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 Let $a,b,c,d$ be real numbers which satisfy $\frac{1}{2}\leq a,b,c,d\leq 2$ and $abcd=1$. Find the maximum value of $$\left(a+\frac{1}{b}\right)\left(b+\frac{1}{c}\right)\left(c+\frac{1}{d}\right)\left(d+\frac{1}{a}\right)\text{.}$$

%V0 Let $x$, $y$, $z$ be real numbers satisfying $x^2+y^2+z^2+9=4(x+y+z)$. Prove that $$x^4+y^4+z^4+16(x^2+y^2+z^2) \ge 8(x^3+y^3+z^3)+27$$ and determine when equality holds.

%V0 For each integer $n\geqslant2$, determine the largest real constant $C_n$ such that for all positive real numbers $a_1, \ldots, a_n$ we have $$\frac{a_1^2+\ldots+a_n^2}{n}\geqslant\left(\frac{a_1+\ldots+a_n}{n}\right)^2+C_n\cdot(a_1-a_n)^2\mbox{.}$$

%V0 Let $a, b, c$ be positive real numbers such that $$\frac{a}{1+a}+\frac{b}{1+b}+\frac{c}{1+c}=2\text{.}$$ Prove that $$\frac{\sqrt a + \sqrt b+\sqrt c}{2} \geq \frac{1}{\sqrt a}+\frac{1}{\sqrt b}+\frac{1}{\sqrt c}\text{.}$$

%V0 Neka su $a$, $b$ i $c$ pozitivni realni brojevi takvi da je $ab + bc + ca = 1$. Pokažite da vrijedi $$\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a} \geqslant \sqrt 3 + \frac{ab}{a+b} + \frac{bc}{b+c} + \frac{ca}{c+a}$$