### IMO 1962

**Problem A2**
Find all real x satisfying: √(3 - x) - √(x + 1) > 1/2.

**Solution**

It is easy to show that the inequality implies |x-1| > √31/8, so x > 1 + √31/8, or x < 1 - √31/8. But the converse is not true.

Indeed, we easily see that x > 1 implies the lhs < 0. Also care is needed to ensure that the expressions under the root signs are not negative, which implies -1 ≤ x ≤ 3. Putting this together, suggests the solution is -1 ≤ x < 1 - √31/8, which we can easily check.

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.

4th IMO 1962

© John Scholes

jscholes@kalva.demon.co.uk

19 Sep 1998

Last corrected/updated 24 Sep 2003