MathDB

Ben has a big blackboard, initially empty, and Francisco has a fair coin. Francisco flips the coin

2013

times. On the

n^{\text{th}}

flip (where

n=1,2,\dots,2013

), Ben does the following if the coin flips heads: (i) If the blackboard is empty, Ben writes

n

on the blackboard. (ii) If the blackboard is not empty, let

m

denote the largest number on the blackboard. If

m^2+2n^2

is divisible by

3

, Ben erases

m

from the blackboard; otherwise, he writes the number

n

. No action is taken when the coin flips tails. If probability that the blackboard is empty after all

2013

flips is

\frac{2u+1}{2^k(2v+1)}

, where

u

v

, and

k

are nonnegative integers, compute

k

.
Proposed by Evan Chen

Problem Statement