VWO 1996

Determine the gcd of all numbers of the form $p^8-1$ , with p a prime above 5.

Consider the points $1,\frac12,\frac13,...$ on the real axis. Find the smallest value $k \in \mathbb{N}_0$ for which all points above can be covered with 5 closed intervals of length $\frac1k$ .

Consider a real poylnomial $p(x)=a_nx^n+...+a_1x+a_0$ . (a) If $\deg(p(x))>2$ prove that $\deg(p(x)) = 2 + deg(p(x+1)+p(x-1)-2p(x))$ . (b) Let $p(x)$ a polynomial for which there are real constants $r,s$ so that for all real $x$ we have $$p(x+1)+p(x-1)-rp(x)-s=0$$Prove $\deg(p(x))\le 2$ . (c) Show, in (b) that $s=0$ implies $a_2=0$ .