HMMT 十一月 2010 · 冲刺赛 · 第 23 题

英文解析

23. [ 12 ] Let a , a , . . . be an infinite sequence of positive integers such that for integers n > 2, a =
    1 2 n
    2012
    3 a − 2 a . How many such sequences { a } are there such that a ≤ 2 ?
    n − 1 n − 2 n 2010
    2009
    Answer: 36 · 2 + 36 Consider the characteristic polynomial for the recurrence a − 3 a +
    n +2 n +1
    2 a = 0, which is x − 3 x + 2. The roots are at 2 and 1, so we know that numbers a must be of 2
    n ii − 1 2009
    the form a = a 2 + b for integers a and b . Therefore a must equal to a 2 + b , where a andi 2010
    b are both integers. If the expression is always positive, it is sufficient to say a is positive and a isnonnegative, or a + b > 0, and a ≥ 0.1
    2012 2009 2012 2009
    For a given value of a , 1 − a ≤ b ≤ 2 − a 2 , so there are 2 − a 2 + a possible values of b foreach a (where the quantity is positive). a can take any value between 0 and 2 , we sum over all such a 3
    2012 2009
    in this range, to attain 9 · 2 − (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8)2 + (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8),
    ( )
    2009
    or 36 2 + 36 , which is our answer. 1