**BALKAN MATHEMATICAL OLYMPIAD
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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 -
n^{2} | n integer and n^{2} <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,... , 2^{1996} - 1 } with
at most 2012 elements such that 1 and 2^{1996} - 1 both belong to
A, and every element of A \ { 1 } is the sum of two not necessarily
distinct elements of A.