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, $$ \frac{1}{a_1^2+a_2^2} + \frac{1}{b_1^2 + b_2^2} \geq (\frac{1}{(a_1 b_2)^2 + (a_1 b_1 + a_2 b_2)^2 + (b_1 a_2)^2})^2 $$
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, $$ \frac{1}{a_1^2+a_2^2} + \frac{1}{b_1^2 + b_2^2} \geq (\frac{1}{(a_1 b_2)^2 + (a_1 b_1 + a_2 b_2)^2 + (b_1 a_2)^2})^2 $$
$a_1=1/2$, $a_2=1/2$, $b_1=1$, $b_2=0$.
The left hand side is $3$, the right hand side is $4$.