Question 60
Let $n$ be a positive integer. Suppose there are infinitely many cards and on each of these cards one integer greater than or equal to $0$ is written. Suppose also for each integer $l$ greater than or equal to $0$, there are exactly $n$ cards which have the number $l$ written on them. Let us consider repeating the process of selecting $2012$ cards from the given infinite collection of cards. Assume that we do not return the cards once selected to the original collection. Find the smallest possible value of $n$ for which it is possible to keep going through the procedures satisfying the following condition:
Condition: For every positive integer $k$, the sums of the numbers written on the $2012$ cards selected at the $k$-th stage equals $k$.
Condition: For every positive integer $k$, the sums of the numbers written on the $2012$ cards selected at the $k$-th stage equals $k$.