Školjka

%V0 Find all of the positive real numbers like $x,y,z,$ such that : 1.) $x + y + z = a + b + c$ 2.) $4xyz = a^2x + b^2y + c^2z + abc$ Proposed to Gazeta Matematica in the 80s by VASILE CÎRTOAJE and then by Titu Andreescu to IMO 1995.

%V0 Suppose that $x_1, x_2, x_3, \ldots$ are positive real numbers for which $$x^n_n = \sum^{n-1}_{j=0} x^j_n$$ for $n = 1, 2, 3, \ldots$ Prove that $\forall n,$ $$2 - \frac{1}{2^{n-1}} \leq x_n < 2 - \frac{1}{2^n}.$$

%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 $n\geq 1$.