返回题库

HMMT 十一月 2008 · 冲刺赛 · 第 19 题

HMMT November 2008 — Guts Round — Problem 19

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 11 ] Let P be a polynomial with P (1) = P (2) = · · · = P (2007) = 0 and P (0) = 2009!. P ( x ) hasleading coefficient 1 and degree 2008. Find the largest root of P ( x ).
解析

英文解析

  1. [ 11 ] Let P be a polynomial with P (1) = P (2) = · · · = P (2007) = 0 and P (0) = 2009!. P ( x ) hasleading coefficient 1 and degree 2008. Find the largest root of P ( x ).
    Answer: 4034072 P (0) is the constant term of P ( x ), which is the product of all the roots of thepolynomial, because its degree is even. So the product of all 2008 roots is 2009! and the product of
    2009!
    the first 2007 is 2007!, which means the last root is = 2009 · 2008 = 4034072.
    2007!