# 1983 IMO Problems/Problem 6

## Problem 6

Let , and be the lengths of the sides of a triangle. Prove that

.

Determine when equality occurs.

## Solution 1

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.

## Solution 2

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.