RIOPLATENSE 1995

Let $n$ and $p$ be two integers with $p$ positive prime, such that $pn + 1$ is a perfect square. Show that $n + 1$ is the sum of $p$ perfect squares, not necessarily distinct.

In a circle of center $O$ and radius $r$ , a triangle $ABC$ of orthocenter $H$ is inscribed. It is considered a triangle $A'B'C'$ whose sides have by length the measurements of the segments $AB, CH$ and $2r$ . Determine the triangle $ABC$ so that the area of the triangle $A'B'C'$ is maximum.

Given a regular tetrahedron with edge $a$ , its edges are divided into $n$ equal segments, thus obtaining $n + 1$ points: $2$ at the ends and $n - 1$ inside. The following set of planes is considered: $\bullet$ those that contain the faces of the tetrahedron, and $\bullet$ each of the planes parallel to a face of the tetrahedron and containing at least one of the points determined above. Now all those points $P$ that belong (simultaneously) to four planes of that set are considered. Determine the smallest positive natural $n$ so that among those points $P$ the eight vertices of a square-based rectangular parallelepiped can be chosen.

Given the natural numbers $a$ and $b$ , with $1 \le a <b$ , prove that there exist natural numbers $n_1<n_2< ...<n_k$ , with $k \le a$ such that $$\frac{a}{b}=\frac{1}{n_1}+\frac{1}{n_2}+...+\frac{1}{n_k}$$

Consider $2n$ points in the plane. Two players $A$ and $B$ alternately choose a point on each move. After $2n$ moves, there are no points left to choose from and the game ends. Add up all the distances between the points chosen by $A$ and add up all the distances between the points chosen by $B$ . The one with the highest sum wins. If $A$ starts the game, describe the winner's strategy.

Clarification: Consider that all the partial sums of distances between points give different numbers.

A convex polygon with $2n$ sides is called rhombic if its sides are equal and all pairs of opposite sides are parallel. A rhombic polygon can be partitioned into rhombic quadrilaterals. For what value of $n$ , a $2n$ -sided rhombic polygon splits into $666$ rhombic quadrilaterals?