4th Junior Balkan 2000


1. x and y are positive reals such that x³ + y³ + (x + y)³ + 30xy = 2000. Show that x + y = 10.
2. Find all positive integers n such that n³ + 3³ is a perfect square.
3. ABC is a triangle. E, F are points on the side BC such that the semicircle diameter EF touches AB at Q and AC at P. Show that the intersection of EP and FQ lies on the altitude from A.
4. n girls and 2n boys played a tennis tournament. Every player played every other player. The boys won 7/5 times as many matches as the girls (and there were no draws). Find n.

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