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a+b+c=1/a+1/b+1/c=3, prove that at least one of a,b,c is negative

Source: 2022 CMO Qualifying Repechage Problem 6

March 15, 2022
inequalities

Problem Statement

Let a,b,ca,b,c be real numbers, which are not all equal, such that a+b+c=1a+1b+1c=3.a+b+c=\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=3. Prove that at least one of a,b,ca, b, c is negative.
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