$5$ points are given in the plane, any three non-collinear and any four non-concyclic. If three points determine a circle that has one of the remaining points inside it and the other one outside it, then the circle is said to be [i]good[/i]. Let the number of good circles be $n$; find all possible values of $n$.

Consider the set of all integer points in $Z^3$. Sasha and Masha play such a game. At first, Masha marks an arbitrary point. After that, Sasha marks all the points on some a plane perpendicular to one of the coordinate axes and at no point, which Masha noted. Next, they continue to take turns (Masha can't to select previously marked points, Sasha cannot choose the planes on which there are points said Masha). Masha wants to mark $n$ consecutive points on some line that parallel to one of the coordinate axes, and Sasha seeks to interfere with it. Find all $n$, in which Masha can achieve the desired result.

Let us call "[i]big[/i]" a triangle with all sides longer than $1$. Given a equilateral triangle with all the sides equal to $5$. Prove that: a) You can cut $100$ [i]big [/i] triangles out of given one. b) You can divide the given triangle onto $100$ [i]big [/i] nonintersecting ones fully covering the initial one. c) The same as b), but the triangles either do not have common points, or have one common side, or one common vertex. d) The same as c), but the initial triangle has the side $3$.

Five points are given in space, no four of which are coplanar. Each of the segments connecting two of them is painted in white, green or red, so that all the colors are used and no three segments of the same color form a triangle. Prove that among these five points there is one at which segments of all the three colors meet.

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)

Let \(n>1\) be a natural number, and let \(K\) be the square of side length \(n\) subdivided into \(n^2\) unit squares. Determine for which values of \(n\) it is possible to dissect \(K\) into \(n\) connected regions of equal area using only the diagonals of those unit squares, subject to the condition that from each unit square at most one of its diagonals is used (some unit squares may have neither diagonal).

Decide whether any rectangle that can be covered by 25 circles of radius 2 can also be covered by 100 circles of radius 1.

In a plane a set of $n\geq 3$ points is given. Each pair of points is connected by a segment. Let $d$ be the length of the longest of these segments. We define a diameter of the set to be any connecting segment of length $d$. Prove that the number of diameters of the given set is at most $n$.

For an integer $n \geq 5,$ two players play the following game on a regular $n$-gon. Initially, three consecutive vertices are chosen, and one counter is placed on each. A move consists of one player sliding one counter along any number of edges to another vertex of the $n$-gon without jumping over another counter. A move is legal if the area of the triangle formed by the counters is strictly greater after the move than before. The players take turns to make legal moves, and if a player cannot make a legal move, that player loses. For which values of $n$ does the player making the first move have a winning strategy?

A point in the plane is called a node if both its coordinates are integers. Consider a triangle with vertices at nodes containing exactly two nodes inside. Prove that the straight line connecting these nodes either passes through a vertex or is parallel to a side of the triangle.

Show that a convex polygon with more than four sides cannot be decomposed into two others, both similar to the first (directly or inversely), by means of a single rectilinear cut. Reasonably specify which are the quadrilaterals and triangles that admit a decomposition of this type.

A square is cut into a finitely number of triangles in an arbitrary way. Show the sum of the diameters of the inscribed circles in these triangles is greater than the side length of the square.

Let $n$ be a natural number. There are $n$ blue points , $n$ red points and one green point on the circle . Prove that it is possible to draw $n$ lengths whose ends are in the given points, so that a maximum of one segment emerges from each point, no more than two segments intersect and the endpoints of none of the segments are blue and red points. [hide=original wording]Нека je ? природан број. На кружници се налази ? плавих, ? црвених и једна зелена тачка. Доказати да је могуће повући ? дужи чији су крајеви у датим тачкама, тако да из сваке тачке излази максимално једна дуж, никоје две дужи се не сијеку и крајње тачке ниједне од дужи нису плава и црвена тачка.[/hide]

For any set of points $A_1, A_2,...,A_n$ on the plane, one defines $r( A_1, A_2,...,A_n)$ as the radius of the smallest circle that contains all of these points. Prove that if $n \ge 3$, there are indices $i,j,k$ such that $r( A_1, A_2,...,A_n)=r( A_i, A_j,A_k)$

Each square of a chessboard is painted either blue or red . Prove that the squares of one colour possess the property that the chess queen can perform a tour of all of them. The rules are that the queen may visit the squares of this colour not necessarily only once each , and may not be placed on squares of the other colour, although she may pass over them ; the queen moves along any horizontal , vertical or diagonal file over any distance. (A . K . Tolpugo , Kiev)

There is a token in one of the nodes of a hexagon with side $n$, divided into regular triangles (see figure). Two players take turns moving it to one of the neighboring nodes, and it is forbidden to go to a node that the token has already visited. The one who loses who can't make a move. Who wins with the right game? [img]https://cdn.artofproblemsolving.com/attachments/2/f/18314fe7f9f4cd8e783037a8e5642e17f4e1be.png[/img]

Each point in the plane with integer coordinates is colored red or blue such that the following two properties hold. For any two red points, the line segment joining them does not contain any blue points. For any two blue points that are distance $2$ apart, the midpoint of the line segment joining them is blue. Prove that if three red points are the vertices of a triangle, then the interior of the triangle does not contain any blue points.

In the convex $(2n+2) $-gon are drawn $n^2$ diagonals. Prove that one of these of diagonals cuts the $(2n+2)$ -gon into two polygons, each of which has an odd number vertices.

A line meets a segment $AB$ at point $C$. Which is the maximal number of points $X$ of this line such that one of angles $AXC$ and $BXC$ is equlal to a half of the second one?

There is a regular hexagon that is cut direct to $6n^2$ equilateral triangles (Fig.). There are arranged $2n$ rooks, neither of which beats each other (the rooks hit in directions parallel to sides of the hexagon). Prove that if we consider chess coloring all $6n^2$ equilateral triangles, then the number of rooks that stand on black triangles will be equal to the number of rooks standing on white triangles. [img]https://cdn.artofproblemsolving.com/attachments/d/0/43ce6c5c966f60a8ec893d5d8cd31e33c43fc0.png[/img] [hide=original wording] Є правильний шестикутник, що розрізаний прямими на 6n^2 правильних трикутників (рис. 2). У них розставлені 2n тур, ніякі дві з яких не б'ють одна одну (тура б'є в напрямках, що паралельні до сторін шестикутника). Доведіть, що якщо розглянути шахове розфарбування всіх 6n^2 правильних трикутників, то тоді кількість тур, що стоять на чорних трикутниках, буде рівна кількості тур, що стоять на білих трикутниках. [/hide]

Prove that an arbitrary triangle can be cut into three polygons, one of which must be an obtuse triangle, so that they can then be folded into a rectangle. (Turning over parts is possible).

Cut an arbitrary triangle into $2019$ pieces so that one of them turns out to be a triangle, one is a quadrilateral, ... one is a $2019$-gon and one is a $2020$-gon. Polygons do not have to be convex.

We consider an $n \times n$ table, with $n\ge1$. Aya wishes to color $k$ cells of this table so that that there is a unique way to place $n$ tokens on colored squares without two tokens are not in the same row or column. What is the maximum value of $k$ for which Aya's wish is achievable?

As the figure is shown, place a $2\times 5$ grid table in horizontal or vertical direction, and then remove arbitrary one $1\times 1$ square on its four corners. The eight different shapes consisting of the remaining nine small squares are called [i]banners[/i]. [asy] defaultpen(linewidth(0.4)+fontsize(10));size(50); pair A=(-1,1),B=(-1,3),C=(-1,5),D=(-3,5),E=(-5,5),F=(-7,5),G=(-9,5),H=(-11,5),I=(-11,3),J=(-11,1),K=(-9,1),L=(-7,1),M=(-5,1),N=(-3,1),O=(-5,3),P=(-7,3),Aa=(-1,7),Ba=(-1,9),Ca=(-1,11),Da=(-3,11),Ea=(-5,11),Fa=(-7,11),Ga=(-9,11),Ha=(-11,11),Ia=(-11,9),Ja=(-11,7),Ka=(-9,7),La=(-7,7),Ma=(-5,7),Na=(-3,7),Oa=(-5,9),Pa=(-7,9); draw(B--C--H--J--N^^B--I^^D--N^^E--M^^F--L^^G--K); draw(Aa--Ca--Ha--Ja--Aa^^Ba--Ia^^Da--Na^^Ea--Ma^^Fa--La^^Ga--Ka); [/asy] [asy] defaultpen(linewidth(0.4)+fontsize(10));size(50); pair A=(-1,1),B=(-1,3),C=(-1,5),D=(-3,5),E=(-5,5),F=(-7,5),G=(-9,5),H=(-11,5),I=(-11,3),J=(-11,1),K=(-9,1),L=(-7,1),M=(-5,1),N=(-3,1),O=(-5,3),P=(-7,3),Aa=(-1,7),Ba=(-1,9),Ca=(-1,11),Da=(-3,11),Ea=(-5,11),Fa=(-7,11),Ga=(-9,11),Ha=(-11,11),Ia=(-11,9),Ja=(-11,7),Ka=(-9,7),La=(-7,7),Ma=(-5,7),Na=(-3,7),Oa=(-5,9),Pa=(-7,9); draw(B--Ca--Ea--M--N^^B--O^^C--E^^Aa--Ma^^Ba--Oa^^Da--N); draw(L--Fa--Ha--J--L^^Ga--K^^P--I^^F--H^^Ja--La^^Pa--Ia); [/asy] Here is a fixed $9\times 18$ grid table. Find the number of ways to cover the grid table completely with 18 [i]banners[/i].

There are $1996$ points marked on a straight line at regular intervals. Petya colors half of them red and the rest blue. Then Vasya divides them into pairs ''red'' - ''blue'' so that the sum distances between points in pairs was maximum. Prove that this maximum does not depend on what coloring Petya made.