Find the maximum value of $ x_{0}$ for which there exists a sequence $ x_{0},x_{1}\cdots ,x_{1995}$ of positive reals with $ x_{0} \equal{} x_{1995}$, such that \[ x_{i \minus{} 1} \plus{} \frac {2}{x_{i \minus{} 1}} \equal{} 2x_{i} \plus{} \frac {1}{x_{i}}, \] for all $ i \equal{} 1,\cdots ,1995$.

Let $a,b,c>0$ such that $a \geq bc^2$ , $b \geq ca^2$ and $c \geq ab^2$ . Find the maximum value that the expression : $$E=abc(a-bc^2)(b-ca^2)(c-ab^2)$$ can acheive.

Find the maximum possible value of $k$ for which there exist distinct reals $x_1,x_2,\ldots ,x_k $ greater than $1$ such that for all $1 \leq i, j \leq k$, $$x_i^{\lfloor x_j \rfloor }= x_j^{\lfloor x_i\rfloor}.$$ [i]Proposed by Morteza Saghafian[/i]

Point $P$ and equilateral triangle $ABC$ satisfy $|AP|=2$, $|BP|=3$. Maximize $|CP|$.

Given an equilateral triangle with sidelength $k$ cm. With lines parallel to it's sides, we split it into $k^2$ small equilateral triangles with sidelength $1$ cm. This way, a triangular grid is created. In every small triangle of sidelength $1$ cm, we place exactly one integer from $1$ to $k^2$ (included), such that there are no such triangles having the same numbers. With vertices the points of the grid, regular hexagons are defined of sidelengths $1$ cm. We shall name as [i]value [/i] of the hexagon, the sum of the numbers that lie on the $6$ small equilateral triangles that the hexagon consists of . Find (in terms of the integer $k>4$) the maximum and the minimum value of the sum of the values of all hexagons .

Let $a$ and $b$ be real numbers bigger than $1$. Find maximal value of $c \in \mathbb{R}$ such that $$\frac{1}{3+\log _{a} b}+\frac{1}{3+\log _{b} a} \geq c$$

Let $\omega \in \mathbb{C}$, and $\left | \omega \right | = 1$. Find the maximum length of $z = \left( \omega + 2 \right) ^3 \left( \omega - 3 \right)^2$.

Let $n$ be a positive integer. Positive numbers $a$, $b$, $c$ satisfy $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$. Find the greatest possible value of $$E(a,b,c)=\frac{a^{n}}{a^{2n+1}+b^{2n} \cdot c + b \cdot c^{2n}}+\frac{b^{n}}{b^{2n+1}+c^{2n} \cdot a + c \cdot a^{2n}}+\frac{c^{n}}{c^{2n+1}+a^{2n} \cdot b + a \cdot b^{2n}}$$

If $x$ and $y$ are real numbers, determine the greatest possible value of the expression $\frac{(x+1)(y+1)(xy+1)}{(x^2+1)(y^2+1)}$.

The numbers $x_i,i=1,2…6\in \mathbb{R}^+$ are such that $x_1+x_2+...+x_6=1$ and $x_1 x_3 x_5+x_2 x_4 x_6\geq \frac{1}{540}$. Let $S=x_1 x_2 x_3+x_2 x_3 x_4+...+x_6 x_1 x_2$. If $max\, S=\frac{p}{q}$ , where $gcd(p,q)=1$, find $p+q$.

There are $2n + 1$ tickets, each with a unique positive integer as the ticket number. It is known that the sum of all ticket numbers is more than $2330$, but the sum of any $n$ ticket numbers is at most $1165$. What is the maximum value of $n$?

Let $a_0,a_1,a_2,\dots $ be a sequence of real numbers such that $a_0=0, a_1=1,$ and for every $n\geq 2$ there exists $1 \leq k \leq n$ satisfying \[ a_n=\frac{a_{n-1}+\dots + a_{n-k}}{k}. \]Find the maximum possible value of $a_{2018}-a_{2017}$.

Find the maximum and minimum of the expression \[ \max(a_1, a_2) + \max(a_2, a_3), + \dots + \max(a_{n-1}, a_n) + \max(a_n, a_1), \] where $(a_1, a_2, \dots, a_n)$ runs over the set of permutations of $(1, 2, \dots, n)$.

The natural numbers $ a_1 $, $ a_2 $, $ \dots $, $ a_n $ satisfy the condition $ 1 / a_1 + 1 / a_2 + \ldots + 1 / a_n = 1 $. Prove that all these numbers do not exceed $$ n ^ {2 ^ n} $$

Let $a_0,a_1,a_2,\dots $ be a sequence of real numbers such that $a_0=0, a_1=1,$ and for every $n\geq 2$ there exists $1 \leq k \leq n$ satisfying \[ a_n=\frac{a_{n-1}+\dots + a_{n-k}}{k}. \]Find the maximum possible value of $a_{2018}-a_{2017}$.

Let $a,b,c$ be positive real numbers such that $a+b+c=3$. Determine, with certainty, the largest possible value of the expression $$ \frac{a}{a^3+b^2+c}+\frac{b}{b^3+c^2+a}+\frac{c}{c^3+a^2+b}$$

Let $a_0,a_1,a_2,\dots $ be a sequence of real numbers such that $a_0=0, a_1=1,$ and for every $n\geq 2$ there exists $1 \leq k \leq n$ satisfying \[ a_n=\frac{a_{n-1}+\dots + a_{n-k}}{k}. \]Find the maximum possible value of $a_{2018}-a_{2017}$.

Given circle $\omega$ and point $D$ outside this circle. Find the following points $A, B$ and $C$ on the circle $\omega$ so that the $ABCD$ quadrilateral is convex and has the maximum possible area. Justify your answer.

There are $n$ stone piles each consisting of $2018$ stones. The weight of each stone is equal to one of the numbers $1, 2, 3, ...25$ and the total weights of any two piles are different. It is given that if we choose any two piles and remove the heaviest and lightest stones from each of these piles then the pile which has the heavier one becomes the lighter one. Determine the maximal possible value of $n$.

Esmeralda writes $2n$ real numbers $x_1, x_2, \dots , x_{2n}$, all belonging to the interval $[0, 1]$, around a circle and multiplies all the pairs of numbers neighboring to each other, obtaining, in the counterclockwise direction, the products $p_1 = x_1x_2$, $p_2 = x_2x_3$, $\dots$ , $p_{2n} = x_{2n}x_1$. She adds the products with even indices and subtracts the products with odd indices. What is the maximum possible number Esmeralda can get?

Let $a_0,a_1,a_2,\dots $ be a sequence of real numbers such that $a_0=0, a_1=1,$ and for every $n\geq 2$ there exists $1 \leq k \leq n$ satisfying \[ a_n=\frac{a_{n-1}+\dots + a_{n-k}}{k}. \]Find the maximum possible value of $a_{2018}-a_{2017}$.

Let $P$ be an interior point of triangle $ABC$. Let $a,b,c$ be the sidelengths of triangle $ABC$ and let $p$ be it's semiperimeter. Find the maximum possible value of $$ \min\left(\frac{PA}{p-a},\frac{PB}{p-b},\frac{PC}{p-c}\right)$$ taking into consideration all possible choices of triangle $ABC$ and of point $P$. by Elton Bojaxhiu, Albania

Let $a_0,a_1,a_2,\dots $ be a sequence of real numbers such that $a_0=0, a_1=1,$ and for every $n\geq 2$ there exists $1 \leq k \leq n$ satisfying \[ a_n=\frac{a_{n-1}+\dots + a_{n-k}}{k}. \]Find the maximum possible value of $a_{2018}-a_{2017}$.

Let $p,q$ be positive integers. For any $a,b\in\mathbb{R}$ define the sets $$P(a)=\bigg\{a_n=a \ + \ n \ \cdot \ \frac{1}{p} : n\in\mathbb{N}\bigg\}\text{ and }Q(b)=\bigg\{b_n=b \ + \ n \ \cdot \ \frac{1}{q} : n\in\mathbb{N}\bigg\}.$$ The [i]distance[/i] between $P(a)$ and $Q(b)$ is the minimum value of $|x-y|$ as $x\in P(a), y\in Q(b)$. Find the maximum value of the distance between $P(a)$ and $Q(b)$ as $a,b\in\mathbb{R}$.

Let $n$ be an integer and $n \geq 2$, $x_1, x_2, \cdots , x_n$ are arbitrary real number, find the maximum value of $$2\sum_{1\leq i<j \leq n}\left \lfloor x_ix_j \right \rfloor-\left ( n-1 \right )\sum_{i=1}^{n}\left \lfloor x_i^2 \right \rfloor $$