DANUBE 2010

Determine all integer numbers $n\ge 3$ such that the regular $n$ -gon can be decomposed into isosceles triangles by non-intersecting diagonals.

Given a triangle $ABC$ , let $A',B',C'$ be the perpendicular feet dropped from the centroid $G$ of the triangle $ABC$ onto the sides $BC,CA,AB$ respectively. Reflect $A',B',C'$ through $G$ to $A'',B'',C''$ respectively. Prove that the lines $AA'',BB'',CC''$ are concurrent.

All sides and diagonals of a convex $n$ -gon, $n\ge 3$ , are coloured one of two colours. Show that there exist $\left[\frac{n+1}{3}\right]$ pairwise disjoint monochromatic segments.

(Two segments are disjoint if they do not share an endpoint or an interior point).

Let $n\ge3$ be a positive integer. Find the real numbers $x_1\ge0,\ldots,x_n\ge 0$ , with $x_1+x_2+\ldots +x_n=n$ , for which the expression $$(n-1)(x_1^2+x_2^2+\ldots+x_n^2)+nx_1x_2\ldots x_n$$ takes a minimal value.