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Easy Chess Tournament Problem
Posted: Tue Feb 02, 2016 4:05 am
by Mallika Prova
On a chess tournament 12 players took part, and any two of them played exactly one match.Any draw gives 0.5 point, win -1 point, loss -0.It turned out after the tournament that the first three together gained three times more points then the last five. How did the match between the seventh and eighth player end?
Re: Easy Chess Tournament Problem
Posted: Wed Feb 03, 2016 12:36 pm
by asif e elahi
Let $A,B,C$ be the set of first $3$,middle $4$ and last $5$ players respectively according to their rank and $U$ is the union set.Let $f(X,Y)$ denote the total score gained by the players of set $X$ against players of set $Y$.Easy to see that every match has total outcome of point $1$.
now $f(A,U)=f(A,A)+f(A,B\cup C) \leq \binom{3}{2} +3 \times 9=30$
and $f(C,U)=f(C,C)+f(C,A\cup B) \geq \binom{5}{2}+0=10$
So $f(A,U)\leq 3f(C,U)$.But we have them equal.
So we must have $f(C,U)=10$.Equality holds when $f(C,A\cup B)=0$.So players of $C$ loses all matches against the players of $A\cup B$.So the$8th$ player will lose against the $7th$ player.