1997 APMO 第 5 题

Suppose that $n$ people $A_1$ , $A_2$ , $\ldots$ , $A_n$ , ( $n \geq 3$ ) are seated in a circle and that $A_i$ has $a_i$ objects such that

$$a_1 + a_2 + \cdots + a_n = nN$$

where $N$ is a positive integer. In order that each person has the same number of objects, each person $A_i$ is to give or to receive a certain number of objects to or from its two neighbours $A_{i-1}$ and $A_{i+1}$ . (Here $A_{n+1}$ means $A_1$ and $A_n$ means $A_0$ .) How should this redistribution be performed so that the total number of objects transferred is minimum?

假设 $n$ 人 $A_1$ 、 $A_2$ 、 $\ldots$ 、 $A_n$ 、 ( $n \geq 3$ ) 围坐成一圈，并且 $A_i$ 有 $a_i$ 物体，使得

$$a_1 + a_2 + \cdots + a_n = nN$$

其中 $N$ 是正整数。为了使每个人拥有相同数量的物体，每个人 $A_i$ 需要向其两个邻居 $A_{i-1}$ 和 $A_{i+1}$ 给予或从其接收一定数量的物体。 (这里 $A_{n+1}$ 表示 $A_1$ ， $A_n$ 表示 $A_0$ 。)应该如何执行这种重新分配，以使传输的对象总数最小？