Difference between revisions of "2006 Romanian NMO Problems/Grade 9/Problem 4"
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<math>\displaystyle 2n</math> students <math>\displaystyle (n \geq 5)</math> participated at table tennis contest, which took <math>\displaystyle 4</math> days. In every day, every student played a match. (It is possible that the same pair meets twice or more times, in different days) Prove that it is possible that the contest ends like this: | <math>\displaystyle 2n</math> students <math>\displaystyle (n \geq 5)</math> participated at table tennis contest, which took <math>\displaystyle 4</math> days. In every day, every student played a match. (It is possible that the same pair meets twice or more times, in different days) Prove that it is possible that the contest ends like this: | ||
| − | + | * there is only one winner; | |
| − | + | * there are <math>\displaystyle 3</math> students on the second place; | |
| − | + | * no student lost all <math>\displaystyle 4</math> matches. | |
How many students won only a single match and how many won exactly <math>\displaystyle 2</math> matches? (In the above conditions) | How many students won only a single match and how many won exactly <math>\displaystyle 2</math> matches? (In the above conditions) | ||
Revision as of 10:51, 27 July 2006
Problem
students 
participated at table tennis contest, which took 
days. In every day, every student played a match. (It is possible that the same pair meets twice 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;
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)
