2010 Putnam Exam

Doable problems (In approximate order of doableness): A1, B1, A2, B2, B3, A4.

The other problems are harder.

Hints:
A1: The common sum can never be less than n.
B1: Cauchy-Schwartz inequality.
A2: Show f'(x+1) = f'(x).
B2: Show that AB cannot equal 0, 1 or 2. Use the triangle inequality.
B3: Pigeonhole principle.
A4: Write n = (2^m)*k, where k is odd. Now work mod 10^(2^m)+1.