# 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...") |
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Determine when equality occurs. | Determine when equality occurs. | ||

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+ | ==Solution 1== | ||

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+ | 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>. | ||

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+ | By Cauchy, we have: | ||

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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. |

## Revision as of 16:37, 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.