3rd Junior Balkan 1999


1. a, b, c are distinct reals and there are reals x, y such that a³ + ax + y = 0, b³ + bx + y = 0 and c³ + cx + y = 0. Show that a + b + c = 0.
2. Let a_n = 2³ⁿ + 3⁶ⁿ⁺² + 5⁶ⁿ⁺². Find gcd(a₀, a₁, a₂, ... , a₁₉₉₉).
3. A square has side 20. S is a set of 1999 points inside the square and the 4 vertices. Show that we can find three points in S which form a triangle with area ≤ 1/10.
4. The triangle ABC has AB = AC. D is a point on the side BC. BB' is a diameter of the circumcircle of ABD, and CC' is a diameter of the circumcircle of ACD. M is the midpoint of B'C'. Show that the area of BCM is independent of the position of D.

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