Difference between revisions of "2004 USAMO Problems/Problem 5"
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== Problem 5 == | == Problem 5 == | ||
− | (''Titu Andreescu'') Let <math> \displaystyle a </math>, <math> \displaystyle b </math>, and <math> \displaystyle c </math> be positive real numbers. Prove that | + | (''Titu Andreescu'') |
+ | Let <math> \displaystyle a </math>, <math> \displaystyle b </math>, and <math> \displaystyle c </math> be positive real numbers. Prove that | ||
<center> | <center> | ||
<math> | <math> | ||
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as desired. | as desired. | ||
− | Unfortunately, it is also possible to solve this inequality by expanding terms and applying brute force, either before or after proving that <math> \displaystyle x^5 - x^2 + 3 \ge x^3 + 2 </math>. | + | |
+ | ''Unfortunately, it is also possible to solve this inequality by expanding terms and applying brute force, either before or after proving that <math> \displaystyle x^5 - x^2 + 3 \ge x^3 + 2 </math>.'' | ||
Revision as of 21:10, 24 March 2007
Problem 5
(Titu Andreescu)
Let ,
, and
be positive real numbers. Prove that
.
Solutions
We first note that for positive ,
. We may prove this in the following ways:
- Since
and
must be both lesser than, both equal to, or both greater than 1, by the rearrangement inequality,
.
- Since
and
have the same sign,
, with equality when
.
- By weighted AM-GM,
and
. Adding these gives the desired inequality. Equivalently, the desired inequality is a case of Muirhead's Inequality.
It thus becomes sufficient to prove that
.
We present two proofs of this inequality.
First, Hölder's Inequality gives us
.
Setting ,
,
, and
when
gives us the desired inequality, with equality when
.
Second, we may apply the Cauchy-Schwarz Inequality twice to obtain
,
as desired.
Unfortunately, it is also possible to solve this inequality by expanding terms and applying brute force, either before or after proving that .
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.