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?

  • $\begingroup$ 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

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.