1974 IMO 第 5 题

I 5 (GBR 3) Let $A_{r}, B_{r}, C_{r}$ be points on the circumference of a given circle $S$. From the triangle $A_{r} B_{r} C_{r}$, called $\triangle_{r}$, the triangle $\triangle_{r+1}$ is obtained by constructing the points $A_{r+1}, B_{r+1}, C_{r+1}$ on $S$ such that $A_{r+1} A_{r}$ is parallel to $B_{r} C_{r}, B_{r+1} B_{r}$ is parallel to $C_{r} A_{r}$, and $C_{r+1} C_{r}$ is parallel to $A_{r} B_{r}$. Each angle of $\triangle_{1}$ is an integer number of degrees and those integers are not multiples of 45 . Prove that at least two of the triangles $\triangle_{1}, \triangle_{2}, \ldots, \triangle_{15}$ are congruent.

I 5 (GBR 3) 令 $A_{r}, B_{r}, C_{r}$ 为给定圆 $S$ 圆周上的点。从三角形 $A_{r} B_{r} C_{r}$(称为 $\triangle_{r}$)出发，通过在 $S$ 上构建点 $A_{r+1}、B_{r+1}、C_{r+1}$ 来获得三角形 $\triangle_{r+1}$，使得 $A_{r+1} A_{r}$ 与 $B_{r} C_{r} 平行， B_{r+1} B_{r}$ 与 $C_{r} A_{r}$ 平行，$C_{r+1} C_{r}$ 与 $A_{r} B_{r}$ 平行。 $\triangle_{1}$ 的每个角度都是整数度数，并且这些整数不是 45 的倍数。证明三角形 $\triangle_{1}, \triangle_{2}, \ldots, \triangle_{15}$ 中至少有两个全等。