Školjka

%V0 Let $a_{0} = 1994$ and $a_{n + 1} = \frac {a_{n}^{2}}{a_{n} + 1}$ for each nonnegative integer $n$. Prove that $1994 - n$ is the greatest integer less than or equal to $a_{n}$, $0 \leq n \leq 998$

%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 $a, b, c > 0$ 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 $r_{1},r_{2},\ldots ,r_{n}$ be real numbers greater than or equal to 1. Prove that $$\frac{1}{r_{1} + 1} + \frac{1}{r_{2} + 1} + \cdots +\frac{1}{r_{n}+1} \geq \frac{n}{ \sqrt[n]{r_{1}r_{2} \cdots r_{n}}+1}.$$

%V0 The numbers from 1 to $n^2$ are randomly arranged in the cells of a $n \times n$ square ($n \geq 2$). For any pair of numbers situated on the same row or on the same column the ratio of the greater number to the smaller number is calculated. Let us call the characteristic of the arrangement the smallest of these $n^2\left(n-1\right)$ fractions. What is the highest possible value of the characteristic ?

%V0 Let $a_{ij}$ (with the indices $i$ and $j$ from the set $\left\{1,\ 2,\ 3\right\}$) be real numbers such that $a_{ij}>0$ for $i = j$; $a_{ij}<0$ for $i\neq j$. Prove the existence of positive real numbers $c_{1}$, $c_{2}$, $c_{3}$ such that the numbers $a_{11}c_{1}+a_{12}c_{2}+a_{13}c_{3}$, $a_{21}c_{1}+a_{22}c_{2}+a_{23}c_{3}$, $a_{31}c_{1}+a_{32}c_{2}+a_{33}c_{3}$ are either all negative, or all zero, or all positive.