PUMaC 2022 · 代数（A 组） · 第 1 题

PUMaC 2022 — Algebra (Division A) — Problem 1

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

, (2 k + 1 , 3 k + 2) for k ∈ Z . These are roots of the linear polynomials 3 x − 2 y , 3 x − 2 y + 2, 3 x − 2 y − 1, and 3 x − 2 y + 1, respectively. It follows that P ( x, y ) is divisible by the product (3 x − 2 y )(3 x − 2 y + 2)(3 x − 2 y − 1)(3 x − 2 y + 1). Letting z = 3 x − 2 y , the product equals 2 4 3 2 z ( z + 2)( z − 1) = z + 2 z − z − 2 z . The coefficient of y is given as − 2( − 2) = 4, hence in 2 2 fact P ( x, y ) equals the product. To find the coefficient of x y , apply the Binomial Theorem 4 2 2 to find · 3 · ( − 2) = 216, our answer. 2