Školjka

%V0 Find all natural numbers $n$ for which $2^8 +2^{11} +2^n$ is a perfect square.

%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 $N >M.$

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

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