KURSCHAK 1994

The ratio of the sides of a parallelogram is $\lambda>1$ . Given $\lambda$ , determine the maximum of the acute angle subtended by the diagonals of the parallelogram.

Prove that if we erase $n-3$ diagonals of a regular $n$ -gon, then we may still choose $n-3$ of the remaining diagonals such that they don't intersect inside the $n$ -gon; but it is possible to erase $n-2$ diagonals such that this statement doesn't hold.

Consider the sets $A_1,A_2,\dots,A_n$ . Set $A_k$ is composed of $k$ disjoint intervals on the real axis ( $k=1,2,\dots,n$ ). Prove that from the intervals contained by these sets, one can choose $\left\lfloor\frac{n+1}2\right\rfloor$ intervals such that they belong to pairwise different sets $A_k$ , and no two of these intervals have a common point.