Let $a, b, c$ be nonzero real numbers such that $a + b + c = 0$. Determine the maximum possible value of $\frac{a^2b^2c^2}{ (a^2 + ab + b^2)(b^2 + bc + c^2)(c^2 + ca + a^2)}$ .

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 $a_{2018}-a_{2017}$.

Each of the numbers $1$ up to and including $2014$ has to be coloured; half of them have to be coloured red the other half blue. Then you consider the number $k$ of positive integers that are expressible as the sum of a red and a blue number. Determine the maximum value of $k$ that can be obtained.

Each of the numbers $1$ up to and including $2014$ has to be coloured; half of them have to be coloured red the other half blue. Then you consider the number $k$ of positive integers that are expressible as the sum of a red and a blue number. Determine the maximum value of $k$ that can be obtained.

Let $a_1,a_2,\dots,a_{2017}$ be reals satisfied $a_1=a_{2017}$, $|a_i+a_{i+2}-2a_{i+1}|\le 1$ for all $i=1,2,\dots,2015$. Find the maximum value of $\max_{1\le i<j\le 2017}|a_i-a_j|$.

In a triangle $ABC$, $T$ is the centroid and $ \angle TAB = \angle ACT$. Find the maximum possible value of $sin \angle CAT +sin \angle CBT$.

Let $a_1,a_2,\cdots,a_{41}\in\mathbb{R},$ such that $a_{41}=a_1, \sum_{i=1}^{40}a_i=0,$ and for any $i=1,2,\cdots,40, |a_i-a_{i+1}|\leq 1.$ Determine the greatest possible value of $(1)a_{10}+a_{20}+a_{30}+a_{40};$ $(2)a_{10}\cdot a_{20}+a_{30}\cdot a_{40}.$

For all integers $ {a}_{0},{a}_{1}, \cdot\cdot\cdot {a}_{100}$, find the maximum of ${a}_{5}-2{a}_{40}+3{a}_{60}-4{a}_{95} $ $\bigstar$ 1) ${a}_{0}={a}_{100}=0$ 2) for all $i=0,1,\cdot \cdot \cdot 99, $ $|{a}_{i+1}-{a}_{i}|\le1$ 3) $ {a}_{10}={a}_{90} $

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 $a_1,a_2,\dots, a_{17}$ be a permutation of $1,2,\dots, 17$ such that $(a_1-a_2)(a_2-a_3)\dots(a_{17}-a_1)=n^{17}$ .Find the maximum possible value of $n$ .

Let $0<\alpha<\beta <\gamma\in \mathbb{R}$. Let $K=\{(x,y,z)\in \mathbb{R}^{3}\;|\; x,y,z\geq 0\; \text{and}\; x^{\beta}+y^{\beta}+z^{\beta}=1\}$. Define $f:K\rightarrow \mathbb{R},\; (x,y,z)\mapsto x^{\alpha}+y^{\beta}+z^{\gamma}$. At what points of $K$ does $f$ assume its minimal and maximal values?

Let $a_1,a_2,\dots, a_{17}$ be a permutation of $1,2,\dots, 17$ such that $(a_1-a_2)(a_2-a_3)\dots(a_{17}-a_1)=2^n$ . Find the maximum possible value of positive integer $n$ .

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 $n\geq 2$ be an integer. For each $k$-tuple of positive integers $a_1, a_2, \ldots, a_k$ such that $a_1+a_2+\cdots +a_k=n$, consider the sums $S_i=1+2+\ldots +a_i$ for $1\leq i\leq k$. Determine, in terms of $n$, the maximum possible value of the product $S_1S_2\cdots S_k$. [i]Proposed by Misael Pelayo[/i]

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 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]

Show that the sum of the (local) maximum and minimum values of the function $\frac{tan(3x)}{tan^3x}$ on the interval $\big(0, \frac{\pi }{2}\big)$ is rational.

Let $x, y, z$ be real numbers such that $$x+y+z=2, \;\;\;\; xy+yz+zx=1$$ Find the maximum possible value of $x-y$.

Determine the maximum value of the sum \[ \sum_{i < j} x_ix_j (x_i \plus{} x_j) \] over all $ n \minus{}$tuples $ (x_1, \ldots, x_n),$ satisfying $ x_i \geq 0$ and $ \sum^n_{i \equal{} 1} x_i \equal{} 1.$

Let $n\geq 2$ be a positive integer and let $a_1,a_2,...,a_n\in[0,1]$ be real numbers. Find the maximum value of the smallest of the numbers: \[a_1-a_1a_2, \ a_2-a_2a_3,...,a_n-a_na_1.\]

Determine the minimal and maximal values the expression $\frac{|a+b|+|b+c|+|c+a|}{|a|+|b|+|c|}$ can take, where $a,b,c$ are real numbers.

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 .

A company has $n$ employees. It is known that each of the employees works at least one of the $7$ days of the week, with the exception of an employee who does not work any of the $7$ days. Furthermore, for any two of these $n$ employees, there are at least $3$ days of the week in which one of the two works that day and the other does not (it is not necessarily the same employee who works those days). Determine the highest possible value of $n$.

For positive numbers $a, b, c$ holds $(a + c) (b^2 + a c) = 4a$. Determine the maximum value of $b + c$ and find all triplets of numbers $(a, b, c)$ for which expression takes this value

Real numbers $x,y,z$ satisfy $$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+x+y+z=0$$ and none of them lies in the open interval $(-1,1)$. Find the maximum value of $x+y+z$. [i]Proposed by Jaromír Šimša[/i]