Difference between revisions of "2006 IMO Problems/Problem 3"

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==Problem==
 
==Problem==
Determine the least real number <math>M</math> such that the inequality <math> \left|ab\left(a^{2}-b^{2}\right)+bc\left(b^{2}-c^{2}\right)+ca\left(c^{2}-a^{2}\right)|\leq M\left(a^{2}+b^{2}+c^{2}\right)^{2} </math> holds for all real numbers <math>a,b</math> and <math>c</math>
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Determine the least real number <math>M</math> such that the inequality <cmath> \left| ab\left(a^{2}-b^{2}\right)+bc\left(b^{2}-c^{2}\right)+ca\left(c^{2}-a^{2}\right)\right|\leq M\left(a^{2}+b^{2}+c^{2}\right)^{2} </cmath> holds for all real numbers <math>a,b</math> and <math>c</math>
  
 
==Solution==
 
==Solution==
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{{solution}}
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==See Also==
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{{IMO box|year=2006|num-b=2|num-a=4}}

Latest revision as of 01:03, 19 November 2023

Problem

Determine the least real number $M$ such that the inequality \[\left| ab\left(a^{2}-b^{2}\right)+bc\left(b^{2}-c^{2}\right)+ca\left(c^{2}-a^{2}\right)\right|\leq M\left(a^{2}+b^{2}+c^{2}\right)^{2}\] holds for all real numbers $a,b$ and $c$

Solution

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See Also

2006 IMO (Problems) • Resources
Preceded by
Problem 2
1 2 3 4 5 6 Followed by
Problem 4
All IMO Problems and Solutions