HMMT 二月 2020 · ALGNT 赛 · 第 1 题

HMMT February 2020 — ALGNT Round — Problem 1

Let P ( x ) = x + x − r x − 2020 be a polynomial with roots r, s, t . What is P (1)?

解析

Let P ( x ) = x + x − r x − 2020 be a polynomial with roots r, s, t . What is P (1)? Proposed by: James Lin Answer: − 4038 2 2 Solution 1: Plugging in x = r gives r = 2020. This means P (1) = 2 − r − 2020 = − 4038. Solution 2: Vieta’s formulas give the following equations: r + s + t = − 1 2 rs + st + tr = − r rst = 2020 . The second equation is ( r + t )( r + s ) = 0. Without loss of generality, let r + t = 0. Then s = r + s + t = − 1. 2 Finally r = rst = 2020, so P (1) = − 4038.