VMO 1968 Problem 1

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 $1$ .