Difference between revisions of "2005 IMO Problems/Problem 3"
(Created page with "Let <math>x, y, z > 0</math> satisfy <math>xyz\ge 1</math>. Prove that <cmath>\frac{x^5-x^2}{x^5+y^2+z^2} + \frac{y^5-y^2}{x^2+y^5+z^2} + \frac{z^5-z^2}{x^2+y^2+z^5} \ge 0.</c...") |
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| + | ==Problem== | ||
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Let <math>x, y, z > 0</math> satisfy <math>xyz\ge 1</math>. Prove that <cmath>\frac{x^5-x^2}{x^5+y^2+z^2} + \frac{y^5-y^2}{x^2+y^5+z^2} + \frac{z^5-z^2}{x^2+y^2+z^5} \ge 0.</cmath> | Let <math>x, y, z > 0</math> satisfy <math>xyz\ge 1</math>. Prove that <cmath>\frac{x^5-x^2}{x^5+y^2+z^2} + \frac{y^5-y^2}{x^2+y^5+z^2} + \frac{z^5-z^2}{x^2+y^2+z^5} \ge 0.</cmath> | ||
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| + | ==Solution== | ||
| + | {{solution}} | ||
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| + | ==See Also== | ||
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| + | {{IMO box|year=2005|num-b=2|num-a=4}} | ||
satisfy 
. Prove that ![\[\frac{x^5-x^2}{x^5+y^2+z^2} + \frac{y^5-y^2}{x^2+y^5+z^2} + \frac{z^5-z^2}{x^2+y^2+z^5} \ge 0.\]](http://latex.artofproblemsolving.com/7/9/a/79af15ecc7549604600f49baf6f88c10a4fe01a3.png)
