• Feb 22nd 2006, 06:56 PM
mong sody
In a mathematical competition 6 problems were posed to the contestants.Each pair of problems was solved by more than 2/5 of the constants.Nobody solved all 6 problem.Show that there were at least 2 contestants who each solved exactly 5 problems.] Thanks for your helping!!! :)
• Feb 23rd 2006, 12:36 PM
ThePerfectHacker
Quote:

Originally Posted by mong sody
Let $n$ be the number of contestants. Let $x_k$ be the number of contestants which solved exactly $k$ questions. Thus,
$x_0+x_1+x_2+x_3+x_4+x_5+x_6=n$
But nobody solved all six thus, $x_6=0$. Thus we have that,
$x_0+x_1+x_2+x_3+x_4+x_5=n$.
Now, I am trying to determine which one of x's has to be greater then $\frac{2}{5}n$, but I do not understand what you mean?