Is the data given sufficient?
The members of a chess club took part in a round robin competition in which each player plays every other player once.All members scored the same number of points, except four juniors. The sum of the score of these juniors is 17.5. Find the number of players.
For every win one point is awarded to the victor.
For every draw half a point is awarded to both players.
For every loss the loser is given no points for that round.
The answer is 27.
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1 solution
Let there be n players. Since the sum of all points scored by everybody is n(n-1)/2 The sum of points scored by other players is a(n-4) where 'a' is the number of points scored by each player. Thus a(n-4)=n(n-1)/2 -17.5
Now we know that 'a' is either of the form I or I+0.5 where I an integer.
Thus 2a is necessarily integral say k
k(n-4)=n(n-1)-35
k=(n^2-n-35)/(n-4)
k=n+3-23/(n-4)
Let (k-n-3) be K which is an integer since n also has to be an integer thus
K=-23/(n-4)
Thus n-4 either is 23 or 1 Thus n is either 27 or 5
In case n=5 k is negative which is impossible
In case n=27 k is positive thus number of players are 27