Školjka

%V0 Let $x_0 = 5$ and $x_{n+1} = x_n + \frac{1}{x_n} \ (n = 0, 1, 2, \ldots )$. Prove that $$45 < x_{1000} < 45. 1.$$

%V0 Prove that from $x + y = 1 \ (x, y \in \mathbb R)$ it follows that $$x^{m+1} \sum_{j=0}^n \binom{m+j}{j} y^j + y^{n+1} \sum_{i=0}^m \binom{n+i}{i} x^i = 1 \qquad (m, n = 0, 1, 2, \ldots ).$$

%V0 There are six ports on a lake. Is it possible to organize a series of routes satisfying the following conditions ? (i) Every route includes exactly three ports; (ii) No two routes contain the same three ports; (iii) The series offers exactly two routes to each tourist who desires to visit two different arbitrary ports.

%V0 The $n$ points $P_1,P_2, \ldots, P_n$ are placed inside or on the boundary of a disk of radius 1 in such a way that the minimum distance $D_n$ between any two of these points has its largest possible value $D_n.$ Calculate $D_n$ for $n = 2$ to 7. and justify your answer.

%V0 Is it possible to choose a set of $100$ (or $200$) points on the boundary of a cube such that this set is fixed under each isometry of the cube into itself? Justify your answer.

%V0 We take $100$ consecutive natural numbers $a_{1},$ $a_{2},$ $...,$ $a_{100}.$ Determine the last two digits of the number $a_{1}^{8}+a_{2}^{8}+...+a_{100}^{8}.$