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Problem 1130

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Algebra Difficulty 5.0 Prove it Romanian Mathematical Olympiad - District Round · Romania

Find all pairs of real numbers $(a, b)$ such that the equality $|ax + by| + |bx + ay| = 2|x| + 2|y|$ holds for all reals $x$ and $y$ .

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

Set $x = y = 1$ to get $|a + b| = 2$ , so $a + b \in \{-2, 2\}$ . For $x = 1$ , $y = -1$ we get $|a - b| = 2$ , hence $a - b \in \{-2, 2\}$ . Combining the two above leads to the pairs $(a, b) \in \{(2, 0), (0, 2), (-2, 0), (0, -2)\}$ , all of them verifying the given equality for any reals $x$ and $y$ .

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.