For all nonempty finite sets of point $S$ on the plane satisfying: $|S|$ is even and for all partitions of $S$ into two subsets $A,B$ of equal size, there is a reflection that maps $A$ to $B$.

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)?

$6$ open disks in the plane are such that the center of no disk lies inside another. Show that no point lies inside all $6$ disks.

Fernanda decided to decorate a square blanket with a ribbon and buttons, placing a button in the center of each square where the ribbon passes and forming the design indicated in the figure. If Fernanda sews the first button in the shaded square on line $0$, on which line does she sew the $2007$th button? [img]https://cdn.artofproblemsolving.com/attachments/2/9/0c9c85ec6448ee3f6f363c8f4bcdd5209f53f6.png[/img]

Prove that any identity that holds for every finite $ n$-distributive lattice also holds for the lattice of all convex subsets of the $ (n\minus{}1)$-dimensional Euclidean space. (For convex subsets, the lattice operations are the set-theoretic intersection and the convex hull of the set-theoretic union. We call a lattice $ n$-$ \textit{distributive}$ if \[ x \wedge (\bigvee_{i\equal{}0}^n y_i)\equal{}\bigvee_{j\equal{}0}^n(x \wedge (\bigvee_{0\leq i \leq n, \;i \not\equal{} j\ }y_i))\] holds for all elements of the lattice.) [i]A. Huhn[/i]

The plane is divided into unit cells, and each of the cells is painted in one of two given colors. Find the minimum possible number of cells in a figure consisting of entire cells which contains each of the $16$ possible colored $2\times2$ squares.

What is the smallest number of $3$-cell corners needed to be painted in a $6\times 6$ square so that it was impossible to paint more than one corner of it? (The painted corners should not overlap.)

There is a rectangular sheet of paper on an infinite blackboard. Marvin secretly chooses a convex $2024$-gon $P$ that lies fully on the piece of paper. Tigerin wants to find the vertices of $P$. In each step, Tigerin can draw a line $g$ on the blackboard that is fully outside the piece of paper, then Marvin replies with the line $h$ parallel to $g$ that is the closest to $g$ which passes through at least one vertex of $P$. Prove that there exists a positive integer $n$, independent of the choice of the polygon, such that Tigerin can always determine the vertices of $P$ in at most $n$ steps.

The eight points below are the vertices and the midpoints of the sides of a square. We would like to draw a number of circles through the points, in such a way that each pair of points lie on (at least) one of the circles. Determine the smallest number of circles needed to do this. [asy] unitsize(1 cm); dot((0,0)); dot((1,0)); dot((2,0)); dot((0,1)); dot((2,1)); dot((0,2)); dot((1,2)); dot((2,2)); [/asy]

Every side and diagonal of a regular $12$-gon is colored in one of $12$ given colors. Can this be done in such a way that, for every three colors, there exist three vertices which are connected to each other by segments of these three colors?

What $n$ is the smallest such that “there is a $n$-gon that can be cut into a triangle, a quadrilateral, ..., a $2006$-gon''?

Suppose there are $2019$ distinct points in a plane and the distances between pairs of them attain $k$ different values. Prove that $k$ is at least $44$.

Is it possible to cut a square into five squares?

An inscribed $n$-gon ($n > 3$), is divided into $n-2$ triangles by diagonals which meet only in vertices. What is the maximum possible number of congruent triangles obtained? (An inscribed $n$-gon is an $n$-gon where all its vertices lie on a circle) [i]Proposed by Boris Frenkin - Russia[/i]

(a) On each of the $1 \times 1$ squares of the top row of an $8 \times 8$ chessboard there is a black pawn, and on each of the $1 \times 1$ squares of the bottom row of this chessboard there is a white pawn. On each move one can shift any pawn vertically or horizontally to any adjacent empty $1 \times 1$ square. What is the smallest number of moves that are needed to move all white pawns to the top row and all black pawns to the bottom one? (b) The same question for a $7 \times 7$ board. (A Shapovalov_

Is it possible to write natural numbers at all points of the plane with integer coordinates so that three points with integer coordinates lie on the same line if and only if the numbers written in them had a common divisor greater than one?

Several triangles are [b]intersecting[/b] if any two of them have non-empty intersections. Show that for any two finite collections of intersecting triangles, there exists a line that intersects all the triangles. [i] Proposed by usjl[/i]

Let $n$ be an integer greater than $3$. Show that it is possible to divide a square into $n^2 + 1$ or more disjointed rectangles and with sides parallel to those of the square so that any line parallel to one of the sides intersects at most the interior of $n$ rectangles. Note: We say that two rectangles are [i]disjointed [/i] if they do not intersect or only intersect at their perimeters.

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?

For which values of $n$ does there exist a convex polyhedron with $n$ edges?

It is known that from these five segments it is possible to form four different right triangles. Find the ratio of the largest segment to the smallest.

Find the greatest number of depicted pieces composed of $4$ unit squares that can be placed without overlapping on an $n \times n$ grid (where n is a positive integer) in such a way that it is possible to move from some corner to the opposite corner via uncovered squares (moving between squares requires a common edge). The shapes can be rotated and reflected. [img]https://cdn.artofproblemsolving.com/attachments/b/d/f2978a24fdd737edfafa5927a8d2129eb586ee.png[/img]

In a circle with a radius of $1$, $64$ mutually different points are selected. Prove that $10$ mutually different points can be selected from them, which lie in a circle with a radius $\frac12$.

Let $P_1$ be a convex polyhedron with vertices $A_1,A_2,\ldots,A_9$. Let $P_i$ be the polyhedron obtained from $P_1$ by a translation that moves $A_1$ to $A_i$. Prove that at least two of the polyhedra $P_1,P_2,\ldots,P_9$ have an interior point in common.

A triangle is constructed with the lengths of the sides chosen from the set $\{2, 3, 5, 8, 13, 21, 34, 55, 89, 144\}$. Show that this triangle must be isosceles. (A triangle is isosceles if it has at least two sides the same length.)