3rd JUNIOR BALKAN MATHEMATICAL OLYMPIAD

1. Let a, b, c, x, y ∈ R so that a^{3} + ax + y = 0, b^{3} + bx + y = 0, c^{3} + cx + y = 0. If a ≠ b ≠ c ≠ a, prove that a + b + c = 0.

2. For n = 0, 1, ..., 1999 define A_{n} = 2^{3n}+3^{6n+2}+5^{6n+2}. Find the greatest common divisor of the numbers A_{0}, A_{1}, ..., A_{1999}.

3. Let S be a square with the side of length 20. Let M be the set of the vertices of S and other 1999 points from inside S. Prove that there exists a triangle with vertices in M and with the area at most equal to \frac{1}{10}.

4. Let ΔABC with [AB] ≡ [AC]. Let D ∈ [BC] be an arbitrary point so that BC > BD > DC > 0. Let k_{1}, k_{2} be the circumscribed circles of ΔABD, ΔADC. Let [BB′], [CC′] be the diameters of k_{1}, k_{2} and M the midpoint of (B′C′). Prove that the area of ΔMBC is a constant, it doesn’t depend on the position of M.

Compiled by Călin Popa.