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Assume the Earth is a perfect sphere. There are infinitely many ways to divide the Earth into hemispheres, besides North and South. Any great circle will play the role of an “equator” between them.

Suppose five cities on Earth are chosen. Show that there is a hemisphere which contains at least four of the cities, possibly on the “equator” between the two hemispheres.

I assume that I will need to use the pigeonhole principle to prove this since I've just learned it, but I don't know how to divide Earth into hemisphere such that four of the any 5 randomly chosen cities will fall under one hemisphere?

bloomsdayforever
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    Start by running a great circle through two of the cities, which is always possible. – Robert Shore Nov 10 '21 at 03:00
  • Can you clarify what running a great circle mean? – bloomsdayforever Nov 10 '21 at 03:03
  • @bloomsdayforever A great circle is a circle on the surface of the sphere that is formed by the intersection of the sphere with a plane that passes through the sphere's center. In other words, a circle whose radius equals the radius of the sphere. – heropup Nov 10 '21 at 03:08
  • I still do not understand why being able to find a point between the two cities and creating a plane that separate the two cities in two different hemispheres proves that at least four cities must fall under hemisphere? – bloomsdayforever Nov 10 '21 at 03:36
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    No, the hint is not to put the great circle so it separates the two cities in two different hemispheres. You put the great circle so both cities are on the great circle. The hint exploits the part of the problem that says a city is in a hemisphere even if it is on the "equator" of the hemisphere. – David K Nov 10 '21 at 04:05
  • Another different way to do it: suppose that the statement is false. Consider the tetrahedron formed by 4 of these points, and notice that for the statement to be false, the center of the sphere must always lie within the tetrahedron. Now consider the 5th point and it should be easy to see that no matter where it goes, we can easily find a plane running through the center of the circle that has 4 points (at least) on one side of it. – kyary Nov 10 '21 at 06:28

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