RU-ALLRUSSIAN 2018 Problem 115

On the table, there're $1000$ cards arranged on a circle. On each card, a positive integer was written so that all $1000$ numbers are distinct. First, Vasya selects one of the card, remove it from the circle, and do the following operation: If on the last card taken out was written positive integer $k$ , count the $k^{th}$ clockwise card not removed, from that position, then remove it and repeat the operation. This continues until only one card left on the table. Is it possible that, initially, there's a card $A$ such that, no matter what other card Vasya selects as first card, the one that left is always card $A$ ?