1. The measures of the angles formed around a point $O$ are expressed (in degrees) by powers of the number 5. Find the minimum number of angles under the given conditions.
1. To have as few angles as possible, their measures need to be as large as possible. We can take at most two angles of measure $125^{\circ}$ (and we will take exactly two!), at most four angles of measure $25^{\circ}$ (and we will take exactly four), and, finally, two more angles of measure $5^{\circ}$. The minimum number of angles under the conditions of the problem is, therefore, eight.