Difference between revisions of "1983 IMO Problems/Problem 6"
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| − | <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. | + | <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 16:49, 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.
