14th Brasil 1992


1. The polynomial x³ + px + q has three distinct real roots. Show that p < 0.
2. Show that there is a positive integer n such that the first 1992 digits of n¹⁹⁹² are 1.
3. Given positive real numbers x₁, x₂, ... , x_n find the polygon A₀A₁..A_n with A₀A₁ = x₁, A₁A₂ = x₂, ... , A_n-1A_n = x_n which has greatest area.
4. ABC is a triangle. Find D on AC and E on AB such that area ADE = area DEBC and DE has minimum possible length.
5. Let d(n) be the number of positive divisors of n. Show that n(1/2 + 1/3 + ... + 1/n) ≤ d(1) + d(2) + ... + d(n) ≤ n(1 + 1/2 + 1/3 + ... + 1/n).
6. Given a set of n elements, find the largest number of subsets such that no subset is contained in any other.
7. Find all solutions in positive integers to n^a + n^b = n^c.
8. In a chess tournament each player plays every other player once. A player gets 1 point for a win, ½ point for a draw and 0 for a loss. Both men and women played in the tournament and each player scored the same total of points against women as against men. Show that the total number of players must be a square.
9. Show that for each n > 5 it is possible to find a convex polyhedron with all faces congruent such that each face has another face parallel to it.

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