1978 AHSME Problems/Problem 30
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Problem 30
In a tennis tournament, women and men play, and each player plays exactly one match with every other player. If there are no ties and the ratio of the number of matches won by women to the number of matches won by men is , then equals
Solution 1
Since there are women, the number of matches between only women is and similarly, there are matches between only men. Since every woman plays every man exactly once, there are matches which are between a man and a woman. Call these matches co-ed matches, and let be the number of co-ed matches won by women.
Then it follows that which can be simplified to
The number of matches won by women must be less than the total number of matches, so we obtain the inequality Rearranging and factoring gives and the only integers which satisfy this inequality are and
Clearly, there could not have been people in the tournament, so If then there would have been only one woman and two men in the tournament, in which case the woman could not have won the majority of the matches.
We can now plug back into the equation and solving for gives Since must be an integer, cannot be It follows that so the answer is (E). (When solving gives )
Solution 2
Since there are women and men, there are a total of players. Hence, the number of total matches must be . We also know that the ratio of the number of matches won by women to the number of matches won by men is and that there were no draws, so the total number of matches must be for some value . This givesSoIt follows that (various ways of splitting into two factors). However, for solutions , isn't an integer, therefore leaving solutions and .
is invalid. This can be shown as when ,The ratio of matches won by gender is , so the number of matches won by women must be at least . The maximum number of matches won by women is equivalent to the number of matches between men (generates 1 match won by men no matter the outcome) subtracted from the total number of matches, which is . As the maximum number of matches won is smaller than , the solution is extraneous/invalid.
Hence, the answer must be , or
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See also
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