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Algebra Difficulty 4.9 AIME Prove it United States

Problem:

Does there exist a two-variable polynomial $P(x, y)$ with real number coefficients such that $P(x, y)$ is positive exactly when $x$ and $y$ are both positive?

Solution

Solution:

Answer: No
For any $\epsilon$ and positive $x$ , $P(x, \epsilon)>0$ and $P(x,-\epsilon) \leq 0$ . Thus by continuity/IVT, $P(x, 0)=0$ for all positive $x$ . Similarly $P(0, y)=0$ for all positive $y$ . This implies $x y \mid P(x, y)$ , and so we can write $P(x, y)=x y Q(x, y)$ . But then this same logic holds for $Q$ , and this cannot continue infinitely unless $P$ is identically $0$ -- in which case the conditions do not hold. So no such polynomial exists.

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