VMO 1969

A graph $G$ has $n + k$ vertices. Let $A$ be a subset of $n$ vertices of the graph $G$ , and $B$ be a subset of other $k$ vertices. Each vertex of $A$ is joined to at least $k - p$ vertices of $B$ . Prove that if $np < k$ then there is a vertex in $B$ that can be joined to all vertices of $A$ .

Find all real $x$ such that $0 < x < \pi$ and $\frac{8}{3 sin x - sin 3x} + 3 sin^2x \le 5$ .

Consider $x_1 > 0, y_1 > 0, x_2 < 0, y_2 > 0, x_3 < 0, y_3 < 0, x_4 > 0, y_4 < 0.$ Suppose that for each $i = 1, ... ,4$ we have $(x_i -a)^2 +(y_i -b)^2 \le c^2$ . Prove that $a^2 + b^2 < c^2$ . Restate this fact in the form of geometric result in plane geometry.

Two circles centers $O$ and $O'$ , radii $R$ and $R'$ , meet at two points. A variable line $L$ meets the circles at $A, C, B, D$ in that order and $\frac{AC}{AD} = \frac{CB}{BD}$ . The perpendiculars from $O$ and $O'$ to $L$ have feet $H$ and $H'$ . Find the locus of $H$ and $H'$ . If $OO'^2 < R^2 + R'^2$ , find a point $P$ on $L$ such that $PO + PO'$ has the smallest possible value. Show that this value does not depend on the position of $L$ . Comment on the case $OO'^2 > R^2 + R'^2$ .