$\underline{\text { by Banov I.S }}$

The height and the radius of the base of a cylinder are equal to 1. What is the smallest number of spheres with a radius of 1 that can completely cover this cylinder?
We will inscribe an equilateral triangle in the cylinder with its vertices at the midpoints of the generators and consider three unit spheres with centers at the midpoints of its sides. Taking the base of the cylinder, our spheres intersect it in three circles of radius $\sqrt{3} / 2$ with centers at the midpoints of the sides of the inscribed equilateral triangle. It is easy to verify that these three circles completely cover the base.

Thus, the cylinder is entirely covered by three cylinders of height 1 built on these circles, and consequently, by our spheres containing these cylinders.

To show that the cylinder cannot be covered by two spheres, it is sufficient to note that if the center of a sphere does not coincide with the center of the base, then the circle where the sphere intersects the base plane intersects the boundary of the base along an arc less than $180^{\circ}$, and two such arcs cannot cover the circumference.

If the centers of the two spheres coincide with the centers of the bases, then the lateral surface of the cylinder remains uncovered.

## Answer

Three.

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