7th Vietnamese Mathematical Olympiad 1968 Problems

7th Vietnamese Mathematical Olympiad 1968 Problems

1. The real numbers a and b satisfy a ≥ b > 0, a + b = 1. Show that am - an ≥ bm - bn > 0 for any positive integers m < n. Show that the quadratic x2 - bnx - an has two real roots in the interval (-1, 1).

2. L and M are two parallel lines a distance d apart. Given r and x, construct a triangle ABC, with A on L, and B and C on M, such that the inradius is r, and angle A = x. Calculate angles B and C in terms of d, r and x. If the incircle touches the side BC at D, find a relation between BD and DC.