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 positive integer $n$ is [i]Danish[/i] if a regular hexagon can be partitioned into $n$ congruent polygons. Prove that there are infinitely many positive integers $n$ such that both $n$ and $2^n+n$ are Danish.

A set of points in the plane is called [i]free [/i] if no three points of the set are the vertices of an equilateral triangle. Prove that any set of $n$ points in the plane has a free subset of at least $\sqrt{n}$ points

In the plane are given two mutually perpendicular lines and $n$ rectangles with sides parallel to the two lines. Show that if every two rectangles have a common point, then all the rectangles have a common point.

The closed broken line $M$ has odd number of vertices -- $A_1,A_2,..., A_{2n+1}$ in sequence. Let us denote with $S(M)$ a new closed broken line with vertices $B_1,B_2,...,B_{2n+1}$ -- the midpoints of the first line links: $B_1$ is the midpoint of $[A_1A_2], ... , B_{2n+1}$ -- of $[A_{2n+1}A_1]$. Prove that in a sequence $M_1=S(M), ... , M_k = S(M_{k-1}), ...$ there is a broken line, homothetic to the $M$.

There are $2017$ lines in the plane such that no three of them go through the same point. Turbo the snail sits on a point on exactly one of the lines and starts sliding along the lines in the following fashion: she moves on a given line until she reaches an intersection of two lines. At the intersection, she follows her journey on the other line turning left or right, alternating her choice at each intersection point she reaches. She can only change direction at an intersection point. Can there exist a line segment through which she passes in both directions during her journey?

There are $11$ points equally spaced on a circle. Some of the segments having endpoints among these vertices are drawn and colored in two colors, so that each segment meets at an internal for it point at most one other segment from the same color. What is the greatest number of segments that could be drawn?

Prove that the squares with sides $\frac{1}{1}, \frac{1}{2}, \frac{1}{3},\ldots$ may be put into the square with side $\frac{3}{2} $ in such a way that no two of them have any interior point in common.

A lattice point in the plane is a point both of whose coordinates are integers. Each lattice point has four neighboring points: upper, lower, left, and right. Let $k$ be a circle with radius $r \geq 2$, that does not pass through any lattice point. An interior boundary point is a lattice point lying inside the circle $k$ that has a neighboring point lying outside $k$. Similarly, an exterior boundary point is a lattice point lying outside the circle $k$ that has a neighboring point lying inside $k$. Prove that there are four more exterior boundary points than interior boundary points.

Given two closed broken lines in the plane with odd numbers of edges. All the lines, containing those edges are different, and not a triple of them intersects in one point. Prove that it is possible to chose one edge from each line such, that the chosen edges will be the opposite sides of a convex quadrangle.

The $2001 \times 2001$ trees in a park form a square grid. What is the largest number of trees that can be cut so that no tree stump can be seen from any other? (Each tree has zero width.)

Let $n_1, n_2$ be positive integers. Consider in a plane $E$ two disjoint sets of points $M_1$ and $M_2$ consisting of $2n_1$ and $2n_2$ points, respectively, and such that no three points of the union $M_1 \cup M_2$ are collinear. Prove that there exists a straightline $g$ with the following property: Each of the two half-planes determined by $g$ on $E$ ($g$ not being included in either) contains exactly half of the points of $M_1$ and exactly half of the points of $M_2.$

The plane is colored with two colors so that the following property holds: for each real $a>0$ there is an equilateral triangle of side length $a$ whose $3$ vertices are of the same color. Prove that for any three numbers $a,b,c>0$ for which the sum of any two is greater than the third there is a triangle with sides $a$, $b$, and $c$ whose $3$ vertices are of the same color.

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.

There are $k$ points in the interior of a pentagon. Together with the vertices of the pentagon they form a $(k + 5)$-element set $M$. The area of the pentagon is defined by connecting lines between the points of $M$ into sub-areas in such a way that it is divided into sub-areas in such a way that no sub-areas have a point on their interior of $M$ and contains exactly three points of $M$ on the boundary of each part. None of the connecting lines has a point in common with any other connecting line or pentagon side, which does not belong to $M$. With such a decomposition of the pentagon, there can be an even number of connecting lines (including the pentagon sides) go out? The answer has to be justified.

The $27$ points $(a,b,c)$ of the space are marked such that $a$, $b$ and $c$ take the values $0$, $1$ or $2$. We will call these points "junctures". Using $54$ rods of length $1$, all the joints that are at a distance of $1$ are joined together. A cubic structure of $2\times 2\times 2$ is thus formed. An ant starts from a juncture $A$ and moves along the rods; When it reaches a juncture it turns $90^\circ$ and changes rod. If the ant returns to $A$ and has not visited any juncture more than once except $A$, which it visited $2$ times, at the beginning of the walk and at the end of it, what is the greatest length that the path of the ant can have?

Let $ k$ and $ n$ be integers with $ 0\le k\le n \minus{} 2$. Consider a set $ L$ of $ n$ lines in the plane such that no two of them are parallel and no three have a common point. Denote by $ I$ the set of intersections of lines in $ L$. Let $ O$ be a point in the plane not lying on any line of $ L$. A point $ X\in I$ is colored red if the open line segment $ OX$ intersects at most $ k$ lines in $ L$. Prove that $ I$ contains at least $ \dfrac{1}{2}(k \plus{} 1)(k \plus{} 2)$ red points. [i]Proposed by Gerhard Woeginger, Netherlands[/i]

A number of unit discs are given inside a square of side $100$ such that (i) no two of the discs have a common interior point, and (ii) every segment of length $10$, lying entirely within the square, meets at least one disc. Prove that there are at least $400$ discs in the square.

The set Φ consists of a finite number of points on the plane. The distance between any two points from Φ is at least $\sqrt{2}$. It is known that a regular triangle with side lenght $3$ cut out of paper can cover all points of Φ. What is the greatest number of points that Φ can consist of?

a) A castle is equilateral triangle with the side of $100$ metres. It is divided onto $100$ triangle rooms. Each wall between the rooms is $10$ metres long and contain one door. You are inside and are allowed to pass through every door not more than once. Prove that you can visit not more than $91$ room (not exiting the castle). b) Every side of the triangle is divided onto $k$ parts by the lines parallel to the sides. And the triangle is divided onto $k^2$ small triangles. Let us call the "chain" such a sequence of triangles, that every triangle in it is included only once, and the consecutive triangles have the common side. What is the greatest possible number of the triangles in the chain?

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 rectangle is divided into $n^2$ smaller rectangle by $n - 1$ horizontal lines and $n - 1$ vertical lines. Between those rectangles there are exactly $5660$ which are not congruent. For what minimum value of $n$ is this possible?

Numbers from $1$ to $1000$ are arranged around a circle. Prove that it is possible to form $500$ non-intersecting line segments, each joining two such numbers, and so that in each case the difference between the numbers at each end (in absolute value) is not greater than $749$. (AA Razborov, Moscow)

Let $n$ distinct points in the plane be given. Prove that fewer than $2 n^{3 \slash 2}$ pairs of them are a unit distance apart.

On a circular billiard table a ball rebounds from the rails as if the rail was the tangent to the circle at the point of impact. A regular hexagon with its vertices on the circle is drawn on a circular billiard table. A (point-shaped) ball is placed somewhere on the circumference of the hexagon, but not on one of its edges. Describe a periodical track of this ball with exactly four points at the rails. With how many different directions of impact can the ball be brought onto such a track?