Tomonigo A.K.

How many different ways are there to break down the number 2004 into natural addends that are approximately equal? There can be one or several addends. Numbers are considered approximately equal if their difference is no more than 1. Ways that differ only in the order of the addends are considered the same.
Each method is uniquely determined by the number of summands. Indeed, let 2004 be split into $n$ summands, $r$ of which are equal to $q+1$, and the rest are equal to $q(0 \leq r<n)$. Then $2004=q n+r$. Thus, the numbers $q$ and $r$ are the quotient and remainder of the division of 2004 by $n$, they are uniquely determined by the choice of $n$. Therefore, by choosing any $n$ from 1 to 2004, we get a unique partition of 2004 into $n$ approximately equal summands.

## Answer

2004 ways.