International Mathematical Olympiad 1973 Problem 6

Let $a_1, a_2, \ldots, a_n$ be $n$ positive numbers, and let $q$ be a given real number such that $0 < q < 1$. Find $n$ numbers $b_1, b_2, \ldots, b_n$ for which

(a) $a_k < b_k$ for $k = 1, 2, \ldots, n$,

(b) $q < \frac{b_{k+1}}{b_k} < \frac{1}{q}$ for $k = 1, 2, \ldots, n - 1$,

(c) $b_1 + b_2 + \cdots + b_n < \frac{1+q}{1-q}(a_1 + a_2 + \cdots + a_n)$.