1983 IMO Problems/Problem 6: Difference between revisions
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By Cauchy, we have: | 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. | <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 <math>x = y = z</math>. Thus the original inequality has equality if and only if the triangle is equilateral. | ||
Revision as of 16:50, 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 
. Thus the original inequality has equality if and only if the triangle is equilateral.
