On a circle with radius $1$ the points $A_1, A_2,..., A_n$ lie such that every arc $A_iA_{i+i}$ has length $\frac{2\pi}{n}= a$. Given that there exists a set $V$ consisting of $ k$ of these points ($k < n$), which has the property that each of the arc lengths $a$, $2a$$,...$, $(n- 1)a$ can be obtained in exactly one way be taken as the length of an arc traversed in a positive sense, beginning and ending in a point of $V$. Express $n$ in terms of $k$ and give the set $V$ for the case $n = 7$.

$100$ red points divide a blue circle into $100$ arcs such that their lengths are all positive integers from $1$ to $100$ in an arbitrary order. Prove that there exist two perpendicular chords with red endpoints.

Find the number of non-isosceles triangles (up to congruence) with integral side lengths, in which the sum of the two shorter sides is $19$.

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]

Each grid point of a cartesian plane is colored with one of three colors, whereby all three colors are used. Show that one can always find a right-angled triangle, whose three vertices have pairwise different colors.

The plane is divided into equilateral triangles of side length $1$. Consider a equilateral triangle of side length $n$ whose sides lie on the grid lines. On every grid point on the edge and inside of this triangle lies a stone. In a move, a unit triangle is selected, which has exactly $2$ corners with is covered with a stone. The two stones are removed, and the third corner is turned a new stone was laid. For which $n$ is it possible that after finitely many moves only one stone left?

A rectangle is tiled with rectangles of size $6\times 1$. Prove that one of its side lengths is divisible by $6$.

Prove that for any natural number $ m $ there exists a natural number $ n $ such that any $ n $ distinct points on the plane can be partitioned into $ m $ non-empty sets whose [i]convex hulls[/i] have a common point. The [i] convex hull [/i] of a finite set $ X $ of points on the plane is the set of points lying inside or on the boundary of at least one convex polygon with vertices in $ X $, including degenerate ones, that is, the segment and the point are considered convex polygons. No three vertices of a convex polygon are collinear. The polygon contains its border.

There are nine points in the data square, of which no three are collinear. Prove that three of them are vertices of a triangle with an area not exceeding $ \frac{1}{8} $ the area of a square.

Pete claims that he can draw $3$ segments of length $1$ and a circle of radius less than $\sqrt3 / 3$ on a piece of paper, such that all segments would lie inside the circle and there would be no line that intersects each of $3 $ segments. Is Pete right?

On a $ 9\times 9$ board, divided into $1\times 1$ squares, pieces of the form Each piece covers exactly $3$ squares. (a) Starting from the empty board, what is the maximum number of pieces that can be placed? (b) Starting from the board with $3$ pieces already placed as shown in the diagram below, what is the maximum number of pieces that can be placed? [img]https://cdn.artofproblemsolving.com/attachments/d/4/3bd010828accb2d1811d49eb17fa69662ff60d.gif[/img]

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 $\sqrt{3}.$

Fix an integer $n \ge 2$, let $Q_n$ be the graph consisting of all vertices and all edges of an $n$-cube, and let $T$ be a spanning tree in $Q_n$. Show that $Q_n$ has an edge whose adjunction to $T$ produces a simple cycle of length at least $2n$.

What is the largest number of obtuse triangles that can be composed of $16$ different segments (each triangle is composed of three segments), if the largest of these segments does not exceed twice the smallest?

(a) A radio lamp has a $7$-contact plug, with the contacts arranged in a circle. The plug is inserted into a socket with $7$ holes. Is it possible to number the contacts and the holes so that for any insertion at least one contact would match the hole with the same number? (b) A radio lamp has a $20$-contact plug, with the contacts arranged in a circle. The plug is inserted into a socket with $20$ holes. Let the contacts in the plug and the socket be already numbered. Is it always possible to insert the plug so that none of the contacts matches its socket?

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)

$60$ points lie on the plane, such that no three points are collinear. Prove that one can divide the points into $20$ groups, with $3$ points in each group, such that the triangles ( $20$ in total) consist of three points in a group have a non-empty intersection.

Each point on a circle is colored with one of $10$ colors. Is it true that for any coloring there are $4$ points of the same color that are vertices of a quadrilateral with two parallel sides (an isosceles trapezoid or a rectangle)?

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]

Inside a square with side $60$, $121$ points are drawn. Prove them are three points that are vertices of a triangle of area not exceeding $30$.

On a straight line lie $100$ points and another point outside the line. Which is the biggest the number of isosceles triangles can be formed from the vertices of these $101$ points?

On a circle $4n$ points are chosen ($n \ge 1$). The points are alternately colored yellow and blue. The yellow points are divided into $n$ pairs and the points in each pair are connected with a yellow line segment. In the same manner the blue points are divided into $n$ pairs and the points in each pair are connected with a blue segment. Assume that no three of the segments pass through a single point. Show that there are at least $n$ intersection points of blue and yellow segments.

For a given set of points in space it is allowed to mirror a point from the set with respect to another point from the set, and to include the image in the set. Starting with a set of seven vertices of a cube, is it possible to include the eight vertex in the set after finitely many such steps?

The enemy ship has landed on a $9\times 9$ board that covers exactly $5$ squares of the board, like this: [img]https://cdn.artofproblemsolving.com/attachments/2/4/ae5aa95f5bb5e113fd5e25931a2bf8eb872dbe.png[/img] The ship is invisible. Each defensive missile covers exactly one square, and destroys the ship if it hits one of the $5$ squares that it occupies. Determine the minimum number of missiles needed to destroy the enemy ship with certainty .

Given a chess-board $99\times 99$ with a set $F$ of fields marked on it (the set is different in three tasks). There is a beetle sitting on every field of the set $F$. Suddenly all the beetles have raised into the air and flied to another fields of the same set. The beetles from the neighbouring fields have landed either on the same field or on the neighbouring ones (may be far from their starting point). (We consider the fields to be neighbouring if they have at least one common vertex.) Consider a statement: [i]"There is a beetle, that either stayed on the same field or moved to the neighbouring one".[/i] Is it always valid if the figure $F$ is: a) A central cross, i.e. the union of the $50$-th row and the $50$-th column? b) A window frame, i.e. the union of the $1$-st, $50$-th and $99$-th rows and the $1$-st, $50$-th and $99$-th columns? c) All the chess-board?