AT-OEMO-FED1 2003

Find all triples of prime numbers $(p, q, r)$ such that $p^q + p^r$ is a perfect square.

Find the greatest and smallest value of $f(x, y) = y-2x$ , if x, y are distinct non-negative real numbers with $\frac{x^2+y^2}{x+y}\leq 4$ .

Given a positive real number $t$ , find the number of real solutions $a, b, c, d$ of the system $$a(1 - b^2) = b(1 -c^2) = c(1 -d^2) = d(1 - a^2) = t.$$

In a parallelogram $ABCD$ , points $E$ and $F$ are the midpoints of $AB$ and $BC$ , respectively, and $P$ is the intersection of $EC$ and $FD$ . Prove that the segments $AP,BP,CP$ and $DP$ divide the parallelogram into four triangles whose areas are in the ratio $1 : 2 : 3 : 4$ .