Problem

Source: 2015 Taiwan TST round 1 Day 1 P2

Tags: algebra, function, functional equation, Taiwan TST 2015



Given a positive integer $n \geq 3$. Find all $f:\mathbb{R}^+ \rightarrow \mathbb{R}^+$ such that for any $n$ positive reals $a_1,...,a_n$, the following condition is always satisfied: $\sum_{i=1}^{n}(a_i-a_{i+1})f(a_i+a_{i+1}) = 0$ where $a_{n+1} = a_1$.