HMMT 二月 2002 · 冲刺赛 · 第 7 题

HMMT February 2002 — Guts Round — Problem 7

[5] A rubber band is 4 inches long. An ant begins at the left end. Every minute, the ant walks one inch along rightwards along the rubber band, but then the band is stretched (uniformly) by one inch. For what value of n will the ant reach the right end during the n th minute?

解析

A rubber band is 4 inches long. An ant begins at the left end. Every minute, the ant walks one inch along rightwards along the rubber band, but then the band is stretched (uniformly) by one inch. For what value of n will the ant reach the right end during the n th minute? Solution: 7 The ant traverses 1 / 4 of the band’s length in the first minute, 1 / 5 of the length in the second minute (the stretching does not affect its position as a fraction of the band’s length), 1 / 6 of the length in the third minute, and so on. Since 1 / 4 + 1 / 5 + · · · + 1 / 9 < 0 . 25 + 0 . 20 + 0 . 167 + 0 . 143 + 0 . 125 + 0 . 112 = 0 . 997 < 1 , the ant does not cover the entire band in six minutes. However, 1 / 4 + · · · + 1 / 10 > 0 . 25 + 0 . 20 + 0 . 16 + 0 . 14 + 0 . 12 + 0 . 11 + 0 . 10 = 1 . 08 > 1 , so seven minutes suffice.