PUMaC 2022 · 代数（B 组） · 第 3 题
PUMaC 2022 — Algebra (Division B) — Problem 3
题目详情
- Given two polynomials f and g satisfying f ( x ) ≥ g ( x ) for all real x , a separating line between f and g is a line h ( x ) = mx + k such that f ( x ) ≥ h ( x ) ≥ g ( x ) for all real x . Consider the 2 2 set of all possible separating lines between f ( x ) = x − 2 x + 5 and g ( x ) = 1 − x . The set of 4 4 slopes of these lines is a closed interval [ a, b ]. Determine a + b .
解析
- Given two polynomials f and g satisfying f ( x ) ≥ g ( x ) for all real x , a separating line between f and g is a line h ( x ) = mx + k such that f ( x ) ≥ h ( x ) ≥ g ( x ) for all real x . Consider the 2 2 set of all possible separating lines between f ( x ) = x − 2 x + 5 and g ( x ) = 1 − x . The set of 4 4 slopes of these lines is a closed interval [ a, b ]. Determine a + b . Proposed by Frank Lu Answer: 184 2 Solution: We consider y = mx + b for our line. To have f ( x ) ≥ mx + b , we need x − ( m + 2