Let $n$ be a strictly positive integer, and $a_1,\cdots a_n,b_1,b_2 \cdots b_n$ strictly positive real numbers.
Prove that
$$\sum_{i=1}^n (\frac{a_i}{b_i})^2 \le 2\sum_{1\le j,i \le n} \frac{a_ia_j}{(b_i+b_j)^2}+k\sqrt{\sum_{1\le j,i \le n} \frac{a_ia_j}{(b_i+b_j)} \sum_{1\le j,i \le n} \frac{a_ia_j}{(b_i+b_j)^3}} $$
for a certain $k$ which we need to determine.
I just need hints, I don't want the answer.