### I remember math

#### Apr 30, 2005, 11:06am EDT

Sean, a friend of mine at work, asked me the following problem during lunch a few days ago:

Given any five points on a sphere, show that some four of them must lie on a closed hemisphere.

The problem is from the 2002 Putnam and has an easy to explain solution.

Looking through the rest of the 2002 eam, I also like problem B1 from the same year:

Shanille O’Keal shoots free throws on a basketball court. She hits the first and misses the second, and thereafter the probability that she hits the next shot is equal to the proportion of shots she has his so far. What is the probability she hits exactly 50 of her first 100 shots

I like this problem because my first thought was to use a computer to solve it (DP). Heh, It doesn’t hurt to know how to prove things by hand.

Mattat Apr 30, 2005, 09:56pm EDTAny three points must lie on a plane. The plane that intersects a pair of points and the center of the sphere separates the sphere into hemispheres. Of the remaining three points, at least 2 of them must be in the same hemisphere (pigeonhole principle).

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tonyat May 01, 2005, 05:38pm EDTThat sounds right. I think it’s sufficient to say, “pick two points and use the great circle that passes through those points.”

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