Authors: Dumanov I., Krumovsky R.

Discussing winter holidays in class, Sasha said: "Now, after I flew to Addis Ababa, I have celebrated New Year's in all possible hemispheres of the Earth, except one!

What is the minimum number of places where Sasha has celebrated New Year's?

Consider the places where Sasha celebrated New Year's as points on a sphere. Points on the boundary of a hemisphere do not count as belonging to that hemisphere.
Evaluation. Suppose three points were enough. They define a plane. If it passes through the center of the sphere, then Sasha has not been in two hemispheres cut by this plane. Contradiction.

If it does not pass, then we can cut off a hemisphere in which Sasha has not been with a parallel plane. In all hemispheres obtained from it by a small movement, Sasha has also not been. Again, a contradiction.

Example. Let's take three points on a certain great circle, forming an acute triangle, and any point outside it (this can be done so that one of the points is the city where Sasha lives, and the other is Addis Ababa). Of the two hemispheres cut by this circle, Sasha has not been in exactly one. If we draw another plane through the center of the Earth, it will cut a diameter in the original circle, on each side of which there will be a point where Sasha has been.

## Answer

In four places.