2018 APMO 第 4 题

Let $A B C$ be an equilateral triangle. From the vertex $A$ we draw a ray towards the interior of the triangle such that the ray reaches one of the sides of the triangle. When the ray reaches a side, it then bounces off following the law of reflection, that is, if it arrives with a directed angle $\alpha$, it leaves with a directed angle $180^{\circ}-\alpha$. After $n$ bounces, the ray returns to $A$ without ever landing on any of the other two vertices. Find all possible values of $n$.

Answer: All $n \equiv 1,5 \bmod 6$ with the exception of 5 and 17

设$A B C$ 为等边三角形。我们从顶点 $A$ 向三角形内部绘制一条射线，使得射线到达三角形的一条边。当光线到达一侧时，它会按照反射定律反弹，也就是说，如果它以定向角 $\alpha$ 到达，那么它会以定向角 $180^{\circ}-\alpha$ 离开。在 $n$ 反弹后，光线返回到 $A$，而不会落在任何其他两个顶点上。找出 $n$ 的所有可能值。

答案：除了 5 和 17 之外，所有 $n \equiv 1,5 \bmod 6$