Prove or find a counterexample.

Let $a_1, a_2, b_1, b_2 \in \mathbb{R}$ satisfy $a_1+a_2=1$ and $b_1+b_2=1$. Then, $$ \frac{1}{a_1^2+a_2^2} + \frac{1}{b_1^2 + b_2^2} \geq (\frac{1}{(a_1 b_2)^2 + (a_1 b_1 + a_2 b_2)^2 + (b_1 a_2)^2})^2 $$

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    $\begingroup$ Is this a Math Olympiad problem? If so, you should probably rather post it on Mathlinks. For a question here, a little bit of motivation, previous work (have you tried 10000 random values on a computer? Is there a counterexample?) and describing the connection to a research problem would be more welcome. $\endgroup$ – Federico Poloni Jan 23 '14 at 23:12
  • $\begingroup$ I posted the research level original question of the question in the followin address. mathoverflow.net/questions/155542/energy-of-repeated-filter $\endgroup$ – Seyong Jan 23 '14 at 23:48

$a_1=1/2$, $a_2=1/2$, $b_1=1$, $b_2=0$.

The left hand side is $3$, the right hand side is $4$.


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