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HMMT 十一月 2011 · 团队赛 · 第 3 题

HMMT November 2011 — Team Round — Problem 3

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 6 ] Find the sum of the coefficients of the polynomial P ( x ) = x − 29 x + ax + bx + c , given that
    P (5) = 11, P (11) = 17, and P (17) = 23. 2
解析

英文解析

  1. [ 6 ] Find the sum of the coefficients of the polynomial P ( x ) = x − 29 x + ax + bx + c , given that
    P (5) = 11, P (11) = 17, and P (17) = 23.
    4 3 2
    Answer: -3193 Define Q ( x ) = P ( x ) − x − 6 = x − 29 x + ax + ( b − 1) x + ( c − 6) and notice that
    Q (5) = Q (11) = Q (17) = 0. Q ( x ) has degree 4 and by Vieta’s Formulas the sum of its roots is 29, soits last root is 29 − 17 − 11 − 5 = − 4, giving us Q ( x ) = ( x − 5)( x − 11)( x − 17)( x + 4). This meansthat P (1) = Q (1) + 7 = ( − 4)( − 10)( − 16)(5) + 7 = − 3200 + 7 = − 3193. 2