42nd International Mathematical Olympiad

IMO 2001 Problems and Solutions

Problem 2

for all positive real numbers a, b and c.

Solution

First we shall prove that

or equivalently, that

The AM-GM 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: