IMAC 2009

Prove for the sidelengths $a,b,c$ of a triangle $ABC$ the inequality $\frac{a^3}{b+c-a}+\frac{b^3}{c+a-b}+\frac{c^3}{a+b-c}\ge a^2+b^2+c^2$

In the triangle $ABC$ , the circle with the center at the point $O$ touches the pages $AB, BC$ and $CA$ in the points $C_1, A_1$ and $B_1$ , respectively. Lines $AO, BO$ and $CO$ cut the inscribed circle at points $A_2, B_2$ and $C_2,$ respectively. Prove that it is the area of the triangle $A_2B_2C_2$ is double from the surface of the hexagon $B_1A_2C_1B_2A_1C_2$ .

(Moldova)

In the interior of the convex polygon $A_1A_2...A_{2n}$ there is point $M$ . Prove that at least one side of the polygon has not intersection points with the lines $MA_i$ , $1\le i\le 2n$ .

(Spain)