$2015$ points are given in a plane such that from any five points we can choose two points with distance less than $1$ unit. Prove that $504$ of the given points lie on a unit disc.

Define a regular $n$-pointed star to be a union of $n$ lines segments $P_1P_2, P_2P_3, ..., P_nP_1$ such that $\bullet$ the points $P_1,P_2,...,P_n$ are coplanar and no three of them are collinear, $\bullet$ each of the $n$ line segments intersects at least one of the other line segments at a point other than an endpoint, $\bullet$ all of the angles at $P_1, P_2,..., P_n$ are congruent , $\bullet$ all of the $n$ line segments $P_1P_2, P_2P_3, ..., P_nP_1$ are congruent, and $\bullet$ the path $P_1P_2...P_nP_1$ turns counterclockwise at an angle less than $180^o$ at each vertex. There are no regular $3$-pointed, $4$-pointed, or $6$-pointed stars. All regular $5$-pointed star are similar, but there are two non-similar regular $7$-pointed stars. Find all possible values of $n$ such that there are exactly $29$ non-similar regular $n$-pointed stars.

We are given an infinite set of rectangles in the plane, each with vertices of the form $(0, 0)$, $(0,m)$, $(n, 0)$ and $ (n,m)$, where $m$ and $n$ are positive integers. Prove that there exist two rectangles in the set such that one contains the other.

A planar polygon $A_1A_2A_3 . . .A_{n-1}A_n$ ($n > 4$) is made of rigid rods that are connected by hinges. Is it possible to bend the polygon (at hinges only!) into a triangle?

Let $X= \{A_1, A_2, A_3, A_4\}$ be a set of four distinct points in the plane. Show that there exists a subset $Y$ of $X$ with the property that there is no (closed) disk $K$ such that $K\cap X = Y$.

Given $2014$ points in the plane, no three of which are collinear, what is the minimum number of line segments that can be drawn connecting pairs of points in such a way that adding a single additional line segment of the same sort will always produce a triangle of three connected points?

Let $A_1 A_2…A_{66}$ be a convex 66-gon. What’s the greatest number of pentagons $A_i A_{i+1} A_{i+2} A_{i+3} A_{i+4},1\leq i\leq 66,$ which have an inscribed circle? ($A_{66+i}\equiv A_i$).

Lines parallel to the sides of an equilateral triangle are drawn so that they cut each of the sides into n equal segments and the triangle into n congruent triangles. Each of these n triangles is called a “cell”. Also lines parallel to each of the sides of the original triangle are drawn through each of the vertices of the original triangle. The cells between any two adjacent parallel lines form a “stripe”. (a) If $n =10$, what is the maximum number of cells that can be chosen so that no two chosen cells belong to one stripe? (b)The same question for $n = 9$. (R Zhenodarov)

Cut each of the equilateral triangles with sides $2$ and $3$ into three parts and construct an equilateral triangle from all received parts.

An ant crawls along the edges of a cube turning only at its vertices. It has visited one of the vertices $25$ times. Is it possible that it has visited each of the other $7$ vertices exactly $20$ times? (S Tokarev)

Supppose five points in a plane are situated so that no two of the straight lines joining them are parallel, perpendicular, or coincident. From each point perpendiculars are drawn to all the lines joining the other four points. Determine the maxium number of intersections that these perpendiculars can have.

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.

Find all rectangles that can be cut into $13$ equal squares.

A $23 \times 23$ square is tiled with $1 \times 1, 2 \times 2$ and $3 \times 3$ squares. What is the smallest possible number of $1 \times 1$ squares?

$D$ is a closed disk radius $R$. Show that among any $8$ points of $D$ one can always find two whose distance apart is less than $R$.

In a plane $5$ points are given such that all triangles having vertices at these points are of area not greater than $1$. Show that there exists a trapezoid which contains all point in the interior (or on the sides) and having the area not exceeding $3$.

There are three polygons and the area of each one is $3$. They are drawn inside a square of area $6$. Find the greatest value of $m$ such that among those three polygons, we can always find two polygons so that the area of their overlap is not less than $m$.

Does there exist a set of $100$ triangles in which not one of the triangles can be covered by the other $99$?

Turki has divided a square into finitely many white and green rectangles, each with sides parallel to the sides of the square. Within each white rectangle, he writes down its width divided by its height. Within each green rectangle, he writes down its height divided by its width. Finally, he calculates $S$, the sum of these numbers. If the total area of white rectangles equals the total area of green rectangles, determine the minimum possible value of $S$.

Let $n$ be a positive integer. For a set of points in the plane $M$, we call $2$ distinct points $A,B \in M$ [i]connected[/i] if the line $AB$ contains exactly $n+1$ points from $M$. Find the minimum value of a positive integer $m$ such that there exists a set $M$ of $m$ points in the plane with the property that any point $A \in M$ is connected with exactly $2n$ other points from $M$.

A rectangle is dissected into several smaller rectangles. Is it possible that for each pair of these rectangles, the line segment connecting their centers intersects some third rectangle?

* Given an infinite cone. The measure of its unfolding’s angle is equal to $\alpha$. A curve on the cone is represented on any unfolding by the union of line segments. Find the number of the curve’s self-intersections.

Let $n\geq3$ be a positive integer. Let $C_1,C_2,C_3,\ldots,C_n$ be unit circles in the plane, with centres $O_1,O_2,O_3,\ldots,O_n$ respectively. If no line meets more than two of the circles, prove that \[ \sum\limits^{}_{1\leq i<j\leq n}{1\over O_iO_j}\leq{(n-1)\pi\over 4}. \]

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 points are placed in the centers of a $4 \times 4$ chess table in the following way: • • • • • • • • • • • • • • • • (a) Prove that one may choose $6$ points such that no isoceles triangle can be drawn with the vertices at these points. (b) Prove that one cannot choose $7$ points with the above property.