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I am studying for the Putnam exam and I have learned that the graders are quite strict and will cut off points for a variety of reasons. I want to know exactly how to write a Putnam proof. How specific do I have to be? Can I skip obvious steps, or do I have to explain every last detail?

In particular, I would like to see some model Putnam proofs, preferably ones released by the graders of the exam.

EXAMPLE: how many points out of 10 will this solution get?

A2. Given any 5 distinct points on the surface of a sphere, show that we can find a closed hemisphere which contains at least 4 of them.

My solution -

The plane formed by the center of the sphere and some two of the points splits the sphere into two closed hemispheres, one of which must contain at least 2 of the remaining 3 points since the hemispheres are closed. QED.

user1299784
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    Other than actually knowing how to solve the problem, look over your solution and ensure that every statement you make is completely and mathematically true. For example, don't say "the plane formed by the center of the sphere and two of the points". Say that given three distinct points, there exists a unique plane in $\mathbb{R}^3$ containing them. – Christopher A. Wong Aug 06 '14 at 03:20

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Two of the given points and the center of the sphere determine a plane. Because the plane goes through the center of the sphere, it splits the sphere into two hemispheres. Because there are five points total, one of the closed hemispheres must contain four points; two on the boundary and two in the interior. QED

Three points don't "form" a plane, but they do determine a plane. Mention the reason the plane spits the sphere into hemispheres; it goes through the center. It helps to emphasize the reasons. I can't tell how many points out of 10 you would get with your answer, perhaps all 10, but perhaps only 8 (?) Nice job by the way.

Paul Sundheim
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  • Dear Paul, two points on the sphere and the center don't necessarily determine a plane since they might be aligned: just write that some plane goes through the three. Also, there might be no point at all from the five in the interior of either hemisphere: the five points might be on the same great circle. So don't write that there are "two in the interior". – Georges Elencwajg Aug 06 '14 at 07:54
  • Dear Georges, the sentence reads "two of the given points and the center of the sphere determine a plane". That is three points. Also, two points are always aligned. Your comment about the points being on the boundary is a good one though. – Paul Sundheim Aug 06 '14 at 13:44