MathDB

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.

Problem Statement