MathDB

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{\displaystyle \sum_{i=1}^n a_i\cdot \sum_{i=1}^n b_i - \left( \sum_{i=1}^n b_i\right) ^2}{\displaystyle\sum_{i=1}^n (a_i+b_i)}

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

Problem Statement