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{a, b, c}={a^2-2, b^2-2, c^2-2} if {a + b, b + c, c + a} = {ab, bc, ca}

Source: 2021 Czech and Slovak Olympiad III A p3

June 6, 2021
algebra

Problem Statement

The different non-zero real numbers a, b, c satisfy the set equality {a+b,b+c,c+a}={ab,bc,ca}\{a + b, b + c, c + a\} = \{ab, bc, ca\}. Prove that the set equality {a,b,c}={a2−2,b2−2,c2−2}\{a, b, c\} = \{a^2 -2, b^2 - 2, c^2 - 2\} also holds. . (Josef Tkadlec)
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