Školjka

%V0 The variables $a,b,c,d,$ traverse, independently from each other, the set of positive real values. What are the values which the expression $$S= \frac{a}{a+b+d} + \frac{b}{a+b+c} + \frac{c}{b+c+d} + \frac{d}{a+c+d}$$ takes?

%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 Determine all real numbers a for which there exists positive reals $x_{1}, \ldots, x_{5}$ which satisfy the relations $\displaystyle \sum_{k=1}^{5} kx_{k}=a,$ $\displaystyle \sum_{k=1}^{5} k^{3}x_{k}=a^{2},$ $\displaystyle \sum_{k=1}^{5} k^{5}x_{k}=a^{3}.$

%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?