Difference between revisions of "2001 IMO Problems/Problem 2"
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===Alternate Solution using Jensen's=== | ===Alternate Solution using Jensen's=== | ||
By Jensen's, | By Jensen's, | ||
− | <math>\frac{a}{\sqrt{a^{2}+8bc}}+\frac{b}{\sqrt{b^{2}+8ca}}+\frac{c}{\sqrt{c^{2}+8ab}}\ge a^{3}+b^{3}+c^{3}+24abc</math>, so we need to prove <math>a^{3}+b^{3}+c^{3}+24abc\ge 1</math>, which is obvious by [[ | + | <math>\frac{a}{\sqrt{a^{2}+8bc}}+\frac{b}{\sqrt{b^{2}+8ca}}+\frac{c}{\sqrt{c^{2}+8ab}}\ge a^{3}+b^{3}+c^{3}+24abc</math>, so we need to prove <math>a^{3}+b^{3}+c^{3}+24abc\ge 1</math>, which is obvious by [[RMS-AM-GM-HM]]. |
Revision as of 18:45, 25 October 2007
Problem
Let be positive real numbers. Prove that
Solution
Solution using Holder's
By Holder's inequality,
Thus we need only show that
Which is obviously true since
.
Alternate Solution using Jensen's
By Jensen's,
, so we need to prove
, which is obvious by RMS-AM-GM-HM.