A [i]zigzag [/i] is a polyline on a plane formed from two parallel rays and a segment connecting the origins of these rays. What is the maximum number of parts a plane can be split into using $ n $ zigzags?

An ant crawled a total distance of $1$ in the plane and returned to its original position (so that its path is a closed loop of length $1$; the width is considered to be $0$). Prove that there is a circle of radius $\frac{1}{4}$ containing the path. Illustration of an example path:

Is it possible to place $2012$ distinct circles with the same diameter on the plane, such that each circle touches at least three others circles?

Is it possible to divide a polygon with $21$ sides into $2022$ triangles in such a way that among all the vertices there are not three collinear?

The rook, standing on the surface of the checkered cube, beats the cells, located in the same row as well as on the continuations of this series through one or even several edges. (The picture shows an example for a $4 \times 4 \times 4$ cube,visible cells that some beat the rook, shaded gray.) What is the largest number do not beat each other rooks can be placed on the surface of the cube $50 \times 50 \times 50$?

The map of a town shows a plane divided into equal equilateral triangles. The sides of these triangles are streets and their vertices are intersections; $6$ streets meet at each junction. Two cars start simultaneously in the same direction and at the same speed from points $A$ and $B$ situated on the same street (the same side of a triangle). After any intersection an admissible route for each car is either to proceed in its initial direction or turn through $120^o$ to the right or to the left. Can these cars meet? (Either prove that these cars won’t meet or describe a route by which they will meet.) [img]https://cdn.artofproblemsolving.com/attachments/2/d/2c934bcd0c7fc3d9dca9cee0b6f015076abbdb.png[/img]

A rectangular building consists of $30$ square rooms situated like the cells of a $2 \times 15$ board. In each room there are three doors, each of which leads to another room (not necessarily different). How many ways are there to distribute the doors between the rooms so that it is possible to get from any room to any other one without leaving the building?

In the plane we consider rectangles whose sides are parallel to the coordinate axes and have positive length. Such a rectangle will be called a [i]box[/i]. Two boxes [i]intersect[/i] if they have a common point in their interior or on their boundary. Find the largest $ n$ for which there exist $ n$ boxes $ B_1$, $ \ldots$, $ B_n$ such that $ B_i$ and $ B_j$ intersect if and only if $ i\not\equiv j\pm 1\pmod n$. [i]Proposed by Gerhard Woeginger, Netherlands[/i]

The Poincare plane is a half-plane bounded by a line $R$. The lines are taken to be (1) the half-lines perpendicular to $R$, and (2) the semicircles with center on $R$. Show that given any line $L$ and any point $P$ not on $L$, there are infinitely many lines through $P$ which do not intersect $L$. Show that if $ ABC$ is a triangle, then the sum of its angles lies in the interval $(0, \pi)$.

Let $n$ be a positive integer. Consider $2n$ distinct lines on the plane, no two of which are parallel. Of the $2n$ lines, $n$ are colored blue, the other $n$ are colored red. Let $\mathcal{B}$ be the set of all points on the plane that lie on at least one blue line, and $\mathcal{R}$ the set of all points on the plane that lie on at least one red line. Prove that there exists a circle that intersects $\mathcal{B}$ in exactly $2n - 1$ points, and also intersects $\mathcal{R}$ in exactly $2n - 1$ points. [i]Proposed by Pakawut Jiradilok and Warut Suksompong, Thailand[/i]

Several strips and a circle of radius $1$ are drawn on the plane. The sum of the widths of the strips is $100$. Prove that one can translate each strip parallel to itself so that together they cover the circle. (M Smurov )

We are given $101$ rectangles with sides of integer lengths not exceeding $100$ . Prove that among these $101$ rectangles there are $3$ rectangles, say $A , B$ and $C$ such that $A$ will fit inside $B$ and $B$ inside $C$. ( N . Sedrakyan, Yerevan)

a) In a convex $13$-gon all diagonals are drawn, dividing it into smaller polygons. What is the greatest number of sides can these polygons have? b) In a convex $1950$-gon all diagonals are drawn, dividing it into smaller polygons. What is the greatest number of sides can these polygons have?

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?

* Given a regular dodecahedron. Find how many ways are there to draw a plane through it so that its section of the dodecahedron is a regular hexagon?

A set of $n$ points in Euclidean 3-dimensional space, no four of which are coplanar, is partitioned into two subsets $\mathcal{A}$ and $\mathcal{B}$. An $\mathcal{AB}$-tree is a configuration of $n-1$ segments, each of which has an endpoint in $\mathcal{A}$ and an endpoint in $\mathcal{B}$, and such that no segments form a closed polyline. An $\mathcal{AB}$-tree is transformed into another as follows: choose three distinct segments $A_1B_1$, $B_1A_2$, and $A_2B_2$ in the $\mathcal{AB}$-tree such that $A_1$ is in $\mathcal{A}$ and $|A_1B_1|+|A_2B_2|>|A_1B_2|+|A_2B_1|$, and remove the segment $A_1B_1$ to replace it by the segment $A_1B_2$. Given any $\mathcal{AB}$-tree, prove that every sequence of successive transformations comes to an end (no further transformation is possible) after finitely many steps.

On the plane are chosen six points. Prove that the ratio of the longest distance between two points to the shortest is at least $\sqrt3$.

Prove that there are different points $A_0 \,\, ,A_1 \,\, , \cdots A_{2550}$ on the $XY$ plane corresponding to the following properties simultaneously. (i) Any three points are not on the same line. (ii) If $ d(A_i,A_j)$ represents the distance between $A_i\,\, , A_j $ then $$ \sum_{0 \leq i < j \leq 2550}\{d(A_i,A_j)\} < 10^{-2008}$$ Note : $ \{x \}$ represents the decimal part of x e.g. $ \{ 3.16\} = 0.16$. [i] (passer-by)[/i]

A convex $n$-side polygon has angles $v_1,v_2,\dots,v_n$ (in degrees), where all $v_k$ ($k = 1,2,\dots,n$) are positive integers divisible by $36$. (a) Determine the largest $n$ for which this is possible. (b) Show that if $n>5$, two of the sides of the $n$-polygon must be parallel.

A circle is divided into $7$ arcs. The sum of the angles subtending any two neighbouring arcs is no more than $103^o$. Find the maximal number $A$ such that any of the $7$ arcs is subtended by no less than $A^o$. Prove that this value $A$ is really maximal. (A. Tolpygo, Kiev)

Alma places spies on some of the squares on a $2020\times 2020$ game board. Now Bertha secretly chooses a quadradic area consisting of $1020 \times 1020$ squares and tells Alma which spies are standing on a square in the secret quadradic area. At least how many spies must Alma have placed in order for her to determine with certainty which area Bertha has chosen?

An segment $S$ of length $50$ is covered by several segments of length $1$ , all of them contained in $S$. If any of these unit segments were removed, $S$ would no longer be completely covered. Find the maximum number of unit segments with this property. Clarification: Assume that the segments include their endpoints.

Show for any positive integer $n$ that there exists a circle in the plane such that there are exactly $n$ grid points within the circle. (A grid point is a point having integer coordinates.)

Suppose we have $2016$ points in a $2$-dimensional plane such that no three lie on a line. Two quadrilaterals are not disjoint if they share an edge or vertex, or if their edges intersect. Show that there are at least $504$ quadrilaterals with vertices among these points such that any two of the quadrilaterals are disjoint.

A convex $n$-vertex polygon is partitioned into triangles by nonintersecting diagonals . The following operation, called perestroyka (=reconstruction) , is allowed: two triangles $ABD$ and $BCD$ with a common side may be replaced by the triangles $ABC$ and $ACD$. By $P(n)$ denote the smallest number of perestroykas needed to transform any partitioning into any other one. Prove that (a) $P ( n ) \ge n - 3$ (b) $P (n) \le 2n - 7$ (c) $P(n) \le 2n - 10$ if $n \ge 13$ . ( D.Fomin , based on ideas of W. Thurston , D . Sleator, R. Tarjan)