There are an odd number of players in the chess game, and each player plays a game with other players. The scoring method is 1 point for winning a game, 0.5 point for each game and 0 point for losing a game. It is known that two of the contestants scored 8 points, and the average scores of others were integers. How many contestants are there?

Solution: suppose there are x+2 players, then each player plays x+ 1

Then the total score of all the players is (x+ 1)(x+2)/2 (because 1 of a chess game is equal to two players). Let the rest of the scores be y.

So the equation is listed: (x+ 1)(x+2)/2=8+xy, (y∈z).

Because x is not equal to 0.

So we get the following relationship: y = (x 2+3x- 14)/2x.

Reordering: y=(x/2)+3/2-(7/x).

Because y is an integer,

So x must be a divisor of 7, that is, 1 and 7.

Because two players scored 8 points and the number of participants was odd,

So x can only be equal to 7.

After testing, x=7.y=4 holds.

So * * * 9 people took part in the competition.