Difference between revisions of "1983 IMO Problems/Problem 6"
(→Solution 2) |
m (→Solution 2) |
||
Line 32: | Line 32: | ||
Expanding this out, we have | Expanding this out, we have | ||
− | <math>a^3b + b^3 + c^3 a \geq a^3 c + b^3 a + c^3 b</math>, | + | <math>a^3b + b^3 c + c^3 a \geq a^3 c + b^3 a + c^3 b</math>, |
which is the desired result. | which is the desired result. |
Revision as of 19:02, 27 August 2017
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.