A single section at a stadium can hold either $7$ adults or $11$ children. When $N$ sections are completely lled, an equal number of adults and children will be seated in them. What is the least possible value of $N$?

Triangle $ABC$ of area $1$ is given. Point $A'$ lies on the extension of side $BC$ beyond point $C$ with $BC = CA'$. Point $B'$ lies on extension of side $CA$ beyond $A$ and $CA = AB'$. $C'$ lies on extension of $AB$ beyond $B$ with $AB = BC'$. Find the area of triangle $A'B'C'$.

Each square on an $8\times 8$ checkers board contains either one or zero checkers. The number of checkers in each row is a multiple of $3$, the number of checkers in each column is a multiple of $5$. Assuming the top left corner of the board is shown below, how many checkers are used in total?

Which digit must be substituted instead of the star so that the following large number $$\underbrace{66...66}_{2023} \star \underbrace{55...55}_{2023}$$is divisible by $7$?

There are $11$ quadratic equations on the board, where each coefficient is replaced by a star. Initially, each of them looks like this $$\star x^2 + \star x + \star= 0.$$Two players are playing a game making alternating moves. In one move each ofthem replaces one star with a real nonzero number. The first player tries to make as many equations as possible without roots and the second player tries to make the number of equations without roots as small as possible. What is the maximal number of equations without roots that the first player can achieve if the second player plays to her best? Describe the strategies of both players.

Suppose there is an infinite sequence of lights numbered $1, 2, 3,...,$ and you know the following two rules about how the lights work: $\bullet$ If the light numbered $k$ is on, the lights numbered $2k$ and $2k + 1$ are also guaranteed to be on. $\bullet$ If the light numbered $k$ is off, then the lights numbered $4k + 1$ and $4k + 3$ are also guaranteed to be off. Suppose you notice that light number $2023$ is on. Identify all the lights that are guaranteed to be on?

In a square of area $1$ there are situated $2024$ polygons whose total area is greater than $2023$. Prove that they have a point in common.

How few numbers is it possible to cross out from the sequence $$1, 2,3,..., 2023$$so that among those left no number is the product of any two (distinct) other numbers?

Quadrillateral $ABCD$ is inscribed in a circle with centre $O$. Diagonals $AC$ and $BD$ are perpendicular. Prove that the distance from the centre $O$ to $AD$ is half the length of $BC$.

Find the maximum of the expression $$||...||x_1 - x_2|- x_3| -... | - x_{2023}|,$$where $x_1,x_2,..., x_{2023}$ are distinct natural numbers between $1$ and $2023$.