Difference between revisions of "2001 IMO Problems/Problem 2"
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Revision as of 12:02, 17 September 2012
Problem
Let be positive real numbers. Prove that
.
Contents
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
This inequality is homogeneous so we can assume without loss of generality and apply Jensen's inequality for
, so we get:
but
by AMGM, and thus the inequality is proven.
Alternate Solution using Isolated Fudging
We claim that
Cross-multiplying, squaring both sides and expanding, we have
After cancelling the
term, we apply AM-GM to RHS and obtain
as desired, completing the proof of the claim.
Similarly and
.
Summing the three inequalities, we obtain the original inequality.
See also
2001 IMO (Problems) • Resources | ||
Preceded by Problem 1 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 3 |
All IMO Problems and Solutions |