Algebra · Rational and irrational numbers · Quadratic irrationals · Symmetric expressions
Let \(x, y \in \mathbb{R}\) be such that \(x + y\) and \(x^2 + y\) are rational numbers.
(a) If \(x + y^2\) is rational as well, must \(x\) and \(y\) be rational?
(b) If \(x^3 + y\) is rational as well, must \(x\) and \(y\) be rational?
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Serbian Regional Competition (Okruzno takmicenje) 2023, high school grade I, category A, problem 5. Organized by the Mathematical Society of Serbia (DMS). Source