12th Brasil 1990


1. Show that a convex polyhedron with an odd number of faces has at least one face with an even number of edges.
2. Show that there are infinitely many positive integer solutions to a³ + 1990b³ = c⁴.
3. Each face of a tetrahedron is a triangle with sides a, b, c and the tetrahedon has circumradius 1. Find a² + b² + c².
4. ABCD is a convex quadrilateral. E, F, G, H are the midpoints of sides AB, BC, CD, DA respectively. Find the point P such that area PHAE = area PEBF = area PFCG = area PGDH.
5. Given that f(x) = (ax+b)/(cx+d), f(0) ≠ 0, f(f(0)) ≠ 0. Put F(x) = f(...(f(x) ... ) (where there are n fs). If F(0) = 0, show that F(x) = x for all x where the expression is defined.

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.