Školjka

%V0 Let $n \geq 2, n \in \mathbb{N}$ and let $p, a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n \in \mathbb{R}$ satisfying $\frac{1}{2} \leq p \leq 1,$ $0 \leq a_i,$ $0 \leq b_i \leq p,$ $i = 1, \ldots, n,$ and $$\sum^n_{i=1} a_i = \sum^n_{i=1} b_i.$$ Prove the inequality: $$\sum^n_{i=1} b_i \prod^n_{j = 1, j \neq i} a_j \leq \frac{p}{(n-1)^{n-1}}.$$

%V0 Prove the following inequality: $$\prod^k_{i=1} x_i \cdot \sum^k_{i=1} x^{n-1}_i \leq \sum^k_{i=1} x^{n+k-1}_i,$$ where $x_i > 0,$ $k \in \mathbb{N}, n \in \mathbb{N}.$

%V0 Prove that for arbitrary positive numbers the following inequality holds $$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} \leq \frac{a^8 + b^8 + c^8}{a^3b^3c^3}.$$

%V0 Prove that for an arbitrary pair of vectors $f$ and $g$ in the space the inequality $$af^2 + bfg +cg^2 \geq 0$$ holds if and only if the following conditions are fulfilled: $$a \geq 0, \quad c \geq 0, \quad 4ac \geq b^2.$$

%V0 Prove the inequality a.) $\left( a_{1}+a_{2}+...+a_{k}\right) ^{2}\leq k\left(a_{1}^{2}+a_{2}^{2}+...+a_{k}^{2}\right) ,$ where $k\geq 1$ is a natural number and $a_{1},$ $a_{2},$ $...,$ $a_{k}$ are arbitrary real numbers. b.) Using the inequality (1), show that if the real numbers $a_{1},$ $a_{2},$ $...,$ $a_{n}$ satisfy the inequality $$a_{1}+a_{2}+...+a_{n}\geq \sqrt{\left( n-1\right) \left(a_{1}^{2}+a_{2}^{2}+...+a_{n}^{2}\right) },$$ then all of these numbers $a_{1},$ $a_{2},$ $\ldots,$ $a_{n}$ are non-negative.

%V0 Let $a_1, a_2, \ldots, a_n$ be positive real numbers. Prove the inequality $$\binom n2 \sum_{i<j} \frac{1}{a_ia_j} \geq 4 \left( \sum_{i<j} \frac{1}{a_i+a_j} \right)^2$$