Školjka

%V0 Let $n$ be an integer,$n \geq 3.$ Let $x_1, x_2, \ldots, x_n$ be real numbers such that $x_i < x_{i+1}$ for $1 \leq i \leq n - 1$. Prove that $$\frac{n(n-1)}{2}\sum_{i < j}x_{i}x_{j}>\left(\sum^{n-1}_{i=1}(n-i)\cdot x_{i}\right)\cdot\left(\sum^{n}_{j=2}(j-1)\cdot x_{j}\right)$$

%V0 Prove the inequality: $\displaystyle \sum_{i < j}{\frac {a_{i}a_{j}}{a_{i} + a_{j}}}\leq \frac {n}{2(a_{1} + a_{2} + ... + a_{n})}\cdot \sum_{i < j}{a_{i}a_{j}}$ for positive reals $a_{1}$, $a_{2}$, ..., $a_{n}$.

%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 $a$, $b$, $c$, $d$ 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

%V0 Let $a$, $b$, $c$ be positive real numbers such that $ab+bc+ca \leqslant 3abc$. Prove that $$\sqrt{\frac{a^2+b^2}{a+b}}+\sqrt{\frac{b^2+c^2}{b+c}}+\sqrt{\frac{c^2+a^2}{c+a}}+3 \leqslant \sqrt{2}\left(\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a}\right) \text{.}$$ Proposed by Dzianis Pirshtuk, Belarus