1983 IMO Problems/Problem 6
Contents
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.
See Also
1983 IMO (Problems) • Resources | ||
Preceded by Problem 5 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Last Problem |
All IMO Problems and Solutions |