KöMaL 2022 K745

In a game, players are collecting points. The players take turns in playing and scoring points. When a certain player is playing, he or she may get any non-negative integer of points (including 0). The points scored by a player in successive turns add up. The game terminates when the total of the points scored by the players reaches 1000 (that is, the last player may only score as many points as needed to make the total equal to 1000). The player with the largest number of points will win the game. In the case of equality, the player reaching the same score earlier will win. The player with the second largest number of points will finish in the second place, and so on. At the moment, the scores of the players are as follows: Kate has 314 points, Sam has 207 points, John has 58 points, Gillian has 31 and Joe has 0. $a)$ If it is Kate's turn now, what is the minimum number of points she needs to gain in this turn in order to be certain that she will finish in the first or second place? $b)$ If it is Sam's turn now, what is the minimum number of points he needs to gain in this turn in order to be certain that he will finish in the first or second place? $c)$ If it is Joe's turn now, what is the minimum number of points he needs to gain in this turn in order to be certain that he will finish in the first or second place?