IMO 1960/A2 solution

IMO 1960

Problem A2

For what real values of x does the following inequality hold:

4x²/(1 - √(1 + 2x))² < 2x + 9 ?

Answer

- 1/2 ≤ x < 45/8.

Solution

We require the first inequality to avoid imaginary numbers. Hence we may set x = -1/2 + a²/2, where a ≥ 0. The inequality now gives immediately a < 7/2 and hence x < 45/8. It is a matter of taste whether to avoid x = 0. I would allow it because the limit as x tends to 0 of the lhs is 4, and the inequality holds.

Thanks to Dave Arthur for the idea of putting x = -1/2 + a, which saves some tedious algebra.

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.