March 18–March 24, 2008
Problem
Let f be a function such that f(n) is an integer whenever n is an integer. Prove that there is some integer n so that f(f(n)) ≠ n+3.
Solution
There were a few clever solutions using some group theory or using some properties of recurrence relations. Here's a more "hands on" solution that requires less background knowledge: solution. David Cohoon provided a very complete solution for the general problem f(f(n)) ≠ n + p. here.
List of solvers
Amir Talebi (undergrad); Dustin Moody (graduate); Steve Checkoway, David Cohoon (faculty); Lloyd Sakazaki, Stefan Sharkansky, Kate Smith, Mike Goodman (outside).
Amir Talebi wins the prize!