Školjka

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

%V0 Let $a,b$ be two natural numbers. When we divide $a^2+b^2$ by $a+b$, we the the remainder $r$ and the quotient $q.$ Determine all pairs $(a, b)$ for which $q^2 + r = 1977.$

%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 Let $n\ge2$ be an integer. Prove that if $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le\sqrt{n\over3}$, then $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le n-2$.(IMO Problem 6) Original Formulation Let $f(x) = x^2 + x + p$, $p \in \mathbb N.$ Prove that if the numbers $f(0), f(1), \cdots , f(\sqrt{p\over 3} )$ are primes, then all the numbers $f(0), f(1), \cdots , f(p - 2)$ are primes. Proposed by Soviet Union.

%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: $a^{b^2} = b^a$.