Non-zero real numbers $a, b$ and $c$ are given such that $ab+bc+ac=0$. Prove that numbers $a+b+c$ and $\dfrac{1}{a+b}+\dfrac{1}{b+c}+\dfrac{1}{a+c}$ are either both positive or both negative. Proposed by Mykhailo Shtandenko
Source: Kyiv City MO 2023 9.2
Tags: algebra
Non-zero real numbers $a, b$ and $c$ are given such that $ab+bc+ac=0$. Prove that numbers $a+b+c$ and $\dfrac{1}{a+b}+\dfrac{1}{b+c}+\dfrac{1}{a+c}$ are either both positive or both negative. Proposed by Mykhailo Shtandenko