18th All Soviet Union Mathematical Olympiad Problems 1984

1.  Show that we can find n integers whose sum is 0 and whose product is n iff n is divisible by 4. 2.  Show that (a + b)2/2 + (a + b)/4 ≥ a√b + b √a for all positive a and b. 3.  ABC and A'B'C' are equilateral triangles and ABC and A'B'C' have the same sense (both clockwise or both counter-clockwise). Take an arbitrary point O and points P, Q, R so that OP is