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In a group of 6 teams (Football) with each team playing the other twice, once home, once away. points as follows:

win=3 tie=1 loss=0

since each team plays 10 games and only the first two teams advance to the next round. What is the minimum points needed to guarantee advancing to the next round?

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We want to find the maximum score the third place team (including ties) can have, then add one. Clearly we start with the top three teams beating the bottom three every time and gaining $18$ points each. We can then have the top three teams split the pair of games between themselves, which gets them each $24$ points total. Whatever tiebreaker is used, one will fail to advance, so it takes at least $25$ points to guarantee advancement. We can't have three teams get at least $25$ points each because they get in total $54$ from the other teams and each of the six games between them yields at most $3$ points, so their total cannot exceed $72$. So $25$ points, which comes from a record of $8-1-1,$ will guarantee advancement.

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