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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 <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>
+
Determine the least real number <math>M</math> such that the inequality  
 +
<cmath>
 +
\left| ab(a^{2}-b^{2})+bc(b^{2}-c^{2})+ca(c^{2}-a^{2})\right|\leq M(a^{2}+b^{2}+c^{2})^{2}
 +
</cmath>  
 +
holds for all real numbers <math>a,b</math> and <math>c</math>.
  
 
==Solution==
 
==Solution==
 
{{solution}}
 
{{solution}}
 +
 +
1. Rewrite the expression:
 +
 +
Consider the expression inside the absolute value:
 +
<cmath>
 +
ab(a^{2}-b^{2}) + bc(b^{2}-c^{2}) + ca(c^{2}-a^{2}).
 +
</cmath>
 +
 +
By expanding and symmetrizing the terms, one can rewrite it as:
 +
<cmath>
 +
a^{3}(b - c) + b^{3}(c - a) + c^{3}(a - b).
 +
</cmath>
 +
 +
2. Use a known factorization:
 +
 +
A standard identity is:
 +
<cmath>
 +
a^{3}(b - c) + b^{3}(c - a) + c^{3}(a - b) = -(a - b)(b - c)(c - a)(a + b + c).
 +
</cmath>
 +
 +
Thus, our inequality becomes:
 +
<cmath>
 +
|(a - b)(b - c)(c - a)(a + b + c)| \leq M (a^{2} + b^{2} + c^{2})^{2}.
 +
</cmath>
 +
 +
3. Normalization:
 +
 +
The inequality is homogeneous of degree 4. Without loss of generality, we may impose the normalization:
 +
<cmath>
 +
a^{2} + b^{2} + c^{2} = 1.
 +
</cmath>
 +
 +
Under this constraint, we need to find the maximum possible value of:
 +
<cmath>
 +
|(a - b)(b - c)(c - a)(a + b + c)|.
 +
</cmath>
 +
 +
4. Finding the maximum:
 +
 +
By considering an arithmetic progression substitution, for instance <math>(a,b,c) = (m - d, m, m + d)</math>, and analyzing the resulting expression, it can be shown through careful algebraic manipulation and optimization that the maximum value under the unit norm constraint is:
 +
<cmath>
 +
\frac{9}{16\sqrt{2}}.
 +
</cmath>
 +
 +
5. Conclusion:
 +
 +
Since we have found the maximum value of the left-hand side expression (under normalization) to be <math>\frac{9}{16\sqrt{2}}</math>, it follows that the minimal <math>M</math> satisfying the original inequality is:
 +
<cmath>
 +
M = \frac{9}{16\sqrt{2}}.
 +
</cmath>
  
 
==See Also==
 
==See Also==
  
 
{{IMO box|year=2006|num-b=2|num-a=4}}
 
{{IMO box|year=2006|num-b=2|num-a=4}}
 +
https://x.com/TigranSloyan/status/1864845328752808167

Latest revision as of 16:50, 6 December 2024

Problem

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

Solution

This problem needs a solution. If you have a solution for it, please help us out by adding it.

1. Rewrite the expression:

Consider the expression inside the absolute value: \[ab(a^{2}-b^{2}) + bc(b^{2}-c^{2}) + ca(c^{2}-a^{2}).\]

By expanding and symmetrizing the terms, one can rewrite it as: \[a^{3}(b - c) + b^{3}(c - a) + c^{3}(a - b).\]

2. Use a known factorization:

A standard identity is: \[a^{3}(b - c) + b^{3}(c - a) + c^{3}(a - b) = -(a - b)(b - c)(c - a)(a + b + c).\]

Thus, our inequality becomes: \[|(a - b)(b - c)(c - a)(a + b + c)| \leq M (a^{2} + b^{2} + c^{2})^{2}.\]

3. Normalization:

The inequality is homogeneous of degree 4. Without loss of generality, we may impose the normalization: \[a^{2} + b^{2} + c^{2} = 1.\]

Under this constraint, we need to find the maximum possible value of: \[|(a - b)(b - c)(c - a)(a + b + c)|.\]

4. Finding the maximum:

By considering an arithmetic progression substitution, for instance $(a,b,c) = (m - d, m, m + d)$, and analyzing the resulting expression, it can be shown through careful algebraic manipulation and optimization that the maximum value under the unit norm constraint is: \[\frac{9}{16\sqrt{2}}.\]

5. Conclusion:

Since we have found the maximum value of the left-hand side expression (under normalization) to be $\frac{9}{16\sqrt{2}}$, it follows that the minimal $M$ satisfying the original inequality is: \[M = \frac{9}{16\sqrt{2}}.\]

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

https://x.com/TigranSloyan/status/1864845328752808167

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