2000 APMO 第 2 题

Given the following arrangement of circles:

Each of the numbers $1,2, \ldots, 9$ is to be written into one of these circles, so that each circle contains exactly one of these numbers and
(i) the sums of the four numbers on each side of the triangle are equal;
(ii) the sums of squares of the four numbers on each side of the triangle are equal.

Find all ways in which this can be done.
Answer: The only solutions are

and the ones generated by permuting the vertices, adjusting sides and exchanging the two middle numbers on each side.

给定以下圆圈排列：

每个数字 $1,2, \ldots, 9$ 都将被写入这些圆圈之一，以便每个圆圈恰好包含这些数字之一，并且
(i) 三角形每条边的四个数字之和相等；
(ii) 三角形每条边上的四个数的平方和相等。

找出所有可以做到这一点的方法。
答案：唯一的解决办法是

以及通过排列顶点、调整边并交换每边的两个中间数字而生成的。