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This is a problem from Tournament of Town competition taking place today. Please don't amswer.

There are 5 non-zero numbers. One has calculated the sum of each two numbers. Among the sums five are negative and five are positive. How many products are positive?

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  • $\begingroup$ Is it possible that all five numbers are of the same sign? $\endgroup$ – bof Oct 8 '17 at 8:38
  • $\begingroup$ It is impossible that they have the same sign. We won't get a positive/negative sum in that case. $\endgroup$ – Student12 Oct 8 '17 at 8:40
  • $\begingroup$ Again, it is impossible because we won't get five negative products. $\endgroup$ – Student12 Oct 8 '17 at 8:42
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We must have $2$ numbers of one sign and $3$ of the other, since otherwise there would be at least ${4\choose2}=6$ pairwise sums of the same sign. The number of negative pairwise products is therefore ${2\choose1}{3\choose1}=6$, which leaves $4$ positive pairwise products.

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