13th Balkan 1996


A1. Let d be the distance between the circumcenter and the centroid of a triangle. Let R be its circumradius and r the radius of its inscribed circle. Show that d² ≤ R(R - 2r).
A2. p > 5 is prime. A = {p - n² where n² < p}. Show that we can find integers a and b in A such that a > 1 and a divides b.
A3. In a convex pentagon consider the five lines joining a vertex to the midpoint of the opposite side. Show that if four of these lines pass through a point, then so does the fifth.
A4. Can we find a subset X of {1, 2, 3, ... , 2¹⁹⁹⁶-1} with at most 2012 elements such that 1 and 2¹⁹⁹⁶-1 belong to X and every element of X except 1 is the sum of two distinct elements of X or twice an element of X?

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