17th Putnam 1957

Problem A1

A surface S in 3-space is such that every normal intersects a fixed line L. Show that we can find a surface of revolution containing S.



Unfortunately, this is false.

It is not hard to paste together pieces of different surfaces of revolution. A simple example is as follows. Take a cylinder terminated at one end by a circle C. For part of the circumference of C continue the cylinder. Leave a gap either side and then on the rest join a portion of a cone (so that the surface is angled outwards away from the central axis).

It is possible, but hard, certainly too hard for a qu. 1, to prove that S is locally a surface of revolution. For details see Gleason et al.



17th Putnam 1957

© John Scholes
25 Feb 2002