1989 USAMO Problems/Problem 2

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Problem

The 20 members of a local tennis club have scheduled exactly 14 two-person games among themselves, with each member playing in at least one game. Prove that within this schedule there must be a set of 6 games with 12 distinct players

Solution

If there are 14 games with two people each, there must be 28 indistinct players. Since there are just 20 members, at most 8 players could have played in more than one game. That leaves at least $14 - 8 = 6$ games with distinct players.

See also

1989 USAMO (ProblemsResources)
Preceded by
Problem 1
Followed by
Problem 3
1 2 3 4 5
All USAMO Problems and Solutions
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