Question

为完成一项大工程，雇用了 $1000$ 名工人，人数刚好足以按期完成。所有工人在完成前四分之一工程时都留在工地，所以前四分之一按时完成。之后解雇了 $100$ 名工人，于是第二个四分之一延误完成。又解雇了 $100$ 名工人，于是第三个四分之一完成时进度进一步落后。已知所有工人的工作效率相同，在完成四分之三工程后，为使整个工程按期或提前完成，至少还必须在剩下的 $800$ 名工人之外再雇用多少名工人？

In order to complete a large job, $1000$ workers were hired, just enough to complete the job on schedule. All the workers stayed on the job while the first quarter of the work was done, so the first quarter of the work was completed on schedule. Then $100$ workers were laid off, so the second quarter of the work was completed behind schedule. Then an additional $100$ workers were laid off, so the third quarter of the work was completed still further behind schedule. Given that all workers work at the same rate, what is the minimum number of additional workers, beyond the $800$ workers still on the job at the end of the third quarter, that must be hired after three-quarters of the work has been completed so that the entire project can be completed on schedule or before?

答案

答案：766

知识点：速率不等式

难度评级：2390

解答：

取时间单位，使 $1000$ 名工人完成四分之一工程需要 $1$ 个单位；总工期为 $4$ 个单位。 $900$ 名工人完成第二个四分之一需要 $\frac{10}{9}$ 个单位， $800$ 名工人完成第三个四分之一需要 $\frac{10}{8} = \frac{5}{4}$ 个单位。已用时间为 $1 + \frac{10}{9} + \frac{5}{4} = \frac{121}{36},$ 剩余时间为 $4 - \frac{121}{36} = \frac{23}{36}$ 个单位。

最后四分之一需要 $1000$ 个工人时间单位的劳动量，所以总工人数 $w$ 必须满足 $w \cdot \frac{23}{36} \ge 1000$ ，即 $w \ge \frac{36000}{23} \approx 1565.2$ ，因此至少需要 $1566$ 名工人。

工地上还剩 $800$ 名工人，所以至少还要雇用 $1566 - 800 = 766$ 名工人。

Measure time so that $1000$ workers complete a quarter of the job in $1$ unit; the schedule allows $4$ units in all. With $900$ workers the second quarter takes $\frac{10}{9}$ units, and with $800$ workers the third quarter takes $\frac{10}{8} = \frac{5}{4}$ units. The time used so far is $1 + \frac{10}{9} + \frac{5}{4} = \frac{121}{36},$ leaving $4 - \frac{121}{36} = \frac{23}{36}$ of a unit for the last quarter.

The last quarter requires $1000$ worker-units of labor, so the workforce $w$ must satisfy $w \cdot \frac{23}{36} \ge 1000,$ that is $w \ge \frac{36000}{23} \approx 1565.2,$ so at least $1566$ workers are needed.

Since $800$ remain on the job, at least $1566 - 800 = 766$ additional workers must be hired.

2004 AIME II 第 5 题

答案：766

难度评级：2390

解答：

其他年份的第 5 题