CZ-SK-MO 1991

Prove that for any real numbers $p,q,r,\phi$ ,: $$\cos^2\phi+q \sin \phi \cos \phi +r\sin^2 \phi \ge \frac12 (p+r-\sqrt{(p-r)^2+q^2})$$

A museum has the shape of a (not necessarily convex) 3 $n$ -gon. Prove that $n$ custodians can be positioned so as to control all of the museum's space.

For any permutation $p$ of the set $\{1,2,...,n\}$ , let us denote $d(p) = |p(1)-1|+|p(2)-2|+...+|p(n)-n|$ . Let $i(p)$ be the number of inversions of $p$ , i.e. the number of pairs $1 \le i < j \le n$ with $p(i) > p(j)$ . Prove that $d(p)\le 2i(p)$ $.

Prove that in all triangles $ABC$ with $\angle A = 2\angle B$ the distance from $C$ to $A$ and to the perpendicular bisector of $AB$ are in the same ratio.

In a group of mathematicians everybody has at least one friend (friendship is a symmetric relation). Show that there is a mathematician all of whose friends have average number of friends not smaller than the average number of friends in the whole group.

The set $N$ is partitioned into three subsets $A_1,A_2,A_3$ . Prove that at least one of them has the following property: There exists a positive number $m$ such that for any $k$ one can find numbers $a_1 < a_2 < ... < a_k$ in that subset satisfying $a_{j+1} -a_j \le m$ for $j = 1,...,k -1$ .