Difference between revisions of "1983 IMO Problems/Problem 6"
(Created page with "==Problem 6== 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</ma...") |
(→Problem 6) |
||
| Line 5: | Line 5: | ||
Determine when equality occurs. | Determine when equality occurs. | ||
| + | |||
| + | ==Solution 1== | ||
| + | |||
| + | By Ravi substitution, let <math>a = y+z</math>, <math>b = z+x</math>, <math>c = x+y</math>. Then, the triangle condition becomes <math>x, y, z > 0</math>. After some manipulation, the inequality becomes: | ||
| + | |||
| + | <math>xy^3 + yz^3 + zx^3 \geq xyz(x+y+z)</math>. | ||
| + | |||
| + | By Cauchy, we have: | ||
| + | |||
| + | <math>(xy^3 + yz^3 + zx^3)(z+x+y) \geq xyz(y+z+x)^2</math> with equality if and only if <math>\frac{xy^3}{z} = frac{yz^3}{x} = frac{zx^3}{y}</math>. 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. | ||
Revision as of 17:37, 22 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.
