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

(Created page with "==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)|...")
 
(Problem)
 
(One intermediate revision by one other user not shown)
Line 1: Line 1:
 
==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>
+
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==
 
.
 
.

Latest revision as of 03:51, 19 July 2015

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

.

Invalid username
Login to AoPS