Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 5.0 AIME, harder Prove it 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 $y$ .

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 $y$ .

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