#### Positioning and Moving Sprinkler Systems for Irrigation

(MCM 2006A)

There are a wide variety of techniques available for irrigating a field. The technologies range from advanced drip systems to periodic flooding. One of the systems that is used on smaller ranches is the use of "hand move" irrigation systems. Lightweight aluminum pipes with sprinkler heads are put in place across fields, and they are moved by hand at periodic intervals to insure that the whole field receives an adequate amount of water. This type of irrigation system is cheaper and easier to maintain than other systems. It is also flexible, allowing for use on a wide variety of fields and crops. The disadvantage is that it requires a great deal of time and effort to move and set up the equipment at regular intervals.

Given that this type of irrigation system is to be used, how can it be configured to minimize the amount of time required to irrigate a field that is 80 meters by 30 meters? For this task you are asked to find an algorithm to determine how to irrigate the rectangular field that minimizes the amount of time required by a rancher to maintain the irrigation system. One pipe set is used in the field. You should determine the number of sprinklers and the spacing between sprinklers, and you should find a schedule to move the pipes, including where to move them.

A pipe set consists of a number of pipes that can be connected together in a straight line. Each pipe has a 10 cm inner diameter with rotating spray nozzles that have a 0.6 cm inner diameter. When put together the resulting pipe is 20 meters long. At the water source, the pressure is 420 Kilo-Pascal's and has a flow rate of 150 liters per minute. No part of the field should receive more than 0.75 cm per hour of water, and each part of the field should receive at least 2 centimeters of water every 4 days. The total amount of water should be applied as uniformly as possible.

#### 灌溉喷洒系统的定位和移动

(美国竞赛2006年A题)

灌溉土地有许多不同的技术，包括先进的喷洒系统和周期灌水。对于较小的土地有一种叫"手动"的灌溉系统。轻型的铝管和喷头安装在田地各处,人们定期地移动设备保证所有的田地都能受到充分的灌溉。这个系统比起其他系统更便宜也更容易维护。这个系统使用灵活，可以用于各种不同的土地和作物。不足的是人们需要花去大量时间精力定期去移动和安装设备。

针对这种系统，考虑怎样用最少的时间去灌溉一个80乘30平方米的一块土地，在这个系统中你需要确定一个算法，讨论如何由一个农场工人用最少的时间操作这个灌溉系统去灌溉一个长方形的土地。这块土地只用一个管道装置，你需要决定喷头的数目和喷头之间的距离， 您需要寻求一个方案怎样去移动管道，包括在那里移动它们。

一个管道系统包括若干个可以连成一条直线的水管，每根水管的内直径10厘米 ，并带有若个个内径是0.6厘米的旋转喷嘴，把它们组装在一块就可连成一根20米长的水管，在水源处的水压力是420千帕，水流量是每分钟150升，任何地方的降水量不能超过每小时0.75厘米，而且4天内的降水量至少要有2厘米。每处的灌溉的总水量应该尽量一致。