42nd International Mathematical OlympiadIMO 2001 Problems and SolutionsWashinton, DC, USAJuly 114, 2001 Problem 2for all positive real numbers a, b and c. Solution First we shall prove that
or equivalently, that
The AMGM inequality yields
Thus
so
Similarly, we have
Adding these three inequalities yields
Comment. The proposer conjectures that for any a, b, c > 0 and >= 8, the following inequality holds:
