VWO 1990

On the standard unit circle, draw 4 unit circles with centers [0,1],[1,0],[0,-1],[-1,0]. You get a figure as below, find the area of the colored part.

Let $a$ and $b$ be two primes having at least two digits, such that $a > b$ .

Show that $$240|\left(a^4-b^4\right)$$ and show that 240 is the greatest positive integer having this property.

We form a decimal code of $21$ digits. the code may start with $0$ . Determine the probability that the fragment $0123456789$ appears in the code.

Let $f:\mathbb{R}^+_0 \rightarrow \mathbb{R}^+_0$ be a strictly decreasing function.

(a) Be $a_n$ a sequence of strictly positive reals so that $\forall k \in \mathbb{N}_0:k\cdot f(a_k)\geq (k+1)\cdot f(a_{k+1})$ Prove that $a_n$ is ascending, that $\lim_{k\rightarrow +\infty} f(a_k)$ = 0and that $\lim_{k\rightarrow +\infty} a_k =+\infty$

(b) Prove that there exist such a sequence ( $a_n$ ) in $\mathbb{R}^+_0$ if you know $\lim_{x\rightarrow +\infty} f(x)=0$ .