HMMT 二月 2002 · 冲刺赛 · 第 57 题

英文解析

57. How many ways, without taking order into consideration, can 2002 be expressedas the sum of 3 positive integers (for instance, 1000 + 1000 + 2 and 1000 + 2 + 1000 areconsidered to be the same way)?
    Solution: Call the three numbers that sum to 2002 A , B , and C . In order to preventredundancy, we will consider only cases where A ≤ B ≤ C . Then A can range from 1 to
    3( A − 1)
    667, inclusive. For odd A , there are 1000 − possible values for B . For each choiceof A and B , there can only be one possible C , since the three numbers must add up to a 2
    fixed value. We can add up this arithmetic progression to find that there are 167167 possible
    3 Acombinations of A, B, C , for odd A . For each even A , there are 1002 − possible valuesfor B . Therefore, there are 166833 possible combinations for even A . In total, this makes 2
    334000 possibilities.
    n − 1 a