The distance between points A and B is 15 cm. How many different points M are there on ray AB such that | BM | = 20 cm.

Since AB is a ray, it is limited by point A and not limited by point B. Therefore, on the side of point A from point B, only such a point can be located whose distance to B is less than the distance from A to B. Otherwise, the point does not belong beam AB. There are no such restrictions on the opposite side of point B. Hence a point M such that | BM | = 20 cm while | AB | = 15 cm can only be on the opposite side from A, since 20 cm> 15 cm. That is, there is only one such point.
Answer: one point.