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?