HMMT 二月 2020 · 冲刺赛 · 第 9 题

HMMT February 2020 — Guts Round — Problem 9

[6] Let P ( x ) be the monic polynomial with rational coefficients of minimal degree such that , 2 1 1 1 √ √ √ , , . . . , are roots of P . What is the sum of the coefficients of P ? 3 4 1000

解析

[6] Let P ( x ) be the monic polynomial with rational coefficients of minimal degree such that , 2 1 1 1 √ √ √ , , . . . , are roots of P . What is the sum of the coefficients of P ? 3 4 1000 Proposed by: Carl Joshua Quines 1 Answer: 16000 1 1 √ √ Solution: For irrational , − must also be a root of P . Therefore r r ( ) ( ) ( ) 2 1 2 1 2 1 x − x − · · · x − 2 3 1000 ( ) ( ) ( ) P ( x ) = . 1 1 1 x + x + · · · x + 2 3 31 We get the sum of the coefficients of P by setting x = 1, so we use telescoping to get 1 2 999 · · · · 1 2 3 1000 P (1) = = . 3 4 32 16000 · · · · 2 3 31