1983 IMO Problems/Problem 6
Let , and be the lengths of the sides of a triangle. Prove that
Determine when equality occurs.
By Ravi substitution, let , , . Then, the triangle condition becomes . After some manipulation, the inequality becomes:
By Cauchy, we have:
with equality if and only if . So the inequality holds with equality if and only if x = y = z. Thus the original inequality has equality if and only if the triangle is equilateral.
Without loss of generality, let . By Muirhead or by AM-GM, we see that .
If we can show that , we are done, since then , and we can divide by .
We first see that, , so .
Factoring, this becomes . This is the same as:
Expanding and refactoring, this is equal to . (This step makes more sense going backwards.)
Expanding this out, we have
which is the desired result.