HMMT 十一月 2009 · 冲刺赛 · 第 29 题

英文解析

29. [ 17 ] For how many integer values of b does there exist a polynomial function with integer coefficientssuch that f (2) = 2010 and f ( b ) = 8?
    2002
    Answer: 32 We can take f ( x ) = − ( x − b ) + 2010 for all divisors d of − 2002. To see that wedcan’t get any others, note that b − 2 must divide f ( b ) − f (2), so b − 2 divides − 2002 (this is becausen n n nb − 2 divides b − 2 and hence any sum of numbers of the form b − 2 ).