BALKAN MATHEMATICAL OLYMPIAD
BACAU, ROMANIA, 1996
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.