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2006 Romanian NMO Problems/Grade 9/Problem 4

Revision as of 10:49, 10 October 2007 by 1=2 (talk | contribs) (→‎Solution)

Problem

$\displaystyle 2n$ students $\displaystyle (n \geq 5)$ participated at table tennis contest, which took $\displaystyle 4$ 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 $\displaystyle 3$ students on the second place;
  • no student lost all $\displaystyle 4$ matches.

How many students won only a single match and how many won exactly $\displaystyle 2$ matches? (In the above conditions)

Solution

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See also

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