OMCC 2018 Problem 6

A dance with 2018 couples takes place in Havana. For the dance, 2018 distinct points labeled $0, 1,\ldots, 2017$ are marked in a circumference and each couple is placed on a different point. For $i\geq1$ , let $s_i=i\ (\textrm{mod}\ 2018)$ and $r_i=2i\ (\textrm{mod}\ 2018)$ . The dance begins at minute $0$ . On the $i$ -th minute, the couple at point $s_i$ (if there's any) moves to point $r_i$ , the couple on point $r_i$ (if there's any) drops out, and the dance continues with the remaining couples. The dance ends after $2018^2$ minutes. Determine how many couples remain at the end.

Note: If $r_i=s_i$ , the couple on $s_i$ stays there and does not drop out.