HMMT 二月 2010 · GEN1 赛 · 第 7 题

HMMT February 2010 — GEN1 Round — Problem 7

[ 6 ] Suppose that a polynomial of the form p ( x ) = x ± x ± · · · ± x ± 1 has no real roots. What is the maximum possible number of coefficients of − 1 in p ?

解析

[ 6 ] Suppose that a polynomial of the form p ( x ) = x ± x ± · · · ± x ± 1 has no real roots. What is the maximum possible number of coefficients of − 1 in p ? Answer: 1005 Let p ( x ) be a polynomial with the maximum number of minus signs. 2010 2009 p ( x ) cannot have more than 1005 minus signs, otherwise p (1) < 0 and p (2) ≥ 2 − 2 − . . . − 2 − 1 = 1, which implies, by the Intermediate Value Theorem, that p must have a root greater than 1. 2011 x + 1 2010 2009 2008 2011 Let p ( x ) = = x − x + x − . . . − x + 1. − 1 is the only real root of x + 1 = 0 but x + 1 p ( − 1) = 2011; therefore p has no real roots. Since p has 1005 minus signs, it is the desired polynomial.