BarcelonaTech Mathcontest

First we observe that if $r$, $s$, and $r + s$ are positive integers then at least one of them must be even. Indeed, if both are odd then their sum $r + s$ is even. Therefore, the number $rs(r + s)$ must also be even. So, if $n$ and $rs(r + s)$ are coprime then $n$ must be odd. Hence, if $n$ is even there are no suitable positive integers $r$ and $s$ that satisfy the given conditions.

* If $n = 1$, then $n$ divides any number and in particular $r - s$. So, $n$ must be odd and greater than 1.
* If $n = 3$ then $r$ and $s$ cannot be multiples of 3, and also they cannot have the same remainder when divided by 3 because of the condition $3 \nmid (r-s)$. Therefore, one of them must leave remainder 1 and the other remainder 2 when divided by 3. This means that $3 \mid (r+s)$. We conclude that there are no suitable numbers $r$ and $s$ for $n = 3$.

If $n > 3$ and is odd, then the numbers $r = 2$ and $s = n - 1$ satisfy
$$\text{gcd}(rs(r+s), n) = \text{gcd}(2(n-1)(n+1), n) = 1$$
and $n \nmid (r-s) = 3-n$.

As a conclusion, we have that the set of positive integers that fulfill the requirements of the statement is
$$\{5, 7, 11, 13, \dots, 2011, 2013\}$$