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Suppose there are $n$ teams playing a tournament. Each team plays exactly one game against each of the other teams. In each game the winner is awarded $1$ point, the loser gets $0$ point and each of the two teams earn $0.5$ point if the match is washed away due to rain. After the completion of the tournament, it is found that exactly half of the points earned by each team were earned against the ten teams with the least number of points. In particular, each of the ten lowest scoring teams earned half their points against the other nine of the ten team. What was the total number of teams in the tournament? I cannot make any progress. Anybody please give me some hint.

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  • $\begingroup$ is it the same thing "ten teams with the least number of points" and "ten lowest scoring teams"? $\endgroup$
    – Salomo
    Commented May 3, 2016 at 17:44

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Call $(n-10)$ teams "good" and $10$ teams "bad"

Since half the points of each team were earned against bad teams, collectively also it must be so.

The bad teams collectively earned $\binom{10}2 = 45$ points among themselves,
and another $45$ points against the good teams.

The good teams earned $\binom{n-10}2$ points among themselves, and ditto against the bad teams.

Equating the total points earned,

$\binom{n}2 = 90 + (n-10)(n-11)$

Solving this will give $ n = 16\;\; or\;\; 25$

A little thought will show that $16$ is infeasible, thus $n = 25$

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