CAPS 2013

Prove that for every real number $x>0$ and each integer $n>0$ we have $$x^n+\frac1{x^n}-2 \ge n^2\left(x+\frac1x-2\right)$$

Triangular grid divides an equilateral triangle with sides of length $n$ into $n^2$ triangular cells as shown in figure for $n=12$ . Some cells are infected. A cell that is not yet infected, ia infected when it shares adjacent sides with at least two already infected cells. Specify for $n=12$ , the least number of infected cells at the start in which it is possible that over time they will infected all the cells of the original triangle.