PUMaC 2017 · 团队赛 · 第 3 题

PUMaC 2017 — Team Round — Problem 3

(4) Let f ( x ) = ( x − 5)( x − 12) and g ( x ) = ( x − 6)( x − 10). Find the sum of all integers n such f ( g ( n )) that is defined and an integer. 2 f ( n )

解析

Note that 2 g ( x ) − 5 = x − 16 x + 55 = ( x − 5)( x − 11) and 2 g ( x ) − 12 = x − 16 x + 48 = ( x − 4)( x − 12) , f ( g ( n )) so that if h ( n ) = , then 2 f ( n ) ( g ( n ) − 5)( g ( n ) − 12) ( n − 4)( n − 11) h ( n ) = = . 2 2 ( n − 5) ( n − 12) ( n − 5)( n − 12) If n > 12, then n − 5 divides neither n − 4 nor n − 11, h ( n ) is not an integer. Similarly, if n < 4, then n − 12 divides neither n − 4 nor n − 11, so h ( n ) is still not an integer. Since h (5) and h (12) are undefined and h (4) = h (11) = 0, we only need to check h (6), h (7), h (8), h (9), and h (10), and it is clear that of these, only h (8) = 1 is an integer. Therefore, the answer is 4 + 8 + 11 = 23 . Problem written by Zack Stier 1