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.