[list=a] [*]The sum of several numbers is equal to one. Can the sum of their cubes be greater than one? [*]The same question as before, for numbers not exceeding one. [*]Can it happen that the series $a_1+a_2+\cdots$ converges, but the series $a_1^3+a_2^3+\cdots$ diverges? [/list]

Find the maximum value of $k$ such that \[\frac{xy}{\sqrt{(x^2 + y^2)(3x^2 + y^2)}}\leq \frac{1}{k}\] holds for all positive numbers $x$ and $y.$

For any prime number $p>2,$ and an integer $a$ and $b,$ if $1+\frac{1}{2^3}+\frac{1}{3^3}+\cdots+\frac{1}{(p-1)^3}=\frac{a}{b},$ prove that $a$ is divisible by $p.$

The kingdom of Anisotropy consists of $n$ cities. For every two cities there exists exactly one direct one-way road between them. We say that a [i]path from $X$ to $Y$[/i] is a sequence of roads such that one can move from $X$ to $Y$ along this sequence without returning to an already visited city. A collection of paths is called [i]diverse[/i] if no road belongs to two or more paths in the collection. Let $A$ and $B$ be two distinct cities in Anisotropy. Let $N_{AB}$ denote the maximal number of paths in a diverse collection of paths from $A$ to $B$. Similarly, let $N_{BA}$ denote the maximal number of paths in a diverse collection of paths from $B$ to $A$. Prove that the equality $N_{AB} = N_{BA}$ holds if and only if the number of roads going out from $A$ is the same as the number of roads going out from $B$. [i]Proposed by Warut Suksompong, Thailand[/i]

Consider a table whose entries are integers. Adding a same integer to all entries on a same row, or on a same column, is called an [i]operation[/i]. It is given that, for infinitely many positive integers $n$, one can obtain, through a finite number of operations, a table having all entries divisible by $n$. Prove that, through a finite number of operations, one can obtain the table whose all entries are zeroes.

Let $ABCD$ be a trapezoid with $AD \parallel BC$, $AD < BC$. Let $E$ be a point on the side $AB$ and $F$ be point on the side $CD$. The circle $(AEF)$ intersects the segment $AD$ again at $A_1$ and the circle $(CEF)$ intersects these segment $BC$ again at $C_1$. Prove that the lines $A_1 C_1$, $BD$ and $EF$ are concurrent. [i]Proposed by A. Kuznetsov[/i]

Determine all triplets $(a,b,c)$ of real numbers such that sets $\{a^2-4c, b^2-2a, c^2-2b \}$ and $\{a-c,b-4c,a+b\}$ are equal and $2a+2b+6=5c$. In every set all elements are pairwise distinct

At a movie theater tickets for adults cost $4$ dollars more than tickets for children. One afternoon the theater sold $100$ more child tickets than adult tickets for a total sales amount of $1475$ dollars. How much money would the theater have taken in if the same tickets were sold, but the costs of the child tickets and adult tickets were reversed?

Let $ABC$ a scalene triangle and $AD, BE, CF$ your angle bisectors, with $D$ in the segment $BC, E$ in the segment $AC$ and $F$ in the segment $AB$. If $\angle AFE = \angle ADC$. Determine $\angle BCA$.

Let $ \left( a_n \right)_{n\ge 1} $ be a sequence of real numbers such that $ a_1>2 $ and $ a_{n+1} =a_1+\frac{2}{a_n} , $ for all natural numbers $ n. $ [b]a)[/b] Show that $ a_{2n-1} +a_{2n} >4 , $ for all natural numbers $ n, $ and $ \lim_{n\to\infty} a_n =2. $ [b]b)[/b] Find the biggest real number $ a $ for which the following inequality is true: $$ \sqrt{x^2+a_1^2} +\sqrt{x^2+a_2^2} +\sqrt{x^2+a_3^2} +\cdots +\sqrt{x^2+a_n^2} > n\sqrt{x^2+a^2}, \quad\forall x\in\mathbb{R} ,\quad\forall n\in\mathbb{N} . $$