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 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 $x,y$ and $z$ be positive real numbers such that $xyz=1$. Prove that $$\frac{x^{3}}{(1 + y)(1 + z)}+\frac{y^{3}}{(1 + z)(1 + x)}+\frac{z^{3}}{(1 + x)(1 + y)} \geq \frac{3}{4}.$$

%V0 A game is played by $n$ girls ($n \geq 2$), everybody having a ball. Each of the $\binom{n}{2}$ pairs of players, is an arbitrary order, exchange the balls they have at the moment. The game is called nice nice if at the end nobody has her own ball and it is called tiresome if at the end everybody has her initial ball. Determine the values of $n$ for which there exists a nice game and those for which there exists a tiresome game.

%V0 If $a, b, c$ are three positive real numbers such that $ab+bc+ca = 1$, prove that $$\sqrt[3]{ \frac{1}{a} + 6b} + \sqrt[3]{\frac{1}{b} + 6c} + \sqrt[3]{\frac{1}{c} + 6a } \leq \frac{1}{abc}.$$

%V0 Let $n$ be a positive integer, and let $x$ and $y$ be a positive real number such that $x^n + y^n = 1.$ Prove that $$ \left(\sum^n_{k = 1} \frac {1 + x^{2k}}{1 + x^{4k}} \right) \cdot \left( \sum^n_{k = 1} \frac {1 + y^{2k}}{1 + y^{4k}} \right) < \frac{1}{(1 - x)(1 - y)} \text{.}$$ Author: unknown author, Estonia