NL-NWO 1993

Show that any subset of $V= \{ 1,2,...,24,25 \}$ with $17$ or more elements contains at least two distinct numbers the product of which is a perfect square.

In a triangle $ABC$ with $\angle A=90^{\circ}$ , $D$ is the midpoint of $BC$ , $F$ that of $AB$ , $E$ that of $AF$ and $G$ that of $FB$ . Segment $AD$ intersects $CE,CF$ and $CG$ in $P,Q$ and $R$ , respectively. Determine the ratio: $\frac{PQ}{QR}$ .

A sequence of numbers is defined by $u_1=a, u_2=b$ and $u_{n+1}=\frac{u_n+u_{n-1}}{2}$ for $n \ge 2$ . Prove that $\lim_{n\to\infty}u_n$ exists and express its value in terms of $a$ and $b$ .

Let $C$ be a circle with center $M$ in a plane $V$ , and $P$ be a point not on the circle $C$ . $(a)$ If $P$ is fixed, prove that $AP^2+BP^2$ is a constant for every diameter $AB$ of the circle $C$ . $(b)$ Let $AB$ be a fixed diameter of $C$ and $P$ a point on a fixed sphere $S$ not intersecting $V$ . Determine the points $P$ on $S$ that minimize $AP^2+BP^2$ .

Eleven distinct points $P_1,P_2,...,P_{11}$ are given on a line so that $P_i P_j \le 1$ for every $i,j$ . Prove that the sum of all distances $P_i P_j, 1 \le i <j \le 11$ , is smaller than $30$ .