VWO 1989

Show that every subset of {1,2,...,99,100} with 55 elements contains at least 2 numbers with a difference of 9.

When drawing all diagonals in a regular pentagon, one gets an smaller pentagon in the middle. What's the ratio of the areas of those pentagons?

Let $D$ be the set of positive reals different from $1$ and let $n$ be a positive integer. If for $f: D\rightarrow \mathbb{R}$ we have $x^n f(x)=f(x^2)$ , and if $f(x)=x^n$ for $0<x<\frac{1}{1989}$ and for $x>1989$ , then prove that $f(x)=x^n$ for all $x \in D$ .