3rd Chinese Mathematical Olympiad 1988 Problems & Solutions

A1.  a1, ... , an are reals, not all 0, such that there exist bi so that ∑1n bi(xi - ai) ≤ √(∑1n xi2) - √(∑1n ai2) for all real xi. Find the bi (in terms of the ai). A2.  ABCD is a cyclic quadrilateral. Its circumcircle has center O and radius R. The rays AB, BC, CD, DA meet the circle center O radius 2R at A', B', C', D' respectively. Show that A'B' + B'C' + C'D' + D'A' ≥ 2(

2nd Chinese Mathematical Olympiad 1987 Problems & Solutions

A1.  n is a positive integer. Show that zn+1 - zn - 1 = 0 has a root on the unit circle |z| = 1 iff n is congruent to 4 mod 6. A2.  An equilateral triangle side n is divided into n2 equilateral triangles of side 1 by lines parallel to its sides. The n(n+1)/2 vertices of the triangles are each labeled with a real number, so that if ABC and BCD are small triangles then the sum

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