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Nikki, Mikey, Akeem, Katie and Davey have all entered a mathematics competition. In each round of the competition, the entrants were put into pairs as far as possible (if there were an odd number of participants, all but one would be paired) and answered a set of questions at the same time as their opponent.

The winner was the entrant who gave correct answers to most of the questions. The winner got two points, one point was awarded for a draw and zero points for a loss. Every entrant played another entrant exactly once.

As we know, Nikki, Mikey, Akeem, Katie and Davey are all very good mathematicians,and they were delighted to find that they had all won the competition with the same score. In fact they had all dropped only 4 points. All of the other entrants only scored 4 points.

When the judges examined the results, there had been no draws.

How many points did Nikki, Mikey, Akeem, Katie and Davey all score?

When they met their friends Sophie, Stevie, Sabina, Marge and Chrissie, they discovered that they had been in a different group of entrants for the same competition. The friends admitted that they had dropped four times as many points even though they all scored the same number, but they noted that in their round the other entrants also scored exactly the same number of points as they had each dropped.

Davey thought that there were the same number of entrants in each round, so he said that he was sorry that his friends were not through to the next round, but, Sophie said that they too got through.

How many entrants were in their round?

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The total number of points available is the number of competitions (C ) times 2. C is equal to the number of combinations of 2 opponents [C = combin(N,2) where N is number of contestants] We know that N-5 contestants scored 4 points each and the remaining 5 contestants scored (N-1)*2 - 4 points each. Putting it all together,

combin(N,2) * 2 = (N - 5) * 4 + 5 * [(N - 1) * 2 - 4]

solving for N, we get two possible answers: N = 5 or N = 10. The OP appears to assume the presence of more than 5 contestants, so I say there were 10.

Each contestant participated in 9 competitions. Our five friends won 7 of their 9, claiming a total of 7*2*5 = 70 points. The remaining 5 each won two contests, scoring 5*4 = 20 points among them. 70+20 = 90, which is equal to combin(10,2) * 2

Edited by HoustonHokie
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The second question is very similar to the first, except you substitute 16 for 4 in the equations so it looks like this:

combin(N,2) * 2 = (N - 5) * 16 + 5 * [(N - 1) * 2 - 16]

there are two possible solutions for N here as well: N = 10 or N = 17

If N = 10, then the 5 friends each lost 8 of their 9 competitions and ended up with just 2 points apiece. That doesn't work, because the OP says the 5 friends advance. Instead, they lose 8 of their 16 competitions and wind up with 16 points apiece. Interestingly, all their opponents do the same, and all 17 tie with 16 points apiece. Everyone from that round advances!

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