OBM 1990

Show that a convex polyhedron with an odd number of faces has at least one face with an even number of edges.

There exists infinitely many positive integers such that $a^3 + 1990b^3 = c^4$ .

Each face of a tetrahedron is a triangle with sides $a, b,$ c and the tetrahedon has circumradius 1. Find $a^2 + b^2 + c^2$ .

$ABCD$ is a quadrilateral, $E,F,G,H$ are midpoints of $AB,BC,CD,DA$ . Find the point P such that $area (PHAE) = area (PEBF) = area (PFCG) = area (PGDH)$ .

Let $f(x)=\frac{ax+b}{cx+d}$ $F_n(x)=f(f(f...f(x)...))$ (with $n\ f's$ )

Suppose that $f(0) \not =0$ , $f(f(0)) \not = 0$ , and for some $n$ we have $F_n(0)=0$ , show that $F_n(x)=x$ (for any valid x).