At the tennis tournament in one circle attended the 8 girls (i.e. every tennis player has played with each other exactly once, draws in tennis does not happen). Oksana took the second place recruited points and there is no other participant with the same number of points. What is the maximum number of games could lose Olesya who won in this tournament?
Solution
Suppose that Olesya lost 2 games. Then she scored 5 points. Oksana could not score 4 points, because then all together all teams had supplied the maximum wins. But in total we have games, a contradiction. Similarly Oksana could not score 3 or less points, or Olesya lose 3 more games. Thus, only Olesya could lose up to 1 game. This option is possible, as evidenced by the table (Fig. 28) example.
| M | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | p |
|---|---|---|---|---|---|---|---|---|---|
| 1 | X | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 6 |
| 2 | 1 | X | 1 | 0 | 1 | 0 | 1 | 0 | 4 |
| 3 | 0 | 0 | X | 1 | 0 | 1 | 0 | 1 | 3 |
| 4 | 0 | 1 | 0 | X | 1 | 0 | 1 | 0 | 3 |
| 5 | 0 | 0 | 1 | 0 | X | 1 | 0 | 1 | 3 |
| 6 | 0 | 1 | 0 | 1 | 0 | X | 1 | 0 | 3 |
| 7 | 0 | 0 | 1 | 0 | 1 | 0 | X | 1 | 3 |
| 8 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | X | 3 |
Fig. 28
Looking for a route rather than an archive? The track puts 2,000
problems in a working order, from AMC 10 level to the IMO shortlist.