Školjka

%V0 Given a segment $AB$ of the length 1, define the set $M$ of points in the following way: it contains two points $A,B,$ and also all points obtained from $A,B$ by iterating the following rule: With every pair of points $X,Y$ the set $M$ contains also the point $Z$ of the segment $XY$ for which $YZ = 3XZ.$

%V0 Is it possible to find a set of $100$ (or $200$) points on the boundary of a cube such that this set remains fixed under all rotations which leave the cube fixed ?

%V0 Find all $x$ for which, for all $n,$ $$\sum^n_{k=1} \sin {k x} \leq \frac{\sqrt{3}}{2}.$$

%V0 In a group of interpreters each one speaks one of several foreign languages, 24 of them speak Japanese, 24 Malaysian, 24 Farsi. Prove that it is possible to select a subgroup in which exactly 12 interpreters speak Japanese, exactly 12 speak Malaysian and exactly 12 speak Farsi.

%V0 A linear binomial $l(z) = Az + B$ with complex coefficients $A$ and $B$ is given. It is known that the maximal value of $|l(z)|$ on the segment $-1 \leq x \leq 1$ $(y = 0)$ of the real line in the complex plane $z = x + iy$ is equal to $M.$ Prove that for every $z$ $$|l(z)| \leq M \rho,$$ where $\rho$ is the sum of distances from the point $P=z$ to the points $Q_1: z = 1$ and $Q_3: z = -1.$

%V0 Determine all positive roots of the equation $x^x = \frac{1}{\sqrt{2}}.$