1991 APMO 第 4 题

During a break, $n$ children at school sit in a circle around their teacher to play a game. The teacher walks clockwise close to the children and hands out candies to some of them according to the following rule. He selects one child and gives him a candy, then he skips the next child and gives a candy to the next one, then he skips 2 and gives a candy to the next one, then he skips 3, and soon. Determine the values of $n$ for which eventually, perhaps after many rounds, all children will have at least one candy each.

Answer: All powers of 2 .

课间休息时，学校里的 $n$ 孩子围着老师坐成一圈玩游戏。老师按顺时针方向走到孩子们身边，并按照以下规则向其中一些孩子分发糖果。他选择一个孩子并给他一颗糖果，然后他跳过下一个孩子并给下一个孩子一颗糖果，然后他跳过 2 个孩子并给下一个孩子一颗糖果，然后他跳过 3 个，很快。确定 $n$ 的值，最终，也许经过多轮之后，所有孩子每人都会至少得到一颗糖果。

答案： 2 的所有幂。