - z)(z - x) ((x - y)^4 + (y - z)^4 + (z - x)^4)} {(x - y)^5 + (y - z)^5 + (z - x)^5} \ge 3$ holds for all triples of distinct integers $x, y, z$ . When does equality hold?

On a circular billiard table a ball rebounds from the rails as if the rail was the tangent to the circle at the point of impact. A regular hexagon with its vertices on the circle is drawn on a circular billiard table. A (point-shaped) ball is placed somewhere on the circumference of the hexagon, but not on one of its edges. Describe a periodical track of this ball with exactly four points at the rails. With how many different directions of impact can the ball be brought onto such a track?

Consider the part of a lattice given by the corners $(0, 0), (n, 0), (n, 2)$ and $(0, 2)$ . From a lattice point $(a, b)$ one can move to $(a + 1, b)$ or to $(a + 1, b + 1)$ or to $(a, b - 1$ ), provided that the second point is also contained in the part of the lattice. How many ways are there to move from $(0, 0)$ to $(n, 2)$ considering these rules?

Two decompositions of a square into three rectangles are called substantially different, if reordering the rectangles does not change one into the other. How many substantially different decompositions of a $2010 \times 2010$ square in three rectangles with integer side lengths are there such that the area of one rectangle is equal to the arithmetic mean of the areas of the other rectangles?

A diagonal of a convex hexagon is called long if it decomposes the hexagon into two quadrangles. Each pair of long diagonals decomposes the hexagon into two triangles and two quadrangles. Given is a hexagon with the property, that for each decomposition by two long diagonals the resulting triangles are both isosceles with the side of the hexagon as base. Show that the hexagon has a circumcircle.