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 Let ${a_1,a_2,\dots,a_n}$ be positive real numbers, ${n>1}$. Denote by $g_n$ their geometric mean, and by $A_1,\,A_2,\,\dots,\,A_n$ the sequence of arithmetic means defined by $$A_k=\frac{a_1+a_2+\cdots+a_k}{k},\qquad k=1,2,\dots,n.$$ Let $G_n$ be the geometric mean of $A_1,A_2,\dots,A_n$. Prove the inequality $$n \root n\of{\frac{G_n}{A_n}}+ \frac{g_n}{G_n}\le n+1 $$ and establish the cases of equality.

%V0 Determine the least real number $M$ such that the inequality $\left|ab\left(a^{2}-b^{2}\right)+bc\left(b^{2}-c^{2}\right)+ca\left(c^{2}-a^{2}\right)\right| \leq M\left(a^{2}+b^{2}+c^{2}\right)^2$ holds for all real numbers $a$, $b$ and $c$.

%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