Difference between revisions of "2001 IMO Shortlist Problems/A6"
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Prove that for all positive real numbers <math>a,b,c</math>, | Prove that for all positive real numbers <math>a,b,c</math>, | ||
<center><math>\frac {a}{\sqrt {a^2 + 8bc}} + \frac {b}{\sqrt {b^2 + 8ca}} + \frac {c}{\sqrt {c^2 + 8ab}} \geq 1.</math></center> | <center><math>\frac {a}{\sqrt {a^2 + 8bc}} + \frac {b}{\sqrt {b^2 + 8ca}} + \frac {c}{\sqrt {c^2 + 8ab}} \geq 1.</math></center> | ||
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| + | === Generalization === | ||
| + | The leader of the Bulgarian team had come up with the following generalization to the inequality: | ||
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| + | <center><math>\frac {a}{\sqrt {a^2 + kbc}} + \frac {b}{\sqrt {b^2 + kca}} + \frac {c}{\sqrt {c^2 + kab}} \geq \frac{3}{\sqrt{1+k}}.</math></center> | ||
== Solution == | == Solution == | ||
{{solution}} | {{solution}} | ||
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== Resources == | == Resources == | ||
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[[Category:Olympiad Algebra Problems]] | [[Category:Olympiad Algebra Problems]] | ||
[[Category:Olympiad Inequality Problems]] | [[Category:Olympiad Inequality Problems]] | ||
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Revision as of 12:50, 13 September 2012
Contents
Problem
Prove that for all positive real numbers 
,
Generalization
The leader of the Bulgarian team had come up with the following generalization to the inequality:
Solution
This problem needs a solution. If you have a solution for it, please help us out by adding it.
