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by shashank.ism » Tue Feb 09, 2010 1:19 pm
N girls and 2N boys played a chess tournament. Every player played every other player exactly once. The boys won 7/5 times as many matches as the girls (and there were no draws). Then which among the following is definitely false? (Assume 1 point for a win and 0 for a loss)

Boys pocketed prime number of points against girls
Girls always won twice or more matches than boys won against them
The sum of the scores of top 3 individual players was not between 25 and 33
The sum of the scores of top 3 individual players was 69
none of these
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by komal » Tue Feb 16, 2010 10:09 am
shashank.ism wrote:N girls and 2N boys played a chess tournament. Every player played every other player exactly once. The boys won 7/5 times as many matches as the girls (and there were no draws). Then which among the following is definitely false? (Assume 1 point for a win and 0 for a loss)

Boys pocketed prime number of points against girls
Girls always won twice or more matches than boys won against them
The sum of the scores of top 3 individual players was not between 25 and 33
The sum of the scores of top 3 individual players was 69
none of these
Total number of matches among boys were 2nC2, among girls were nC2 and between boys and
girls were n*2n. Please note that 2nC2 + nC2 + 2n^2 = 3nC2. Assume 1 point for a win and 0 for
a loss.
=> Girls pocketed nC2 points amongst themselves and boys pocketed 2nC2 points among
themselves. Let boys take k points from their matches against girls => girls take 2n^2 - k from
their matches gainst boys.
=> 2nC2 + k = 7/5*(nC2 + 2n^2 - k), solving we get 8k = n(5n+1). for n = 3, k = 6. For n = 8,
k = 41, For n = 11, k = 77.
(a) can be true as for n = 8, k = 41. (b) can be true as can be seen for for n = 3, 8, 11, ... (c) is
true as for n = 3, top 3 can score 8+7+6 = 21 points and for n = 11, when 33 matches are
played top 3 will always score more than 16+15+14 = 45. (d) is true, For n = 11, we can have
the top 3 score as 23+23+23 = 69.

Hence, choice (e) is the right answer
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