Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

2006 Romanian NMO Problems/Grade 9/Problem 4

Revision as of 11:49, 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

This problem needs a solution. If you have a solution for it, please help us out by adding it.

See also

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2006_Romanian_NMO_Problems/Grade_9/Problem_4&oldid=17790"