Let $a_1,a_2,\ldots,a_n,b_1,b_2,\ldots,b_n$ be positive numbers with $a_1+a_2+\cdots+a_n = b_1+b_2+\cdots+b_n$. $$\text{Prove that} \dfrac{a_1^2}{a_1+b_1}+\cdots+\dfrac{a_n^2}{a_n+b_n} \geq \dfrac{1}{2}(a_1+\cdots+a_n).$$
Attempt
It seems like I should use AM-GM on the bottom of each fraction. We then get $\dfrac{a_i^2}{a_i+b_i} \leq \dfrac{a_i^2}{2\sqrt{a_ib_i}}$. But this doesn't seem to help as we get an upper bound. Since there is so much about $a_1+\cdots+a_n$ in this problem, I think a substitution for that might work.