HMMT 十一月 2016 · 冲刺赛 · 第 14 题

英文解析

14. [ 9 ] The very hungry caterpillar lives on the number line. For each non-zero integer i , a fruit sits on the point with coordinate i . The caterpillar moves back and forth; whenever he reaches a point with food,
    he eats the food, increasing his weight by one pound, and turns around. The caterpillar moves at a − wspeed of 2 units per day, where w is his weight. If the caterpillar starts off at the origin, weighingzero pounds, and initially moves in the positive x direction, after how many days will he weigh 10
    pounds?
    Proposed by: Eshaan Nichani
    Answer: 9217
    On the n th straight path, the caterpillar travels n units before hitting food and his weight is n − 1.
    1 − n
    Then his speed is 2 . Then right before he turns around for the n th time, he has traveled a total time
    ∑ ∑
    n ni 1
    of = i · 2 . We want to know how many days the catepillar moves before his weighti
    1 − ii =1 i =1
    2 2
    is 10, so we want to take n = 10 so that his last straight path was taken at weight 9. Hence we want to
    ( )
    ∑ ∑ ∑
    10 11 10
    1 i 1 i 1 11 ievaluate S = i · 2 . Note that 2 S = ( i − 1) · 2 , so S = 2 S − S = 11 · 2 − 2 =
    2 i =1 2 i =2 2 i =1
    ( )
    11 111
    10 · 2 − 2 + 2 = 9217 . 2