IMO 1973/B3 solution

IMO 1973

Problem B3

a₁, a₂, ... , a_n are positive reals, and q satisfies 0 < q < 1. Find b₁, b₂, ... , b_n such that:

(a) a_i < b_i for i = 1, 2, ... , n,

(b) q < b_i+1/b_i < 1/q for i = 1, 2, ... , n-1,

Solution

We notice that the constraints are linear, in the sense that if b_i is a solution for a_i, q, and b_i' is a solution for a_i', q, then for any k, k' > 0 a solution for ka_i + k'a_i', q is kb_i + k'b_i'. Also a "near" solution for a_h = 1, other a_i = 0 is b₁ = q^h-1, b₂ = q^h-2, ... , b_h-1 = q, b_h = 1, b_h+1 = q, ... , b_n = q^n-h. "Near" because the inequalities in (a) and (b) are not strict.

However, we might reasonably hope that the inequalities would become strict in the linear combination, and indeed that is true. Define b_r = q^r-1a₁ + q^r-2a₂ + ... + qa_r-1 + a_r + qa_r+1 + ... + q^n-ra_n. Then we may easily verify that (a) - (c) hold.

Solutions are also available in: Samuel L Greitzer, International Mathematical Olympiads 1959-1977, MAA 1978, and in István Reiman, International Mathematical Olympiad 1959-1999, ISBN 189-8855-48-X.

15th IMO 1973