HMMT 十一月 2019 · 团队赛 · 第 3 题

HMMT November 2019 — Team Round — Problem 3

[30] The coefficients of the polynomial P ( x ) are nonnegative integers, each less than 100. Given that P (10) = 331633 and P ( − 10) = 273373, compute P (1).

解析

[30] The coefficients of the polynomial P ( x ) are nonnegative integers, each less than 100. Given that P (10) = 331633 and P ( − 10) = 273373, compute P (1). Proposed by: Carl Joshua Quines Answer: 100 Let 2 P ( x ) = a + a x + a x + . . . 0 1 2 Then 1 ( P (10) + P ( − 10)) = a + 100 a + . . . 0 2 2 and 1 ( P (10) − P ( − 10)) = 10 a + 1000 a + . . . 1 3 2 Since all the coefficients are nonnegative integers, these expressions give us each of the coefficients by just taking two digits in succession. Thus we have a = 3, a = 13, a = 25, a = 29, a = 30 and a = 0 for 0 1 2 3 4 n n > 4. Thus P (1) = a + a + a + · · · = 100 . 0 1 2