Difference between revisions of "1997 USAMO Problems/Problem 5"
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<math>(a^3+b^3+abc)^{-1}+(b^3+c^3+abc)^{-1}+(a^3+c^3+abc)^{-1}\le(abc)^{-1}</math>. | <math>(a^3+b^3+abc)^{-1}+(b^3+c^3+abc)^{-1}+(a^3+c^3+abc)^{-1}\le(abc)^{-1}</math>. | ||
| − | == Solution == | + | == Solution 1 == |
Because the inequality is homogenous (i.e. <math>(a, b, c)</math> can be replaced with <math>(ka, kb, kc)</math> without changing the inequality other than by a factor of <math>k^n</math> for some <math>n</math>), without loss of generality, let <math>abc = 1</math>. | Because the inequality is homogenous (i.e. <math>(a, b, c)</math> can be replaced with <math>(ka, kb, kc)</math> without changing the inequality other than by a factor of <math>k^n</math> for some <math>n</math>), without loss of generality, let <math>abc = 1</math>. | ||
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<cmath>\frac{1}{a^3 + b^3 + abc} + \frac{1}{b^3 + c^3 + abc} + \frac{1}{c^3 + a^3 + abc}</cmath> | <cmath>\frac{1}{a^3 + b^3 + abc} + \frac{1}{b^3 + c^3 + abc} + \frac{1}{c^3 + a^3 + abc}</cmath> | ||
<cmath>\le \frac{c}{a + b + c} + \frac{a}{a + b + c} + \frac{b}{a + b + c} = 1 = \frac{1}{abc}.</cmath> | <cmath>\le \frac{c}{a + b + c} + \frac{a}{a + b + c} + \frac{b}{a + b + c} = 1 = \frac{1}{abc}.</cmath> | ||
| + | |||
| + | == Solution 2 == | ||
| + | Rearranging the AM-HM inequality, we get <math>\frac{1}{x}+\frac{1}{y}+\frac{1}{z} \le \frac{9}{x+y+z}</math>. | ||
==See Also == | ==See Also == | ||
Revision as of 23:02, 21 August 2020
Contents
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:
![\[\frac{1}{a^3 + b^3 + 1} \le \frac{c}{a + b + c}.\]](http://latex.artofproblemsolving.com/4/0/6/40699bc0defba4331f503edc329983f1d2048e7f.png)
Proof: Rearranging gives 
, which is a simple consequence of 
and
![\[(a^2 - ab + b^2)c \ge (2ab - ab)c = abc = 1.\]](http://latex.artofproblemsolving.com/d/7/3/d73b9c3c44c70799a4efe54c36b8834d174e99eb.png)
Thus, by :
![\[\frac{1}{a^3 + b^3 + abc} + \frac{1}{b^3 + c^3 + abc} + \frac{1}{c^3 + a^3 + abc}\]](http://latex.artofproblemsolving.com/9/8/e/98ee2820cb96c8bf7f543afcbde3eabaf90751e5.png)
![\[\le \frac{c}{a + b + c} + \frac{a}{a + b + c} + \frac{b}{a + b + c} = 1 = \frac{1}{abc}.\]](http://latex.artofproblemsolving.com/9/6/5/9658d126aac55adb471e760d4bd7def3ddb71a72.png)
Solution 2
Rearranging the AM-HM inequality, we get 
.
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. 
