Difference between revisions of "2012 USAJMO Problems/Problem 3"
(Created page with "== Problem == Let <math>a</math>, <math>b</math>, <math>c</math> be positive real numbers. Prove that <cmath>\frac{a^3 + 3b^3}{5a + b} + \frac{b^3 + 3c^3}{5b + c} + \frac{c^3 +...") |
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==Solution== | ==Solution== | ||
| + | |||
| + | By the Cauchy-Schwarz inequality, | ||
| + | <cmath>[a(5a + b) + b(5b + c) + c(5c + a)] \left( \frac{a^3}{5a + b} + \frac{b^3}{5b + c} + \frac{c^3}{5c + a} \right) \ge (a^2 + b^2 + c^2)^2,</cmath> | ||
| + | so | ||
| + | <cmath>\frac{a^3}{5a + b} + \frac{b^3}{5b + c} + \frac{c^3}{5c + a} \ge \frac{(a^2 + b^2 + c^2)^2}{5a^2 + 5b^2 + 5c^2 + ab + ac + bc}.</cmath> | ||
| + | Since <math>a^2 + b^2 + c^2 \ge ab + ac + bc</math>, | ||
| + | <cmath>\frac{(a^2 + b^2 + c^2)^2}{5a^2 + 5b^2 + 5c^2 + ab + ac + bc} \ge \frac{(a^2 + b^2 + c^2)^2}{6a^2 + 6b^2 + 6c^2} = \frac{1}{6} (a^2 + b^2 + c^2).</cmath> | ||
| + | Hence, | ||
| + | <cmath>\frac{a^3}{5a + b} + \frac{b^3}{5b + c} + \frac{c^3}{5c + a} \ge \frac{1}{6} (a^2 + b^2 + c^2).</cmath> | ||
| + | |||
| + | By the same argument, | ||
| + | <cmath>\frac{b^3}{5a + b} + \frac{c^3}{5b + c} + \frac{a^3}{5c + a} \ge \frac{1}{6} (a^2 + b^2 + c^2).</cmath> | ||
| + | Therefore, | ||
| + | <cmath>\frac{a^3 + 3b^3}{5a + b} + \frac{b^3 + 3c^3}{5b + c} + \frac{c^3 + 3a^3}{5c + a} \ge \frac{1 + 3}{6} (a^2 + b^2 + c^2) = \frac{2}{3} (a^2 + b^2 + c^2).</cmath> | ||
==See Also== | ==See Also== | ||
, 
, 
be positive real numbers. Prove that
![\[\frac{a^3 + 3b^3}{5a + b} + \frac{b^3 + 3c^3}{5b + c} + \frac{c^3 + 3a^3}{5c + a} \ge \frac{2}{3} (a^2 + b^2 + c^2).\]](http://latex.artofproblemsolving.com/4/4/8/448b311f5688c5c18e520bfe1a9c56906700c3d2.png)
![\[[a(5a + b) + b(5b + c) + c(5c + a)] \left( \frac{a^3}{5a + b} + \frac{b^3}{5b + c} + \frac{c^3}{5c + a} \right) \ge (a^2 + b^2 + c^2)^2,\]](http://latex.artofproblemsolving.com/c/8/5/c853ed04ce97a5d53eb2c179cc962c2d9e0af755.png)
so
![\[\frac{a^3}{5a + b} + \frac{b^3}{5b + c} + \frac{c^3}{5c + a} \ge \frac{(a^2 + b^2 + c^2)^2}{5a^2 + 5b^2 + 5c^2 + ab + ac + bc}.\]](http://latex.artofproblemsolving.com/c/7/0/c70b85f9d08ce18cabc92505778450fa22084b5d.png)
Since 
,
![\[\frac{(a^2 + b^2 + c^2)^2}{5a^2 + 5b^2 + 5c^2 + ab + ac + bc} \ge \frac{(a^2 + b^2 + c^2)^2}{6a^2 + 6b^2 + 6c^2} = \frac{1}{6} (a^2 + b^2 + c^2).\]](http://latex.artofproblemsolving.com/c/6/7/c67539d05946214142c90df1baebf0bcc6fc3875.png)
Hence,
![\[\frac{a^3}{5a + b} + \frac{b^3}{5b + c} + \frac{c^3}{5c + a} \ge \frac{1}{6} (a^2 + b^2 + c^2).\]](http://latex.artofproblemsolving.com/2/5/b/25b8b46bbae4a04cad9f782369cbd6b469179254.png)
![\[\frac{b^3}{5a + b} + \frac{c^3}{5b + c} + \frac{a^3}{5c + a} \ge \frac{1}{6} (a^2 + b^2 + c^2).\]](http://latex.artofproblemsolving.com/5/4/a/54abb3c7c80aa1733717f8d78fa9680a3cf52dc2.png)
Therefore,
![\[\frac{a^3 + 3b^3}{5a + b} + \frac{b^3 + 3c^3}{5b + c} + \frac{c^3 + 3a^3}{5c + a} \ge \frac{1 + 3}{6} (a^2 + b^2 + c^2) = \frac{2}{3} (a^2 + b^2 + c^2).\]](http://latex.artofproblemsolving.com/9/7/3/9736e742df8602ada135197c276e1cdd5c4d108c.png)
