Problem 1

Prove that in any triangle the center of the circumscribed circle is closer to the barycenter of the triangle than it is to the inscribed circle.

Problem 2

Let p be a prime number greater than 5. Prove that the set X = { p - n2 | n integer and n2 <p} contains two different elements x and y, x different from 1, such that x divides y.

Problem 3

Let ABCDE be a convex pentagon and let M, N, P, Q, S be the midpoints of its sides, namely AB, BC, CD, DE, EA. If the lines DM, EN, AP and BQ have a common point, then this point belongs to CS.

Problem 4

Prove or disprove the following statement:
There is a subset A of the set { 1, 2, 3,... , 21996 - 1 } with at most 2012 elements such that 1 and 21996 - 1 both belong to A, and every element of A \ { 1 } is the sum of two not necessarily distinct elements of A.