DK-GEORGMOHR 2023 Problem 5

Georg has a circular game board with 100 squares labelled $1, 2, . . . , 100$ . Georg chooses three numbers $a, b, c$ among the numbers $1, 2, . . . , 99$ . The numbers need not be distinct. Initially there is a piece on the square labelled $100$ . First, Georg moves the piece $a$ squares forward $33$ times and puts a caramel on each of the squares the piece lands on. Then he moves the piece $b$ squares forward $33$ times and puts a caramel on each of the squares the piece lands on. Finally, he moves the piece $c$ squares forward $33$ times and puts a caramel on each of the squares the piece lands on. Thus he puts a total of $99$ caramels on the board. Georg wins all the caramels on square number $1$ . How many caramels can Georg win, at most?