IMC 2016 Problem 3

Let $n$ be a positive integer. Also let $a_1, a_2, \dots, a_n$ and $b_1,b_2,\dots, b_n$ be real numbers such that $a_i+b_i>0$ for $i=1,2,\dots, n$ . Prove that $$\sum_{i=1}^n \frac{a_ib_i-b_i^2}{a_i+b_i}\le\frac{ \sum_{i=1}^n a_i\cdot \sum_{i=1}^n b_i - \left( \sum_{i=1}^n b_i\right) ^2}{ \sum_{i=1}^n (a_i+b_i)}$$ .

(Proposed by Daniel Strzelecki, Nicolaus Copernicus University in Toruń, Poland)