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 Suppose that $n \geq 2$ and $x_1, x_2, \ldots, x_n$ are real numbers between 0 and 1 (inclusive). Prove that for some index $i$ between $1$ and $n - 1$ the inequality $$x_i (1 - x_{i+1}) \geq \frac{1}{4} x_1 (1 - x_{n})$$

%V0 Let $w, x, y, z$ are non-negative reals such that $wx + xy + yz + zw = 1$. Show that $$\frac {w^3}{x + y + z} + \frac {x^3}{w + y + z} + \frac {y^3}{w + x + z} + \frac {z^3}{w + x + y}\geq \frac {1}{3}$$.

%V0 Prove the trigonometric inequality $\displaystyle \cos x < 1 - \frac{x^2}{2} + \frac{x^4}{16},$ when $\displaystyle x \in \left(0, \frac{\pi}{2} \right)\text{.}$

%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 $$\binom n2 \sum_{i<j} \frac{1}{a_ia_j} \geq 4 \left( \sum_{i<j} \frac{1}{a_i+a_j} \right)^2$$