Difference between revisions of "1983 IMO Problems/Problem 6"
(→Problem 6) |
|||
Line 2: | Line 2: | ||
Let <math>a</math>, <math>b</math> and <math>c</math> be the lengths of the sides of a triangle. Prove that | Let <math>a</math>, <math>b</math> and <math>c</math> be the lengths of the sides of a triangle. Prove that | ||
− | <math>a^2 b(a-b) + b^2 c(b-c) + c^2 (c-a) \geq 0</math>. | + | <math>a^2 b(a-b) + b^2 c(b-c) + c^2 a(c-a) \geq 0</math>. |
Determine when equality occurs. | Determine when equality occurs. |
Revision as of 22:42, 15 July 2021
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 |