CZ-SK-MO 2021 Problem 3

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