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The claim is false.

Choose a diameter, and draw a point at each end.

If neither point is on the dividing equator, they must lie in opposite hemispheres.

If you have two such pairs, each hemisphere must have at least 2 points.

Since there are only 5 points total, each hemisphere can have at most 3 points.

The claim is

false.

Choose a diameter, and draw a point at each end.

If neither point is on the dividing equator, they must lie in opposite hemispheres.

If you have two such pairs, each hemisphere must have at least 2 points.

Since there are only 5 points total, each hemisphere can have at most 3 points.

Edit:

If we allow that points can lie on the dividing equator, the above will not suffice as a counterexample. See the comments below.

The claim is false.

Choose a diameter, and draw a point at each end.

If the equator does not cut through either point, they must lie in opposite hemispheres.

If you have two such pairs, each hemisphere must have at least 2 points.

The claim is false.

Choose a diameter, and draw a point at each end.

If neither point is on the dividing equator, they must lie in opposite hemispheres.

If you have two such pairs, each hemisphere must have at least 2 points.

Since there are only 5 points total, each hemisphere can have at most 3 points.

The claim is false.

Draw two points at diametrically opposite ends of the sphere.

If the equator does not cut through the points, they must lie in opposite hemispheres.

If you have two such pairs, each hemisphere must have at least 2 points.

The claim is false.

Choose a diameter, and draw a point at each end.

If the equator does not cut through either point, they must lie in opposite hemispheres.

If you have two such pairs, each hemisphere must have at least 2 points.