by Frank Morgan
Some 25 Williams students took the notorious national Putnam Exam mathematics competition on Saturday, December 7, 2019. The morning session (10-1) and the afternoon session (3-6 pm) each had six problems. Results should be announced end of February, 2020.
Morning problem 4 asked if a continuous function on R3 whose integral over every sphere of radius 1 is 0 has to be identically 0. A short counterexample uses the fact known to the ancient Greeks that the surface area of a sphere equals the area of the circumscribed say vertical cylinder, horizontal slice by horizontal slice. So any periodic function of z with period 2 and integral 0 integrates to 0 over every unit sphere.