Školjka

%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 $a_{1}^{8}+a_{2}^{8}+...+a_{100}^{8}.$

%V0 A figure with area $1$ is cut out of paper. We divide this figure into $10$ parts and color them in $10$ different colors. Now, we turn around the piece of paper, divide the same figure on the other side of the paper in $10$ parts again (in some different way). Show that we can color these new parts in the same $10$ colors again (hereby, different parts should have different colors) such that the sum of the areas of all parts of the figure colored with the same color on both sides is $\geq \frac{1}{10}.$

%V0 Consider $n$ students with numbers $1, 2, \ldots, n$ standing in the order $1, 2, \ldots, n.$ Upon a command, any of the students either remains on his place or switches his place with another student. (Actually, if student $A$ switches his place with student $B,$ then $B$ cannot switch his place with any other student $C$ any more until the next command comes.) Is it possible to arrange the students in the order $n,1, 2, \ldots, n-1$ after two commands ?

%V0 Solve the equation $\frac{1}{\sin x}+\frac{1}{\cos x}=\frac 1p$ where $p$ is a real parameter. Discuss for which values of $p$ the equation has at least one real solution and determine the number of solutions in $[0, 2\pi)$ for a given $p.$