5th Junior Balkan 2001


1. Find all positive integers a, b, c such that a³ + b³ + c³ = 2001.
2. ABC is a triangle with ∠C = 90^o and CA ≠ CB. CH is an altitude and CL is an angle bisector. Show that for X ≠ C on the line CL, we have ∠XAC ≠ ∠XBC. Show that for Y ≠ C on the line CH we have ∠YAC ≠ ∠YBC.
3. ABC is an equilateral triangle. D, E are points on the sides AB, AC respectively. The angle bisector of ∠ADE meets AE at F, and the angle bisector of ∠AED meets AD at G. Show that area DEF + area DEG ≤ area ABC. When do we have equality?
4. N is a convex polygon with 1415 vertices and perimeter 2001. Prove that we can find three vertices of N which form a triangle of area < 1.

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