HMMT 二月 2017 · 团队赛 · 第 2 题
HMMT February 2017 — Team Round — Problem 2
题目详情
- [ 25 ] 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?
解析
- [ 25 ] 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? Proposed by: Alexander Katz Answer: No For any and positive x , P ( x, ) > 0 and P ( x, − ) ≤ 0. Thus by continuity/IVT, P ( x, 0) = 0 for all positive x . Similarly P (0 , y ) = 0 for all positive y . This implies xy | P ( x, y ), and so we can write P ( x, y ) = xyQ ( 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.