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At least two of a, b, c are equal

Source: 7th European Mathematical Cup , Junior Category, Q1

December 25, 2018
algebraemc

Problem Statement

Let a,b,ca, b, c be non-zero real numbers such that a2+b+c=1a,b2+c+a=1b,c2+a+b=1c.a^2+b+c=\frac{1}{a}, b^2+c+a=\frac{1}{b}, c^2+a+b=\frac{1}{c}. Prove that at least two of a,b,ca, b, c are equal.
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