We are given a set $P$ of points and a set $L$ of straight lines. At the beginning there are 4 points, no three of which are collinear, and $L=\emptyset $. Two players are taking turns adding one or two lines to $L$, where each of these lines has to pass through at least two of the points in $P$. After that all intersection points of the lines in $L$ are added to $P$, if they are not already part of it. A player wins, if after his turn there are three collinear points from $P$, which lie on a line that isn’t from $L$. Find who of the two players has a winning strategy.

Six points in general position are given in the space. For each two of them color red the common points (if they exist) of the segment between these points and the surface of the tetrahedron formed by four remaining points. Prove that the number of red points is even.

Projector emits rays in space. Consider all acute angles between the rays. It is known that no matter what ray we remove, the number of acute angles decreases by exactly $2$ What is the maximal number of rays the projector can emit? [i]M. Karpuk, E. Barabanov[/i]

We have $n$ guinea pigs placed on the vertices of a regular polygon with $n$ sides inscribed in a circumference, one guinea pig in each vertex. Each guinea pig has a direction assigned, such direction is either "clockwise" or "anti-clockwise", and a velocity between $1 km/h$, $2km/h$,..., and $n km/h$, each one with a distinct velocity, and each guinea pig has a counter starting from $0$. They start moving along the circumference with the assigned direction and velocity, everyone at the same time, when 2 or more guinea pigs meet a point, all of the guinea pigs at that point follow the same direction of the fastest guinea pig and they keep moving (with the same velocity as before); each time 2 guinea pigs meet for the first time in the same point, the fastest guinea pig adds 1 to its counter. Prove that, at some moment, for each $1\leq i\leq n$ we have that the $i-$th guinea pig has $i-1$ in its counter.

Consider a positive integer $n$ and two points $A$ and $B$ in a plane. Starting from point $A$, $n$ rays and starting from point $B$, $n$ rays are drawn so that all of them are on the same half-plane defined by the line $AB$ and that the angles formed by the $2n$ rays with the segment $AB$ are all acute. Define circles passing through points $A$, $B$ and each meeting point between the rays. What is the smallest number of [b]distinct [/b] circles that can be defined by this construction?

Let $S$ be a set of 1980 points in the plane such that the distance between every pair of them is at least 1. Prove that $S$ has a subset of 220 points such that the distance between every pair of them is at least $\sqrt{3}.$

Prove that among any $51$ vertices of the $101$-regular polygon there are three that are the vertices of an isosceles triangle.

Each vertex of a regular $17$-gon is colored red, blue, or green in such a way that no two adjacent vertices have the same color. Call a triangle “multicolored” if its vertices are colored red, blue, and green, in some order. Prove that the $17$-gon can be cut along nonintersecting diagonals to form at least two multicolored triangles. (A diagonal of a polygon is a a line segment connecting two nonadjacent vertices. Diagonals are called nonintersecting if each pair of them either intersect in a vertex or do not intersect at all.)

Some of the vertices of a convex $n$-gon are connected by segments, such that any two of them have no common interior point. Prove that, for any $n$ points in general position, there exists a one-to-one correspondence between the points and the vertices of the $n$ gon, such that any two segments between the points, corresponding to the respective segments from the $n$ gon, have no common interior point.

Luke chose a set of three different dates $a,b,c$ in the month of May, where in any year, if one makes a calendar with a sheet of grid paper the centers of the cells with dates $a,b,c$ would form an isosceles right triangle or a straight line. How many sets can be chosen? [img]https://cdn.artofproblemsolving.com/attachments/7/3/dbf90fdc81fc0f0d14c32020b69df53b67b397.png[/img]

Sixteen squares of an $8\times 8$ chessboard are chosen so that there are exactly lwo in each row and two in each column. Prove that eight white pawns and eight black pawns can be placed on these sixteen squares so that there is one white pawn and one black pawn in each row and in cach colunm.

Let $n$ and $c$ be coprime positive integers. For any integer $i$, denote by $i' $ the remainder of division of product $ci$ by $n$. Let $A_o.A_1,A_2,...,A_{n-1}$ be a regular $n$-gon. Prove that a) if $A_iA_j \parallel A_kA_i$ then $A_{i'}A_{j'} \parallel A_{k'}A_{i'}$ b) if $A_iA_j \perp A_kA_l$ then $A_{i'}A_{j'} \perp A_{k'}A_{l'}$

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) Two legs of an angle $\alpha$ on a plane are mirrors. Prove that after several reflections in the mirrors any ray leaves in the direction opposite the one from which it came if and only if $\alpha = \frac{90^o}{n}$ for an integer $n$. Find the number of reflections. b) Given three planar mirrors in space forming an octant (trihedral angle with right planar angles), prove that any ray of light coming into this mirrored octant leaves it, after several reflections in the mirrors, in the direction opposite to the one from which it came. Find the number of reflections.

There are $n$ pieces on the squares of a $5 \times 9$ board, at most one on each square at any time during the game. A move in the game consists of simultaneously moving each piece to a neighboring square by side, under the restriction that a piece having been moved horizontally in the previous move must be moved vertically and vice versa. Find the greatest value of $n$ for which there exists an initial position starting at which the game can be continued until the end of the world.

There are $1998$ rectangular pieces $2$ cm wide and $3$ cm long and with them squares are assembled (without overlapping or gaps). What is the greatest number of different squares that can be had at the same time?

Given $4$ points in three-dimensional space, not lying in one plane. What is the number of such a parallelepipeds (bricks), that each point is a vertex of each parallelepiped?

The points of the plane have been colored by $2013$ different colors. We say that a triangle $\vartriangle ABC$ has the color $X$ if its three vertices $A,B,C$ has the color $X$. Prove that there are innitely many triangles with the same color and the same area.

Show that it is possible to color each point of a circle red or blue so that no right-angled triangle inscribed in the circle has its vertices all the same color.

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]

There are $n$ blue points and $n$ red points on a straight line. Prove that the sum of all distances between pairs of points of the same colour is less than or equal to the sum of all distances between pairs of points of different colours

a) What is the least number of points that can be chosen on a circle of length $1956$, so that for each of these points there is exactly one chosen point at distance $1$, and exactly one chosen point at distance $2$ (distances are measured along the circle)? b) On a circle of length $15$ there are selected $n$ points such that for each of them there is exactly one selected point at distance $1$ from it, and exactly one is selected point at distance $2$ from it. (All distances are measured along the circle.) Prove that $n$ is divisible by $10$.

Can $77$ blocks each $3 \times 3 \times1$ be assembled to form a $7 \times 9 \times 11$ block?

In space, $n$ wire triangles are situated so that any two of them have a common vertex and each vertex is the vertex of $k$ triangles. Find all $n$ and $k$ for which this is possible.

In the plane, six circles are given so that none of the circles contain one the center of the other. Show that there is no point that lies in all the circles.