Difference between revisions of "1997 USAMO Problems/Problem 5"
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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> | ||
− | <math> | + | <math>\frac{1}{a^3+b^3+abc}+\frac{1}{b^3+c^3+abc}+\frac{1}{a^3+c^3+abc} \le \frac{1}{abc}</math>. |
== Solution 1 == | == Solution 1 == |
Revision as of 13:31, 12 April 2023
Contents
[hide]Problem
Prove that, for all positive real numbers
.
Solution 1
Because the inequality is homogenous (i.e. can be replaced with
without changing the inequality other than by a factor of
for some
), without loss of generality, let
.
Lemma:
Proof: Rearranging gives
, which is a simple consequence of
and
Thus, by :
Solution 2
Rearranging the AM-HM inequality, we get . Letting
,
, and
, we get
By AM-GM on
,
, and
, we have
So,
.
-Tigerzhang
See Also
1997 USAMO (Problems • Resources) | ||
Preceded by Problem 4 |
Followed by Problem 6 | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAMO Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.