Show that there exists an integer $N$ such that for all $n \ge N$ a square can be partitioned into $n$ smaller squares.

Quadrilaterals may be obtained from an octagon by cutting along its diagonals (in $8$ different ways) . Can it happen that among these $8$ quadrilaterals (a) four (b ) five possess an inscribed circle? (P. M . Sedrakyan , Yerevan)

a) On a circumference $8$ points are marked. We say that Juliana does an “T-operation ” if she chooses three of these points and paint the sides of the triangle that they determine, so that each painted triangle has at most one vertex in common with a painted triangle previously. What is the greatest number of “T-operations” that Juliana can do? b) If in part (a), instead of considering $8$ points, $7$ points are considered, what is the greatest number of “T operations” that Juliana can do?

Alice and Bob play a game, in which they take turns drawing segments of length $1$ in the Euclidean plane. Alice begins, drawing the first segment, and from then on, each segment must start at the endpoint of the previous segment. It is not permitted to draw the segment lying over the preceding one. If the new segment shares at least one point - except for its starting point - with one of the previously drawn segments, one has lost. a) Show that both Alice and Bob could force the game to end, if they don’t care who wins. b) Is there a winning strategy for one of them?

There are $100$ points on the plane, all pairwise distances between which are different. Is there always a polyline with vertices at these points, passing through each point once, in which the link lengths increase monotonously?

We are given $666$ points on the plane such that they cannot be covered by $10$ lines. Show that we can choose $66$ out of these points such that they can not be covered by $10$ lines.

Given a positive integer $n$, determine the largest real number $\mu$ satisfying the following condition: for every set $C$ of $4n$ points in the interior of the unit square $U$, there exists a rectangle $T$ contained in $U$ such that $\bullet$ the sides of $T$ are parallel to the sides of $U$; $\bullet$ the interior of $T$ contains exactly one point of $C$; $\bullet$ the area of $T$ is at least $\mu$.

There is a cube of $3 \times 3 \times 3$ formed by the union of $27$ cubes of $1 \times 1 \times 1$. Some cubes are removed in such a way that those that remain continue to form a solid made up of cubes that are united by at least one facing the rest of the solid. When a cube is removed, those that remain do so in the same place they were. What is the maximum number of cubes that can be removed so that the area of the resulting solid is equal to the area of the original cube?

Faith has four different integer length segments. It turned out that any three of them can form a triangle. What is the smallest total length of this set of segments?

The $X{}$ pentomino consists of five $1\times1$ squares where four squares are all adjacent to the fifth one. Is it possible to cut nine such pentominoes from an $8\times 8$ chessboard, not necessarily cutting along grid lines? (The picture shows how to cut three such $X{}$ pentominoes.) [i]Alexandr Gribalko[/i]

$n$ straight lines are drawn in a plane. They divide the plane onto several parts. Some of the parts are painted. Not a pair of painted parts has non-zero length common bound. Prove that the number of painted parts is not more than $\frac{n^2 + n}{3}$.

There are $n$ straight lines in the plane such that each intersects exactly $1999$ of the others . Find all posssible values of $n$. (R Zhenodarov)

Find the minimal $k$ such that every set of $k$ different lines in $\mathbb R^3$ contains either $3$ mutually parallel lines or $3$ mutually intersecting lines or $3$ mutually skew lines.

Placing $n \in {\mathbb N}$ circles with radius $1$ $unit$ inside a square with side $100$ $unit$ such that, whichever line segment with lenght $10$ $unit$ intersect at least one circle. Prove that $$n \geq 416$$

Let $\mathcal{P}$ be a convex polygon and $\textbf{T}$ be a triangle with vertices among the vertices of $\mathcal{P}$. By removing $\textbf{T}$ from $\mathcal{P}$, we end up with $0, 1, 2,$ or $3$ smaller polygons (possibly with shared vertices) which we call the effect of $\textbf{T}$. A triangulation of $P$ is a way of dissecting it into some triangles using some non-intersecting diagonals. We call a triangulation of $\mathcal{P}$ $\underline{\text{beautiful}}$, if for each of its triangles, the effect of this triangle contains exactly one polygon with an odd number of vertices. Prove that a triangulation of $\mathcal{P}$ is beautiful if and only if we can remove some of its diagonals and end up with all regions as quadrilaterals.

Each integer coordinates are colored with one color and at least 5 colors are used to color every integer coordinates. Two integer coordinates $(x, y)$ and $(z, w)$ are colored in the same color if $x-z$ and $y-w$ are both multiples of 3. Prove that there exists a line that passes through exactly three points when five points with different colors are chosen randomly.

Is it possible to cut a square into five squares?

Does there exist a set of $100$ triangles in which not one of the triangles can be covered by the other $99$?

Let $\,S\,$ be a finite set of points in three-dimensional space. Let $\,S_{x},\,S_{y},\,S_{z}\,$ be the sets consisting of the orthogonal projections of the points of $\,S\,$ onto the $yz$-plane, $zx$-plane, $xy$-plane, respectively. Prove that \[ \vert S\vert^{2}\leq \vert S_{x} \vert \cdot \vert S_{y} \vert \cdot \vert S_{z} \vert, \] where $\vert A \vert$ denotes the number of elements in the finite set $A$. [hide="Note"] Note: The orthogonal projection of a point onto a plane is the foot of the perpendicular from that point to the plane. [/hide]

Is it is possible to choose 24 distinct points in the space such that no three of them lie on the same line and choose 2019 distinct planes in a way that each plane passes through at least 3 of the chosen points and each triple belongs to one of the chosen planes?

The convex set $F$ does not cover a semi-circle of radius $R$. Is it possible that two sets, congruent to $F$, cover the circle of radius $R$ ? What if $F$ is not convex? ( N . B . Vasiliev , A. G . Samosvat)

An $8\times 8$ chessboard is made of unit squares. We put a rectangular piece of paper with sides of length 1 and 2. We say that the paper and a single square overlap if they share an inner point. Determine the maximum number of black squares that can overlap the paper.

Bob plotted $2003$ green points on the plane, so all triangles with three green vertices have area less than $1$. Prove that the $2003$ green points are contained in a triangle $T$ of area less than $4$.

A regular hexagon is cut up into $N$ parallelograms of equal area. Prove that $N$ is divisible by three. (V. Prasolov, I. Sharygin, Moscow)

Given seven points on a plane, with no three points collinear. Prove that it is always possible to divide these points into the vertices of a triangle and a convex quadrilateral, with no shared parts between the two shapes.