CL-MO 2014

Let $a, b,c$ real numbers that are greater than $0$ and less than $1$ . Show that there is at least one of these three values $ab(1-c)^2$ , $bc(1-a)^2$ , $ca(1- b)^2$ which is less than or equal to $\frac{1}{16}$ .

In the plane there are $2014$ plotted points, such that no $3$ are collinear. For each pair of plotted points, draw the line that passes through them. prove that for every three of marked points there are always two that are separated by an amount odd number of lines.

Prove that for every integer $n$ the expression $n^3-9n + 27$ is not divisible by $81$ .