2nd Brasil 1980


1. Box A contains black balls and box B contains white balls. Take a certain number of balls from A and place them in B. Then take the same number of balls from B and place them in A. Is the number of white balls in A then greater, equal to, or less than the number of black balls in B?
2. Show that for any positive integer n > 2 we can find n distinct positive integers such that the sum of their reciprocals is 1.
3. Given a triangle ABC and a point P₀ on the side AB. Construct points P_i, Q_i, R_i as follows. Q_i is the foot of the perpendicular from P_i to BC, R_i is the foot of the perpendicular from Q_i to AC and P_i is the foot of the perpendicular from R_i-1 to AB. Show that the points P_i converge to a point P on AB and show how to construct P.
4. Given 5 points of a sphere radius r, show that two of the points are a distance ≤ r √2 apart.

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