Školjka

%V0 Let $a_{1},a_{2},\ldots ,a_{n}$ be positive real numbers such that $a_{1}+a_{2}+\cdots +a_{n}<1$. Prove that $$\frac{a_{1} a_{2} \cdots a_{n} \left[ 1 - (a_{1} + a_{2} + \cdots + a_{n}) \right] }{(a_{1} + a_{2} + \cdots + a_{n})( 1 - a_1)(1 - a_2) \cdots (1 - a_n)} \leqslant \frac{1}{n^{n+1}}$$

%V0 Let $a_0, a_1, a_2, \ldots$ be an arbitrary infinite sequence of positive numbers. Show that the inequality $1 + a_n > a_{n-1} \sqrt[n]{2}$ holds for infinitely many positive integers $n$.

%V0 The sequence $c_{0}, c_{1}, . . . , c_{n}, . . .$ is defined by $c_{0}= 1, c_{1}= 0$, and $c_{n+2}= c_{n+1}+c_{n}$ for $n \geq 0$. Consider the set $S$ of ordered pairs $(x, y)$ for which there is a finite set $J$ of positive integers such that $x=\sum_{j \in J}{c_{j}}$, $y=\sum_{j \in J}{c_{j-1}}$. Prove that there exist real numbers $\alpha$, $\beta$, and $M$ with the following property: An ordered pair of nonnegative integers $(x, y)$ satisfies the inequality $m < \alpha x+\beta y < M$ if and only if $(x, y) \in S$. Remark: A sum over the elements of the empty set is assumed to be $0$.

%V0 Prove the inequality: $\displaystyle \sum_{i < j}{\frac {a_{i}a_{j}}{a_{i} + a_{j}}}\leq \frac {n}{2(a_{1} + a_{2} + ... + a_{n})}\cdot \sum_{i < j}{a_{i}a_{j}}$ for positive reals $a_{1}$, $a_{2}$, ..., $a_{n}$.

%V0 Let $S\subseteq\mathbb{R}$ be a set of real numbers. We say that a pair $(f, g)$ of functions from $S$ into $S$ is a Spanish Couple on $S$, if they satisfy the following conditions: (i) Both functions are strictly increasing, i.e. $f(x) < f(y)$ and $g(x) < g(y)$ for all $x$, $y\in S$ with $x < y$; (ii) The inequality $f\left(g\left(g\left(x\right)\right)\right) < g\left(f\left(x\right)\right)$ holds for all $x\in S$. Decide whether there exists a Spanish Couple on the set $S = \mathbb{N}$ of positive integers; on the set $S = \{a - \frac {1}{b}: a, b\in\mathbb{N}\}$ Proposed by Hans Zantema, Netherlands

%V0 Neka je $\{ F_i \}$, $i=0,1, \ldots $ niz brojeva definiran na sljedeći način: $$ F_0=0, \ F_1=1,\ F_{i+2}=F_{i+1}+F_{i}, \ i=0,1, \ldots$$ Za prirodan broj $n \geq 2$ neka su $a_0, a_1, \ldots a_n$ nenegativni brojevi koji zadovoljavaju uvjet $$ a_0=1, \ a_i \leq a_{i+1} + a_{i+2}, \ i=0,1, \ldots, n-2. $$ Dokažite da je $a_0+a_1+\ldots+a_n \geq \frac{F_{n+2}-1}{F_{n}}$. Da li se postiže jednakost?