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.