Question 62
Each of the $2014^2$ squares of a $2014\times2014$ board contains a lamp, which can be on or off. Initially some lamps are on. In a move, choose a row or column with at least $1007$ lamps on and switch the state of all $2014$ lamps in it (on to off and off to on). Find the smallest nonnegative integer $k$ such that from every initial state one can reach, in finitely many moves, a state with at most $k$ lamps on.