Difference between revisions of "2004 JBMO Problems/Problem 1"
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<math>7m^2 -46m + 72 \geq 0</math> | <math>7m^2 -46m + 72 \geq 0</math> | ||
or, <math>7(m-4)^2 +10(m-4) \geq 0</math> | or, <math>7(m-4)^2 +10(m-4) \geq 0</math> | ||
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The above is always true since <math>m \geq 4</math>. | The above is always true since <math>m \geq 4</math>. | ||
<math>Kris17</math> | <math>Kris17</math> | ||
Revision as of 23:15, 16 December 2018
Problem
Prove that the inequality ![\[\frac{ x+y}{x^2-xy+y^2 } \leq \frac{ 2\sqrt 2 }{\sqrt{ x^2 +y^2 } }\]](http://latex.artofproblemsolving.com/2/f/6/2f6d7020dcfe5de9ab70585979d81494601dc93a.png)
holds for all real numbers 
and 
, not both equal to 0.
Solution
Since the inequality is homogeneous, we can assume WLOG that xy = 1.
Now, substituting 
, we have:
, thus we have 
Now squaring both sides of the inequality, we get:
![\[\frac{m}{(m-3)^2 } \leq \frac{8}{m-2}\]](http://latex.artofproblemsolving.com/2/8/3/2830e26564a146df2043cf2c863e5820fb254fa1.png)
after cross multiplication and simplification we get:
or, 
The above is always true since .
