Problem

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Tags: combinatorics



Let $n$ be a positive integer. There is an infinite number of cards, each one of them having a non-negative integer written on it, such that for each integer $l \geq 0$, there are exactly $n$ cards that have the number $l$ written on them. A move consists of picking $100$ cards from the infinite set of cards and discarding them. Find the least possible value of $n$ for which there is an infinitely long series of moves such that for each positive integer $k$, the sum of the numbers written on the $100$ chosen cards during the $k$-th move is equal to $k$.