Školjka

%V0 Suppose that $a, b, c > 0$ such that $abc = 1$. Prove that $$\frac{ab}{ab + a^5 + b^5} + \frac{bc}{bc + b^5 + c^5} + \frac{ca}{ca + c^5 + a^5} \leq 1.$$

%V0 Let $a_{1}, a_{2}...a_{n}$ be non-negative reals, not all zero. Show that that (a) The polynomial $p(x) = x^{n} - a_{1}x^{n - 1} + ... - a_{n - 1}x - a_{n}$ has preceisely 1 positive real root $R$. (b) let $A = \sum_{i = 1}^n a_{i}$ and $B = \sum_{i = 1}^n ia_{i}$. Show that $A^{A} \leq R^{B}$.

%V0 Let $a_1,a_2,\ldots$ be an infinite sequence of real numbers, for which there exists a real number $c$ with $0\leq a_i\leq c$ for all $i$, such that $$\left|\,a_i-a_j\,\right|\geq{1\over i+j}{\rm \forall}i,j \quad \textnormal{with} \quad i\ne j.$$ Prove that $c\geq1$.

%V0 Let $a_0$, $a_1$, $a_2$, ... be an infinite sequence of real numbers satisfying the equation $a_n=\left|a_{n+1}-a_{n+2}\right|$ for all $n\geq 0$, where $a_0$ and $a_1$ are two different positive reals. Can this sequence $a_0$, $a_1$, $a_2$, ... be bounded? Remark This one is from the IMO Shortlist 2004, but it's already published on the official BWM website und thus I take the freedom to post it here:

%V0 $a_{0},\ a_{1},\ a_{2},\dots$ is a sequence of real numbers such that $$a_{n + 1} = \left[a_{n}\right]\cdot \left\{a_{n}\right\}$$ prove that exist $j$ such that for every $i\geq j$ we have $a_{i + 2} = a_{i}$.

%V0 Let $a_{0}$, $a_{1}$, $a_{2}$, $...$ be a sequence of reals such that $a_{0} = - 1$ and $a_{n} + \frac {a_{n - 1}}{2} + \frac {a_{n - 2}}{3} + ... + \frac {a_{1}}{n} + \frac {a_{0}}{n + 1} = 0$ for all $n\geq 1$. Show that $a_{n} > 0$ for all $n\geq 1$.