Školjka

%V0 Let $\mathbb{R}$ denote the set of all real numbers and $\mathbb{R}^+$ the subset of all positive ones. Let $\alpha$ and $\beta$ be given elements in $\mathbb{R},$ not necessarily distinct. Find all functions $f: \mathbb{R}^+ \mapsto \mathbb{R}$ such that $$f(x)f(y) = y^{\alpha} f \left( \frac{x}{2} \right) + x^{\beta} f \left( \frac{y}{2} \right) \forall x,y \in \mathbb{R}^+.$$

%V0 Let $S$ be the set of all pairs $(m,n)$ of relatively prime positive integers $m,n$ with $n$ even and $m < n.$ For $s = (m,n) \in S$ write $n = 2^k \cdot n_o$ where $k, n_0$ are positive integers with $n_0$ odd and define $$f(s) = (n_0, m + n - n_0).$$ Prove that $f$ is a function from $S$ to $S$ and that for each $s = (m,n) \in S,$ there exists a positive integer $t \leq \frac{m+n+1}{4}$ such that $$f^t(s) = s,$$ where $$f^t(s) = \underbrace{ (f \circ f \circ \cdots \circ f) }_{t \text{ times}}(s).$$ If $m+n$ is a prime number which does not divide $2^k - 1$ for $k = 1,2, \ldots, m+n-2,$ prove that the smallest value $t$ which satisfies the above conditions is $\left [\frac{m+n+1}{4} \right ]$ where $\left[ x \right]$ denotes the greatest integer $\leq x.$

%V0 Solve the following system of equations, in which $a$ is a given number satisfying $|a| > 1$: $\begin{matrix}x_{1}^{2}= ax_{2}+1\\ x_{2}^{2}= ax_{3}+1\\ \ldots\\ x_{999}^{2}= ax_{1000}+1\\ x_{1000}^{2}= ax_{1}+1\\ \end{matrix}$

%V0 Prove that $$\frac{a}{b+2c+3d} +\frac{b}{c+2d+3a} +\frac{c}{d+2a+3b}+ \frac{d}{a+2b+3c} \geq \frac{2}{3}$$ for all positive real numbers $a,b,c,d$.

%V0 Show that there exists a finite set $A \subset \mathbb{R}^2$ such that for every $X \in A$ there are points $Y_1, Y_2, \ldots, Y_{1993}$ in $A$ such that the distance between $X$ and $Y_i$ is equal to 1, for every $i.$

%V0 Define a sequence $<f(n)>^{\infty}_{n=1}$ of positive integers by $$f(1) = 1$$ and $f(n) =\left\{\begin{array}{ll}f(n-1)-n &\text{ if }f(n-1) > n;\\ f(n-1)+n &\text{ if }f(n-1)\leq n,\end{array}\right$ for $n \geq 2.$ Let $S = \{n \in \mathbb{N} | f(n) = 1993\}.$ (i) Prove that $S$ is an infinite set. (ii) Find the least positive integer in $S.$ (iii) If all the elements of $S$ are written in ascending order as $$n_1 < n_2 < n_3 < \ldots ,$$ show that $$\lim_{i\rightarrow\infty} \frac{n_{i+1}}{n_i} = 3.$$