Consider infinite sequences {x_{n}} of positive reals such that x_{0} = 1 and x_{0} ≥= x_{1} ≥ x_{2} ≥ ... .

(a) Prove that for every such sequence there is an n ≥ 1 such that:

x_{0}^{2}/x_{1} + x_{1}^{2}/x_{2} + ... + x_{n-1}^{2}/x_{n} ≥ 3.999.

(b) Find such a sequence for which:

x_{0}^{2}/x_{1} + x_{1}^{2}/x_{2} + ... + x_{n-1}^{2}/x_{n} < 4 for all n.

**Solution**

(a) It is sufficient to show that the sum of the (infinite) sequence is at least 4. Let k be the greatest lower bound of the limits of all such sequences. Clearly k ≥ 1. Given any ε > 0, we can find a sequence {x_{n}} with sum less than k + ε. But we may write the sum as:

x_{0}^{2}/x_{1} + x_{1}( (x_{1}/x_{1})^{2}/(x_{2}/x_{1}) + (x_{2}/x_{1})^{2}/(x_{3}/x_{1}) + ... + (x_{n}/x_{1})^{2}/(x_{n+1}/x_{1}) + ... ).

The term in brackets is another sum of the same type, so it is at least k. Hence k + ε > 1/x_{1} + x_{1}k. This holds for all ε > 0, and so k ≥ 1/x_{1} + x_{1}k. But 1/x_{1} + x_{1}k ≥ 2√k, so k ≥ 4.

(b) Let x_{n} = 1/2^{n}. Then x_{0}^{2}/x_{1} + x_{1}^{2}/x_{2} + ... + x_{n-1}^{2}/x_{n} = 2 + 1 + 1/2 + ... + 1/2^{n-2} = 4 - 1/2^{n-2} < 4.

Solutions are also available in Murray S Klamkin, International Mathematical Olympiads 1978-1985, MAA 1986, and in István Reiman, International Mathematical Olympiad 1959-1999, ISBN 189-8855-48-X.

© John Scholes

jscholes@kalva.demon.co.uk

14 Oct 1998