Let $a,b,c$ be real numbers, which are not all equal, such that $$a+b+c=\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=3.$$Prove that at least one of $a, b, c$ is negative.
Source: 2022 CMO Qualifying Repechage Problem 6
Tags: inequalities
Let $a,b,c$ be real numbers, which are not all equal, such that $$a+b+c=\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=3.$$Prove that at least one of $a, b, c$ is negative.