Ratings and Rankings

Chef organised a chess tournament, which spanned over $M$ months. There were $N$ players, and player $i$ was rated $R_i$ before the start of the tournament. To see the progress of the players, he noted their rating changes at the end of each month. After the tournament, FIDE asked Chef to find the number of players whose peak rating and peak ranking did not occur in the same month. In other wor