VMO 1968

Let $a$ and $b$ satisfy $a \ge b >0, a + b = 1$ . i) Prove that if $m$ and $n$ are positive integers with $m < n$ , then $a^m - a^n \ge b^m- b^n > 0$ . ii) For each positive integer $n$ , consider a quadratic function $f_n(x) = x^2 - b^nx- a^n$ . Show that $f(x)$ has two roots that are in between $-1$ and $1$ .

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.