We consider the ways to divide a $1$ by $1$ square into rectangles (of which the sides are parallel to those of the square). All rectangles must have the same circumference, but not necessarily the same shape. a) Is it possible to divide the square into 20 rectangles, each having a circumference of $2:5$? b) Is it possible to divide the square into 30 rectangles, each having a circumference of $2$?

There are $100$ circles of radius one in the plane. A triangle formed by the centres of any three given circles has area at most $2017$. Prove that there is a line intersecting at least three of the circles.

There are $12$ points on a circle. Four checkers, one red, one yellow, one green and one blue sit at neighboring points. In one move any checker can be moved four points to the left or right, onto the fifth point, if it is empty. If after several moves the checkers appear again at the four original points, how might their order have changed?

The distance between any two of six given points in the plane is at least $1$. Prove that the distance between some two points is at least $\sqrt{\frac{5+\sqrt5}{2}}$

Let $a$ and $S$ be the length of the side and the area of regular triangle inscribed in a circle of radius $1$. A closed broken line $A_1A_2...A_{51}A_1$ consisting of $51$ segments of the same length $a$ is placed inside the circle. Prove that the sum of areas of the $ 51$ triangles between the neighboring segments $$A_1A_2A_3, A_2A_3A_4,..., A_{49}A_{50}A_{51}, A_{50}A_{51}A_1, A_{51}A_1A_2$$ is not less than $3S$. (A. Berzinsh, Riga)

Show that we cannot tile a $10 x 10$ board with $25$ pieces of type $A$, or with $25$ pieces of type $B$, or with $25$ pieces of type $C$.

A convex $n$-gon is divided into triangles by diagonals which do not intersect except at vertices of the n-gon. Each vertex belongs to an odd number of triangles. Show that $n$ must be a multiple of $3$.

Using several white edge cubes of side $ 1$, Guille builds a large cube. Then he chooses $4$ faces of the big cube and paints them red. Finally, he takes apart the large cube and observe that the cubes with at least a face painted red is $431$. Find the number of cubes that he used to assemble the large cube. Analyze all the possibilities.

Let $C$ consist of a regular polygon and its interior. Show that for each positive integer $n$, there exists a set of points $S(n)$ in the plane such that every $n$ points can be covered by $C$, but $S(n)$ cannot be covered by $C.$

There are three straight roads. Three pedestrians are moving along those roads, and they are NOT on one line in the initial moment. Prove that they will be one line not more than twice

On an infinite “squared” sheet six squares are shaded as in the diagram. On some squares there are pieces. It is possible to transform the positions of the pieces according to the following rule: if the neighbour squares to the right and above a given piece are free, it is possible to remove this piece and put pieces on these free squares. The goal is to have all the shaded squares free of pieces. Is it possible to reach this goal if (a) In the initial position there are $6$ pieces and they are placed on the $6$ shaded squares? (b) In the initial position there is only one piece, located in the bottom left shaded square? [img]https://cdn.artofproblemsolving.com/attachments/2/d/0d5cbc159125e2a84edd6ac6aca5206bf8d83b.png[/img] (M Kontsevich, Moscow)

Into how many parts can a convex $n$-gon be divided by its diagonals if no three diagonals meet at one point?

Let $B$ be a set of more than $\tfrac{2^{n+1}}{n}$ distinct points with coordinates of the form $(\pm 1, \pm 1, \cdots, \pm 1)$ in $n$-dimensional space with $n \ge 3$. Show that there are three distinct points in $B$ which are the vertices of an equilateral triangle.

A rectangle is divided into $n^2$ smaller rectangle by $n - 1$ horizontal lines and $n - 1$ vertical lines. Between those rectangles there are exactly $5660$ which are not congruent. For what minimum value of $n$ is this possible?

An investor has two rectangular pieces of land of size $120\times 100$. a. On the first land, she want to build a house with a rectangular base of size $25\times 35$ and nines circular flower pots with diameter $5$ outside the house. Prove that even the flower pots positions are chosen arbitrary on the land, the remaining land is still sufficient to build the desired house. b. On the second land, she want to construct a polygonal fish pond such that the distance from an arbitrary point on the land, outside the pond, to the nearest pond edge is not over $5$. Prove that the perimeter of the pond is not smaller than $440-20\sqrt{2}$.

Consider a closed broken line of perimeter $1$ on a plane. Prove that a disc of radius $\frac14$ can cover this line.

We say a finite set $S$ of points with $|S|\ge3$ is [i]good[/i] if for any three distinct elements of $S$, they are non-collinear and the orthocenter of them is also in $S$. Find all good sets.

Let $n$ be a positive integer. E. Chen and E. Chen play a game on the $n^2$ points of an $n \times n$ lattice grid. They alternately mark points on the grid such that no player marks a point that is on or inside a non-degenerate triangle formed by three marked points. Each point can be marked only once. The game ends when no player can make a move, and the last player to make a move wins. Determine the number of values of $n$ between $1$ and $2013$ (inclusive) for which the first player can guarantee a win, regardless of the moves that the second player makes. [i]Ray Li[/i]

A ray of light reflects from the rays of a given angle. A ray that enters the vertex of the angle is absorbed. Prove that there is a natural number $n$ such that any ray can reflect at most $n$ times

Show how to divide space into (a) congruent tetrahedra, (b) congruent “equifaced” tetrahedra. (A tetrahedron is called equifaced if all its faces are congruent triangles.) (NB Vassiliev)

Let $A{}$ be a point in the Cartesian plane. At each step, Ann tells Bob a number $0\leqslant a\leqslant 1$ and he then moves $A{}$ in one of the four cardinal directions, at his choice, by a distance of $a{}.$ This process cotinues as long as Ann wishes. Amongst every 100 consecutive moves, each of the four possible moves should have been made at least once. Ann's goal is to force Bob to eventually choose a point at a distance greater than 100 from the initial position of $A.{}$ Can Ann achieve her goal? [i]Selected from an Argentine Olympiad[/i]

Given five segments. It is possible to construct a triangle of every subset of three of them. Prove that at least one of those triangles is acute-angled.

Given are $50$ points in the plane, no three of them belonging to a same line. Each of these points is colored using one of four given colors. Prove that there is a color and at least $130$ scalene triangles with vertices of that color.

A set of lines in the plan is called [i]nice [/i]i f every line in the set intersects an odd number of other lines in the set. Determine the smallest integer $k \ge 0$ having the following property: for each $2018$ distinct lines $\ell_1, \ell_2, ..., \ell_{2018}$ in the plane, there exist lines $\ell_{2018+1},\ell_{2018+2}, . . . , \ell_{2018+k}$ such that the lines $\ell_1, \ell_2, ..., \ell_{2018+k}$ are distinct and form a [i]nice [/i] set.

Three (not necessarily distinct) points in the plane which have integer coordinates between $ 1$ and $2020$, inclusive, are chosen uniformly at random. The probability that the area of the triangle with these three vertices is an integer is $a/b$ in lowest terms. If the three points are collinear, the area of the degenerate triangle is $0$. Find $a + b$.