Školjka

%V0 Let $n$ be a positive integer, and let $x$ and $y$ be a positive real number such that $x^n + y^n = 1.$ Prove that $$ \left(\sum^n_{k = 1} \frac {1 + x^{2k}}{1 + x^{4k}} \right) \cdot \left( \sum^n_{k = 1} \frac {1 + y^{2k}}{1 + y^{4k}} \right) < \frac{1}{(1 - x)(1 - y)} \text{.}$$ Author: unknown author, Estonia

%V0 A nonempty set $A$ of real numbers is called a $B_3$-set if the conditions $a_1, a_2, a_3, a_4, a_5, a_6 \in A$ and $a_1 + a_2 + a_3 = a_4 + a_5 + a_6$ imply that the sequences $(a_1, a_2, a_3)$ and $(a_4, a_5, a_6)$ are identical up to a permutation. Let $$A = \{a(0) = 0 < a(1) < a(2) < \ldots \}, B = \{b(0) = 0 < b(1) < b(2) < \ldots \}$$ be infinite sequences of real numbers with $D(A) = D(B),$ where, for a set $X$ of real numbers, $D(X)$ denotes the difference set $\{|x-y| | x, y \in X \}.$ Prove that if $A$ is a $B_3$-set, then $A = B.$

%V0 A game is played by $n$ girls ($n \geq 2$), everybody having a ball. Each of the $\binom{n}{2}$ pairs of players, is an arbitrary order, exchange the balls they have at the moment. The game is called nice nice if at the end nobody has her own ball and it is called tiresome if at the end everybody has her initial ball. Determine the values of $n$ for which there exists a nice game and those for which there exists a tiresome game.

%V0 Let $x,y$ and $z$ be positive real numbers such that $xyz=1$. Prove that $$\frac{x^{3}}{(1 + y)(1 + z)}+\frac{y^{3}}{(1 + z)(1 + x)}+\frac{z^{3}}{(1 + x)(1 + y)} \geq \frac{3}{4}.$$

%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 $$\lim_{i\rightarrow\infty} \frac{n_{i+1}}{n_i} = 3.$$