Školjka

%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 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 Let $n \geq 2$ be a natural number. Find a way to assign natural numbers to the vertices of a regular $2n$-gon such that the following conditions are satisfied: (1) only digits $1$ and $2$ are used; (2) each number consists of exactly $n$ digits; (3) different numbers are assigned to different vertices; (4) the numbers assigned to two neighboring vertices differ at exactly one digit.

%V0 Determine whether there exist distinct real numbers $a, b, c, t$ for which: (i) the equation $ax^2 + btx + c = 0$ has two distinct real roots $x_1, x_2,$ (ii) the equation $bx^2 + ctx + a = 0$ has two distinct real roots $x_2, x_3,$ (iii) the equation $cx^2 + atx + b = 0$ has two distinct real roots $x_3, x_1.$

%V0 A broken line $A_1A_2 \ldots A_n$ is drawn in a $50 \times 50$ square, so that the distance from any point of the square to the broken line is less than $1$. Prove that its total length is greater than $1248.$

%V0 Natural numbers from $1$ to $99$ (not necessarily distinct) are written on $99$ cards. It is given that the sum of the numbers on any subset of cards (including the set of all cards) is not divisible by $100$. Show that all the cards contain the same number.