Let $P_1P_2...P_n$ be an oriented closed polygonal line with no three segments passing through a single point. Each point $P_i$ is assinged the angle $180^o - \angle P_{i-1}P_iP_{i+1} \ge 0$ if $P_{i+1}$ lies on the left from the ray $P_{i-1}P_i$, and the angle $-(180^o -\angle P_{i-1}P_iP_{i+1}) < 0$ if $P_{i+1}$ lies on the right. Prove that if the sum of all the assigned angles is a multiple of $720^o$, then the number of self-intersections of the polygonal line is odd

On the plane a square is given, and $1993$ equilateral triangles are inscribed in this square. All vertices of any of these triangles lie on the border of the square. Prove that one can find a point on the plane belonging to the borders of no less than $499$ of these triangles. (N Sendrakyan)

Let $S$ be a finite set of points on a plane, where no three points are collinear, and the convex hull of $S$, $\Omega$, is a $2016-$gon $A_1A_2\ldots A_{2016}$. Every point on $S$ is labelled one of the four numbers $\pm 1,\pm 2$, such that for $i=1,2,\ldots , 1008,$ the numbers labelled on points $A_i$ and $A_{i+1008}$ are the negative of each other. Draw triangles whose vertices are in $S$, such that any two triangles do not have any common interior points, and the union of these triangles is $\Omega$. Prove that there must exist a triangle, where the numbers labelled on some two of its vertices are the negative of each other.

A rectangle $R$ is partitioned into smaller rectangles whose sides are parallel with the sides of $R$. Let $B$ be the set of all boundary points of all the rectangles in the partition, including the boundary of $R$. Let S be the set of all (closed) segments whose points belong to $B$. Let a maximal segment be a segment in $S$ which is not a proper subset of any other segment in $S$. Let an intersection point be a point in which $4$ rectangles of the partition meet. Let $m$ be the number of maximal segments, $i$ the number of intersection points and $r$ the number of rectangles. Prove that $m + i = r + 3$.

Given is a convex polygon $ P$ with $ n$ vertices. Triangle whose vertices lie on vertices of $ P$ is called [i]good [/i] if all its sides are unit length. Prove that there are at most $ \frac {2n}{3}$ [i]good[/i] triangles. [i]Author: Vyacheslav Yasinskiy, Ukraine[/i]

Some points with the integer coordinates are marked on the coordinate plane. Given a set of nonzero vectors. It is known, that if you apply the beginnings of those vectors to the arbitrary marked point, than there will be more marked ends of the vectors, than not marked. Prove that there is infinite number of marked points.

Let $n \geq 5$ be a given integer. Determine the greatest integer $k$ for which there exists a polygon with $n$ vertices (convex or not, with non-selfintersecting boundary) having $k$ internal right angles. [i]Proposed by Juozas Juvencijus Macys, Lithuania[/i]

Let $S_5$ be the set of permutations of $\{1,2,3,4,5\}$, and let $C$ be the convex hull of the set $$\{(\sigma(1),\sigma(2),\ldots,\sigma(5))\,|\,\sigma\in S_5\}.$$ Then $C$ is a polyhedron. What is the total number of $2$-dimensional faces of $C$?

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.

The target is a triangle divided by three families of parallel lines into $100$ equal regular triangles with single sides. A sniper shoots at a target. He aims at triangle and hits either it or one of the sides adjacent to it. He sees the results of his shooting and can choose when stop shooting. What is the greatest number of triangles he can with a guarantee of hitting five times?

In a square field of side length $12$ there is a source that contains a system of straight irrigation ditches. This is laid out in such a way that for every point of the field the distance to the next ditch is at most $1$. Here, the source is as a point and are the ditches to be regarded as stretches. It must be verified that the total length of the irrigation ditches is greater than $70$ m. The sketch shows an example of a trench system of the type indicated. [img]https://cdn.artofproblemsolving.com/attachments/6/5/5b51511da468cf14b5823c6acda3c4d2fe8280.png[/img]

You are given $2024$ yellow and $2024$ blue points on the plane, and no three of the points are on the same line. We call a pair of nonnegative integers $(a, b)$ [i]good[/i] if there exists a half-plane with exactly $a$ yellow and $b$ blue points. Find the smallest possible number of good pairs. The points that lie on the line that is the boundary of the half-plane are considered to be outside the half-plane. [i]Proposed by Anton Trygub[/i]

A set of $1985$ points is distributed around the circumference of a circle and each of the points is marked with $1$ or $-1$. A point is called “good” if the partial sums that can be formed by starting at that point and proceeding around the circle for any distance in either direction are all strictly positive. Show that if the number of points marked with $-1$ is less than $662$, there must be at least one good point.

There are $4n$ segments of unit length inside a circle radius $n$. Show that given any line $L$ there is a chord of the circle parallel or perpendicular to $L$ which intersects at least two of the $4n$ segments.

Does there exist a convex polyhedron which has a triangular section (by a plane not passing through the vertices) and each vertex of the polyhedron belonging to (a) no less than $ 5$ faces? (b) exactly $5$ faces? (G. Galperin)

Let two ants stand on the perimeter of a regular $2019$-gon of unit side length. One of them stands on a vertex and the other one is on the midpoint of the opposite side. They start walking along the perimeter at the same speed counterclockwise. The locus of their midpoints traces out a figure $P$ in the plane with $N$ corners. Let the area enclosed by the convex hull of $P$ be $\tfrac{A}{B}\tfrac{\sin^m\left(\tfrac{\pi}{4038}\right)}{\tan\left(\tfrac{\pi}{2019}\right)}$, where $A$ and $B$ are coprime positive integers, and $m$ is the smallest possible positive integer such that this formula holds. Find $A+B+m+N$. [i]Note:[/i] The [i]convex hull[/i] of a figure $P$ is the convex polygon of smallest area which contains $P$.

A $10 \times 10$ square on a grid is split by $80$ unit grid segments into $20$ polygons of equal area (no one of these segments belongs to the boundary of the square). Prove that all polygons are congruent. [i]($6$ points)[/i]

Is there a six-link broken line in space that passes through all the vertices of a given cube?

Every point on a line is painted either red or blue. Prove that there always exist three points $A,B,C$ that are painted the same color and are such that the point $B$ is the midpoint of the segment $AC$.

Each side face of a dodecahedron lies in a uniquely determined plane. Those planes cut the space in a finite number of disjoint [i]regions[/i]. Find the number of such regions.

Let $n$ be a natural number and $L = Z^2$ the set of points on the plane with integer coordinates. Every point in $L$ is colored now in one of the colors red or green. Show that there are $n$ different points $x_1,...,x_n \in L$ all of which have the same color and whose center of gravity is also in $L$ and is of the same color.

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

On a plane is given a finite set of points. Prove that the points can be covered by open squares $Q_1,Q_2,...,Q_n$ such that $1 \le\frac{N_j}{S_j} \le 4$ for $j = 1,...,n,$ where $N_j$ is the number of points from the set inside square $Q_j$ and $S_j$ is the area of $Q_j$.

A man gets lost in a large forest, the boundary of which is a straight line. (We can assume that the forest fills the half-plane.) It is known that the distance from a person to Granina forest does not exceed $2$ km. a) Suggest a path along which he will certainly be able to get out of the forest after walking no more than $14$ km. (Of course, a person does not know in which direction the border of the forest is, BUT he has the opportunity to move along any pre-selected curve. It is believed that a person left the forest as soon as he reached its border, while the border of the forest is invisible to him, no matter how close he would have approached it.) b) Find a path with the same property and length no more than $13$ km.

Given a cube with a side of $4$. Is it possible to completely cover $3$ of its faces, which have a common vertex, with sixteen rectangular paper strips measuring $1 \times3$?