Školjka

%V0 Prove that $\cos{\frac{\pi}{7}}-\cos{\frac{2\pi}{7}}+\cos{\frac{3\pi}{7}}=\frac{1}{2}$

%V0 Let $n$ be a given number greater than 2. We consider the set $V_n$ of all the integers of the form $1 + kn$ with $k = 1, 2, \ldots$ A number $m$ from $V_n$ is called indecomposable in $V_n$ if there are not two numbers $p$ and $q$ from $V_n$ so that $m = pq.$ Prove that there exist a number $r \in V_n$ that can be expressed as the product of elements indecomposable in $V_n$ in more than one way. (Expressions which differ only in order of the elements of $V_n$ will be considered the same.)

%V0 Is it possible to choose $1983$ distinct positive integers, all less than or equal to $10^5$, no three of which are consecutive terms of an arithmetic progression?

%V0 Prove that for each positive integer $n$ there exist $n$ consecutive positive integers none of which is an integral power of a prime number.

%V0 Given an initial integer $n_0 > 1$, two players, ${\mathcal A}$ and ${\mathcal B}$, choose integers $n_1$, $n_2$, $n_3$, $\ldots$ alternately according to the following rules : I.) Knowing $n_{2k}$, ${\mathcal A}$ chooses any integer $n_{2k + 1}$ such that $$n_{2k} \leq n_{2k + 1} \leq n_{2k}^2.$$ II.) Knowing $n_{2k + 1}$, ${\mathcal B}$ chooses any integer $n_{2k + 2}$ such that $$\frac {n_{2k + 1}}{n_{2k + 2}}$$ is a prime raised to a positive integer power. Player ${\mathcal A}$ wins the game by choosing the number 1990; player ${\mathcal B}$ wins by choosing the number 1. For which $n_0$ does : a.) ${\mathcal A}$ have a winning strategy? b.) ${\mathcal B}$ have a winning strategy? c.) Neither player have a winning strategy?

%V0 Find all pairs $(a,b)$ of positive integers that satisfy the equation: $a^{b^2} = b^a$.