Školjka

%V0 Let $n$ be an integer, $n \geq 3.$ Let $a_1, a_2, \ldots, a_n$ be real numbers such that $2 \leq a_i \leq 3$ for $i = 1, 2, \ldots, n.$ If $s = a_1 + a_2 + \ldots + a_n,$ prove that $$\frac{a^{2}_{1}+a^{2}_{2}-a^{2}_{3}}{a_{1}+a_{2}-a_{3}}+\frac{a^{2}_{2}+a^{2}_{3}-a^{2}_{4}}{a_{2}+a_{3}-a_{4}}+\ldots+\frac{a^{2}_{n}+a^{2}_{1}-a^{2}_{2}}{a_{n}+a_{1}-a_{2}}\leq 2s-2n.$$

%V0 Find all $x,y$ and $z$ in positive integer: $z + y^{2} + x^{3} = xyz$ and $x = \gcd(y,z)$.

%V0 Let $a_1, a_2, \ldots, a_{100}$ be nonnegative real numbers such that $a^2_1 + a^2_2 + \ldots + a^2_{100} = 1.$ Prove that $$a^2_1 \cdot a_2 + a^2_2 \cdot a_3 + \ldots + a^2_{100} \cdot a_1 < \frac {12}{25}.$$ Author: Marcin Kuzma, Poland

%V0 Let $S\subseteq\mathbb{R}$ be a set of real numbers. We say that a pair $(f, g)$ of functions from $S$ into $S$ is a Spanish Couple on $S$, if they satisfy the following conditions: (i) Both functions are strictly increasing, i.e. $f(x) < f(y)$ and $g(x) < g(y)$ for all $x$, $y\in S$ with $x < y$; (ii) The inequality $f\left(g\left(g\left(x\right)\right)\right) < g\left(f\left(x\right)\right)$ holds for all $x\in S$. Decide whether there exists a Spanish Couple on the set $S = \mathbb{N}$ of positive integers; on the set $S = \{a - \frac {1}{b}: a, b\in\mathbb{N}\}$ Proposed by Hans Zantema, Netherlands

%V0 Let $a$, $b$, $c$, $d$ be positive real numbers such that $abcd = 1$ and $a + b + c + d > \dfrac{a}{b} + \dfrac{b}{c} + \dfrac{c}{d} + \dfrac{d}{a}$. Prove that $$a + b + c + d < \dfrac{b}{a} + \dfrac{c}{b} + \dfrac{d}{c} + \dfrac{a}{d}$$ Proposed by Pavel Novotný, Slovakia

%V0 Prove that for any four positive real numbers $a$, $b$, $c$, $d$ the inequality $$\frac {(a - b)(a - c)}{a + b + c} + \frac {(b - c)(b - d)}{b + c + d} + \frac {(c - d)(c - a)}{c + d + a} + \frac {(d - a)(d - b)}{d + a + b} \geqslant 0$$ holds. Determine all cases of equality. Author: Darij Grinberg (Problem Proposal), Christian Reiher (Solution), Germany