Difference between revisions of "2006 IMO Problems/Problem 3"
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{{solution}} | {{solution}} | ||
| − | 1. | + | 1. Rewrite the expression: |
Consider the expression inside the absolute value: | Consider the expression inside the absolute value: | ||
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</cmath> | </cmath> | ||
| − | 2. | + | 2. Use a known factorization: |
A standard identity is: | A standard identity is: | ||
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</cmath> | </cmath> | ||
| − | 3. | + | 3. Normalization: |
The inequality is homogeneous of degree 4. Without loss of generality, we may impose the normalization: | The inequality is homogeneous of degree 4. Without loss of generality, we may impose the normalization: | ||
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</cmath> | </cmath> | ||
| − | 4. | + | 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: | 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: | ||
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</cmath> | </cmath> | ||
| − | 5. | + | 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: | 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: | ||
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M = \frac{9}{16\sqrt{2}}. | M = \frac{9}{16\sqrt{2}}. | ||
</cmath> | </cmath> | ||
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==See Also== | ==See Also== | ||
Latest revision as of 16:50, 6 December 2024
Problem
Determine the least real number 
such that the inequality
![\[\left| ab(a^{2}-b^{2})+bc(b^{2}-c^{2})+ca(c^{2}-a^{2})\right|\leq M(a^{2}+b^{2}+c^{2})^{2}\]](http://latex.artofproblemsolving.com/9/7/3/973b3ca7a4e5262cdd8bb6d67febe1dd03a72543.png)
holds for all real numbers 
and 
.
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:
![\[ab(a^{2}-b^{2}) + bc(b^{2}-c^{2}) + ca(c^{2}-a^{2}).\]](http://latex.artofproblemsolving.com/1/7/8/1786bb2051d82bd32efd7b45c74c3c48667fecdd.png)
By expanding and symmetrizing the terms, one can rewrite it as:
![\[a^{3}(b - c) + b^{3}(c - a) + c^{3}(a - b).\]](http://latex.artofproblemsolving.com/f/2/8/f28dc68144800fe3d64dee5ca7ae7f34506265d7.png)
2. Use a known factorization:
A standard identity is:
![\[a^{3}(b - c) + b^{3}(c - a) + c^{3}(a - b) = -(a - b)(b - c)(c - a)(a + b + c).\]](http://latex.artofproblemsolving.com/0/7/6/07632bbf66b92ec1bd03be6ae5dedfb8960299f9.png)
Thus, our inequality becomes:
![\[|(a - b)(b - c)(c - a)(a + b + c)| \leq M (a^{2} + b^{2} + c^{2})^{2}.\]](http://latex.artofproblemsolving.com/f/9/2/f92f04df3513d5b580f3058c244539a7f667386e.png)
3. Normalization:
The inequality is homogeneous of degree 4. Without loss of generality, we may impose the normalization:
![\[a^{2} + b^{2} + c^{2} = 1.\]](http://latex.artofproblemsolving.com/d/7/3/d739820ec28d17b68100198328b613bcafa106a8.png)
Under this constraint, we need to find the maximum possible value of:
![\[|(a - b)(b - c)(c - a)(a + b + c)|.\]](http://latex.artofproblemsolving.com/7/9/4/79468224c59236a579856a55b28f1d7b4d43a53f.png)
4. Finding the maximum:
By considering an arithmetic progression substitution, for instance 
, 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:
![\[\frac{9}{16\sqrt{2}}.\]](http://latex.artofproblemsolving.com/0/a/5/0a5dc0c686ccc0203cf881d3212f95889fc9cdb6.png)
5. Conclusion:
Since we have found the maximum value of the left-hand side expression (under normalization) to be 
, it follows that the minimal satisfying the original inequality is:
![\[M = \frac{9}{16\sqrt{2}}.\]](http://latex.artofproblemsolving.com/a/a/b/aab3630a55f8f018a3234d35a1e3e19828b2cb6e.png)
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 | ||
