7th Junior Balkan 2003


1. Let A = 44...4 (2n digits) and B = 88...8 (n digits). Show that A + 2B + 4 is a square.
2. A₁, A₂, ... , A_n are points in the plane, so that if we take the points in any order B₁, B₂, ... , B_n, then the broken line B₁B₂...B_n does not intersect itself. What is the largest possible value of n?
3. ABC is a triangle. D is the midpoint of the arc BC not containing A. Similarly E, F. DE meets BC at G and AC at H. M is the midpoint of GH. DF meets BC at I and AB at J, and N is the midpoint of IJ. Find the angles of DMN in terms of the angles of ABC. AD meets EF at P. Show that the circumcenter of DMN lies on the circumcircle of PMN.
4. Show that (1+x²)/(1+y+z²) + (1+y²)/(1+z+x²) + (1+z²)/(1+x+y²) ≥ 2 for reals x, y, z > -1.

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