Difference between revisions of "2001 IMO Problems/Problem 2"
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<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]]. | <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]]. | ||
{{IMO box|year=2001|num-b=1|num-a=3}} | {{IMO box|year=2001|num-b=1|num-a=3}} | ||
− | [[Category:Olympiad | + | [[Category:Olympiad Number Theory Problems]] |
Revision as of 19:57, 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.
2001 IMO (Problems) • Resources | ||
Preceded by Problem 1 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 3 |
All IMO Problems and Solutions |