Given $n>4$ points in the plane, no three collinear. Prove that there are at least $\frac{(n-3)(n-4)}{2}$ convex quadrilaterals with vertices amongst the $n$ points.

All possible vertical lines $x = k$ and horizontal lines $y = m$ are drawn on the coordinate plane, where $k$ and $m$ are integers. Let's imagine that all these straight lines are black. A red straight line is also drawn, the equation of which is $19x+96y= c$. Let us denote by $M$ the number of segments of different lengths formed on the red line when intersecting with the black ones.(The ends of each segment are the intersection points of the red and black lines. There are no such intersection points inside the segment.) What values can $M$ take when $c$ changes?

Four equal right-angled triangles are given. We are allowed to cut any triangle into two new ones along the altitude dropped on to the hypotenuse. This operation may be repeated with any of the triangles from the new set. Prove that after any number of such operations there will be congruent triangles among those obtained. (AV Shapovalov)

Given points $O$ and $A$ in the plane. Every point in the plane is colored with one of a finite number of colors. Given a point $X$ in the plane, the circle $C(X)$ has center $O$ and radius $OX+{\angle AOX\over OX}$, where $\angle AOX$ is measured in radians in the range $[0,2\pi)$. Prove that we can find a point $X$, not on $OA$, such that its color appears on the circumference of the circle $C(X)$.

The vertices of a convex $n$-gon are colored so that adjacent vertices have different colors. Prove that if $n$ is odd, then the polygon can be divided into triangles with non-intersecting diagonals such that no diagonal has its endpoints the same color.

2. A unit square paper has a triangle-shaped hole (vertices of the hole are not on the border of the paper). Prove that a triangle with area of $1 / 6$ can be cut from the remaining paper. Alexandr Yuran

Ten vertices of a regular $20$-gon $A_1A_2....A_{20}$ are painted black and the other ten vertices are painted blue. Consider the set consisting of diagonal $A_1A_4$ and all other diagonals of the same length. 1. Prove that in this set, the number of diagonals with two black endpoints is equal to the number of diagonals with two blue endpoints. 2. Find all possible numbers of the diagonals with two black endpoints.

A [i]beautiful configuration[/i] of points is a set of $n$ colored points, such that if a triangle with vertices in the set has an angle of at least $120$ degrees, then exactly 2 of its vertices are colored with the same color. Determine the maximum possible value of $n$.

$40$ cells were marked on an infinite chessboard. Is it always possible to find a rectangle that contains $20$ marked cells? M. Evdokimov

On each side of a triangle, $5$ points are chosen (other than the vertices of the triangle) and these $15$ points are colored red. How many ways are there to choose four red points such that they form the vertices of a quadrilateral?

Andy, Bess, Charley and Dick play on a $1000 \times 1000$ board. They make moves in turn: Andy first, then Bess, then Charley and finally Dick, after that Andy moves again and so on. At each move a player must paint several unpainted squares forming $2 \times 1, 1 \times 2, 1 \times 3$, or $3 \times 1$ rectangle. The player that cannot move loses. Prove that some three players can cooperate to make the fourth player lose.

Let $C$ be a unit circle and $n \ge 1$ be a fixed integer. For any set $A$ of $n$ points $P_1,..., P_n$ on $C$ define $D(A) = \underset{d}{max}\, \underset{i}{min}\delta (P_i, d)$, where $d$ goes over all diameters of $C$ and $\delta (P, \ell)$ denotes the distance from point $P$ to line $\ell$. Let $F_n$ be the family of all such sets $A$. Determine $D_n = \underset{A\in F_n}{min} D(A)$ and describe all sets $A$ with $D(A) = D_n$.

On the plane with a rectangular coordinate system, a set of infinitely many rectangles is given. Every rectangle has the origin as one of its vertices. The sides of all rectangles are parallel to the coordinate axes, and all sides have integer lengths. Prove that there are at least two rectangles in the set, one of which completely covers the other.

Given a regular hexagon with a side of $100$. Each side is divided into one hundred equal parts. Through the division points and vertices of the hexagon, all sorts of straight lines parallel to its sides are drawn. These lines divided the hexagon into single regular triangles. Consider covering a hexagon with equal rhombuses. Each rhombus is made up of two triangles. (These rhombuses cover the entire hexagon and do not overlap.) Among the lines that form our grid, we select those that intersect exactly to the rhombuses (intersect diagonally). How many such lines will there be if: a) $k = 101$; b) $k = 100$; c) $k = 87$?

A side of an arbitrary triangle has a length greater than $1$. Prove that the given triangle it can be cut into at least $2$ triangles, so that each of them has a side of length equal to $1$.

Let $P_{1}, P_{2}, \ldots, P_{2021}$ be 2021 points in the quarter plane $\{(x, y): x \geq 0, y \geq 0\}$. The centroid of these 2021 points lies at the point $(1,1)$. Show that there are two distinct points $P_{i}, P_{j}$ such that the distance from $P_{i}$ to $P_{j}$ is no more than $\sqrt{2} / 20$.

Given a square side $1$ and $2n$ positive reals $a_1, b_1, ... , a_n, b_n$ each $\le 1$ and satisfying $\sum a_ib_i \ge 100$. Show that the square can be covered with rectangles $R_i$ with sides length $(a_i, b_i)$ parallel to the square sides.

Let $n$ be a natural number. A regular hexagon with edge length $n$ gets split into equilateral exploded triangles whose edges are $1$ in length and parallel to one side of the hexagon. Find the number of regular hexagons, the angles of which are all angles of these triangles are.

On the Cartesian plane, all points with both integer coordinates are painted blue. Two blue points are said to be [i]mutually visible[/i] if the line segment that connects them has no other blue points . Prove that there is a set of $ 2019$ blue points that are mutually visible two by two. [hide=official wording]No plano cartesiano, todos os pontos com ambas coordenadas inteiras são pintados de azul. Dois pontos azuis são ditos mutuamente visíveis se o segmento de reta que os conecta não possui outros pontos azuis. Prove que existe um conjunto de 2019 pontos azuis que são mutuamente visíveis dois a dois.[/hide] PS. There is a comment about problem being wrong / incorrect [url=https://artofproblemsolving.com/community/c6h1957974p14780265]here[/url]

Prove that given $n$ points in the plane, not all aligned, there exists a line that passes through exactly two of them. [hide=original wording]Demostrar que dados n puntos en el plano, no todos alineados, existe una recta que pasa por exactamente dos de ellos.[/hide]

On the numerical line, arrange a system of closed segments of length $1$ without common points (endpoints included) so that any infinite arithmetic progression with any non zero difference and any first term has a common point with a segment of the system.

Let eight points be given in a unit cube. Prove that two of these points are on a distance not greater than $1$.

The points in the plane are painted in 100 colors. Prove that there are three points of the same color that are the vertices of a triangle of area 1. [i]Proposed by V. Bragin[/i]

A cube with edge length $2n+ 1$ is dissected into small cubes of size $1\times 1\times 1$ and bars of size $2\times 2\times 1$. Find the least possible number of cubes in such a dissection.

Given a convex polyhedron and a sphere intersecting each its edge at two points so that each edge is trisected (divided into three equal parts). Is it necessarily true that all faces of the polyhedron are (a) congruent polygons? (b) regular polygons?