HMMT 二月 2023 · 冲刺赛 · 第 29 题

HMMT February 2023 — Guts Round — Problem 29

[23] Let P ( x ), P ( x ), . . . , P ( x ) be monic polynomials of degree 13 with integer coefficients. Suppose 1 2 k there are pairwise distinct positive integers n , n , . . . , n for which, for all positive integers i and j 1 2 k less than or equal to k , the statement “ n divides P ( m ) for every integer m ” holds if and only if i = j . i j Compute the largest possible value of k .

解析

[23] Let P ( x ), P ( x ), . . . , P ( x ) be monic polynomials of degree 13 with integer coefficients. Suppose 1 2 k there are pairwise distinct positive integers n , n , . . . , n for which, for all positive integers i and j 1 2 k less than or equal to k , the statement “ n divides P ( m ) for every integer m ” holds if and only if i = j . i j Compute the largest possible value of k . Proposed by: Reagan Choi Answer: 144 Solution: We first consider which integers can divide a polynomial P ( x ) for all x . Assume that i c | P ( x ) for all x . Then, c must also divide the finite difference Q ( x ) = Q ( x + 1) − Q ( x ). Since Q ( x ) i i i i 13 13 12 is degree 13 and monic, the leading term of Q ( x ) is the leading term of ( x + 1) − x , which is 13 x . Continuing this process finding finite differences, we see that c must divide R ( x ) = Q ( x + 1) − Q ( x ), 11 which has a leading term 13 · 12 x . At the end, we will see that c | 13!, so these are the only possible values of c . To show that all of these values of c work, consider the polynomial P ( x ) = x ( x + 1) · · · ( x + i