2006 Romanian NMO Problems/Grade 9/Problem 4
students participated at table tennis contest, which took days. Every day, every student played a match. (It is possible that the same pair meets two or more times, in different days). Prove that it is possible that the contest ends like this:
- there is only one winner;
- there are students on the second place;
- no student lost all matches.
How many students won only a single match and how many won exactly matches? (In the above conditions)
Note that the 3 second place students obviously could not have only won one match, or won all 4 matches. I now claim that they could not have won exactly two matches, either.
Each day there were matches, so at the end of the contest there were total points. Now if the three people in second place won exactly two matches, then people would have to had won exactly one match. The winner of the contest would have won at most 4 matches, so we have the inequality
Solving for yields , which is clearly false. This is a contradiction in logic, so the three people in second place could not have won exactly two matches.
This shows that the three second-place finishers each won exactly three matches. Therefore the winner of the contest won all 4 matches. Now let be the number of people who won two matches. It follows that people won one match. We now have the equation
Solving for yields , so students won exactly two matches. It then follows that people won a single match.