2000 JBMO Problems/Problem 4
At a tennis tournament there were boys and girls participating. Every player played every other player. The boys won times as many matches as the girls. It is knowns that there were no draws. Find .
The total number of games played in the tournament is Since the boys won as many matches as the girls, the boys won of all the games played, so the total number of games that a boy won is
Since the number of games that a boy won is a whole number, must be a multiple of 8. Testing each residue, we find that
For to be valid, the number of games the boys won must be less than or equal to the number of games where a boy has played. The number of games with only girls is so the number of games where there is at least one boy playing is This means we can write and solve the following inequality. Since we do not have to change the inequality sign when we divided by Thus, we can confirm that all positive integers congruent to 0 or 3 modulo 8 satisfy the conditions. In summary,
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