Školjka

%V0 Prove that $$\frac{1}{3}n^2 + \frac{1}{2}n + \frac{1}{6} \geq (n!)^{\dfrac{2}{n}},$$ and let $n \geq 1$ be an integer. Prove that this inequality is only possible in the case $n = 1.$

%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 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 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 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 $n \geq 2, n \in \mathbb{N}$ 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: $$\sum^n_{i=1} b_i \prod^n_{j = 1, j \neq i} a_j \leq \frac{p}{(n-1)^{n-1}}.$$