1st Junior Balkan 1997


1. Show that given any 9 points inside a square side 1 we can always find three which form a triangle with area < 1/8.
2. Given reals x, y with (x² + y²)/(x² - y²) + (x² - y²)/(x² + y²) = k, find (x⁸ + y⁸)/(x⁸ - y⁸) + (x⁸ - y⁸)/(x⁸ + y⁸) in terms of k.
3. I is the incenter of ABC. N, M are the midpoints of sides AB, CA. The lines BI, CI meet MN at K, L respectively. Prove that AI + BI + CI > BC + KL.
4. A triangle has circumradius R and sides a, b, c with R(b+c) = a √(bc). Find its angles.
5. n₁, n₂, ... , n₁₉₉₈ are positive integers such that n₁² + n₂² + ... + n₁₉₉₇² = n₁₉₉₈². Show that at least two of the numbers are even.

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