1st Chinese Mathematical Olympiad 1986 Problems & Solutions

A1.  a1, a2, ... , an are reals. Show that if the sum of any two is non-negative, then for any non-negative real x1, x2, ... , xn with sum 1, we have a1x1 + a2x2 + ... + anxn ≥ a1x12 + a2x22 + ... + anxn2. Show that the converse is also true. A2.  ABC is a triangle. The altitude from A has length 12, the angle bisector from A has length 13. What is are the possible lengths for