Školjka

%V0 Let $R$ be a set of exactly $6$ elements. A set $F$ of subsets of $R$ is called an $S$-family over $R$ if and only if it satisfies the following three conditions: (i) For no two sets $X, Y$ in $F$ is $X \subseteq Y$ ; (ii) For any three sets $X, Y,Z$ in $F$, $X \cup Y \cup Z \neq R,$ (iii) $\bigcup_{X \in F} X = R$

%V0 Show that $\frac{20}{60} <\sin 20^{\circ} < \frac{21}{60}.$

%V0 Find all bases of logarithms in which a real positive number can be equal to its logarithm or prove that none exist.

%V0 Let $K$ denote the set $\{a, b, c, d, e\}$. $F$ is a collection of $16$ different subsets of $K$, and it is known that any three members of $F$ have at least one element in common. Show that all $16$ members of $F$ have exactly one element in common.

%V0 Given the integer $n > 1$ and the real number $a > 0$ determine the maximum of $\sum_{i=1}^{n-1} x_i x_{i+1}$ taken over all nonnegative numbers $x_i$ with sum $a.$

%V0 Let $N$ be the number of integral solutions of the equation $$x^2 - y^2 = z^3 - t^3$$ satisfying the condition $0 \leq x, y, z, t \leq 10^6$, and let $M$ be the number of integral solutions of the equation $$x^2 - y^2 = z^3 - t^3 + 1$$ satisfying the condition $0 \leq x, y, z, t \leq 10^6$. Prove that $N >M.$