KURSCHAK 1985

We have triangulated a convex $(n+1)$ -gon $P_0P_1\dots P_n$ (i.e., divided it into $n-1$ triangles with $n-2$ non-intersecting diagonals). Prove that the resulting triangles can be labelled with the numbers $1,2,\dots,n-1$ such that for any $i\in\{1,2,\dots,n-1\}$ , $P_i$ is a vertex of the triangle with label $i$ .

For every $n\in\mathbb{N}$ , define the power sum of $n$ as follows. For every prime divisor $p$ of $n$ , consider the largest positive integer $k$ for which $p^k\le n$ , and sum up all the $p^k$ 's. (For instance, the power sum of $100$ is $2^6+5^2=89$ .) Prove that the power sum of $n$ is larger than $n$ for infinitely many positive integers $n$ .

We reflected each vertex of a triangle on the opposite side. Prove that the area of the triangle formed by these three reflection points is smaller than the area of the initial triangle multiplied by five.