Consider a set of $2016$ distinct points in the plane, no four of which are collinear. Prove that there is a subset of $63$ points among them such that no three of these $63$ points are collinear.

One day, Professor Umzar Azum decided to fry dumplings for dinner. He took out a frying pan, opened a pack of dumplings, but suddenly thought about the question: how many dumplings could he fit in his frying pan? Measuring the sizes of the frying pan and dumplings, the professor came to the conclusion that the dumplings have the shape of a semicircle, the diameter of which is four times smaller than the diameter of the frying pan. Show how on the frying pan it is possible to place (without overlap): a) $20$ pieces of dumplings; b) $24$ pieces of dumplings; . (The problem boils down to placing, without overlapping, the appropriate number of identical semicircles inside a circle with a diameter four times larger.) [i]Note: We (the authors of the problem) do not know the answer to the question whether it is possible to place 25 semicircles in a circle with a diameter four times smaller, and even more so we do not know what the largest number of such semicircles is. We will welcome any progress in solving the problem and evaluate it accordingly. [/i]

Can one paint four vertices of a cube red and the other four points black so that any plane passing through three points of the same colour contains a vertex of the other colour? (Mebius, Sharygin)

A $100\times \sqrt{3}$ rectangular table is given. What is the minimum number of disk-shaped napkins of radius $1$ required to cover the table completely? [i]Remark:[/i] The napkins are allowed to overlap and protrude the table's edges.

For which positive integers $n \ge 3$ is it possible to mark $n$ points of a plane in such a way that, starting from one marked point and moving on each step to the marked point which is the second closest to the current point, one can walk through all the marked points and return to the initial one? For each point, the second closest marked point must be uniquely determined.

Determine alle positive integers $n>1$ with the following property: For each colouring of the lattice points in the plane with $n$ colours, there are three lattice points of the same colour forming an isosceles right triangle with legs parallel to the coordinate axes.

A cube with the edge of length $n$ is divided onto $n^3$ unit ones. Let us choose some of them and draw three lines parallel to the edges through their centres. What is the least possible number of the chosen small cubes necessary to make those lines cross all the smaller cubes? a) Find the answer for the small $n$ ($n = 2,3,4$). b) Try to find the answer for $n = 10$. c) If You can not solve the general problem, try to estimate that value from the upper and lower side. d) Note, that You can reformulate the problem in such a way: Consider all the triples $(x_1,x_2,x_3)$, where $x_i$ can be one of the integers $1,2,...,n$. What is the minimal number of the triples necessary to provide the property: [i]for each of the triples there exist the chosen one, that differs only in one coordinate. [/i] Try to find the answer for the situation with more than three coordinates, for example, with four.

Sebastian has a certain number of rectangles with areas that sum up to 3 and with side lengths all less than or equal to $1$. Demonstrate that with each of these rectangles it is possible to cover a square with side $1$ in such a way that the sides of the rectangles are parallel to the sides of the square. [b]Note:[/b] The rectangles can overlap and they can protrude over the sides of the square.

Let $S$ be a set of $n$ points in the plane such that for any two points $(a, b), (c, d) \in S$, we have that $| a - c | \cdot | b - d | \ge 1$. Show that [list] [*] (a) If $S = \{ Q_1, Q_2, Q_3\}$ such that for any point $Q_i$ in $S$, this point doesn't lie in the axis-aligned rectangle with corners as the other two points, show that the area of $\triangle Q_1Q_2Q_3$ is at least $\frac{\sqrt{5}}{2}$. [*] (b) If all points in $S$ lie in an axis-aligned square with sidelength $a$, then $|S| \le \frac{a^2}{\sqrt{5}} + 2a + 1$. [/list]

Let's call it a [i] staircase of height [/i]$n$, a figure consisting from all square cells $n\times n$ lying no higher diagonals (the figure shows a [i]staircase of height [/i] $4$ ). In how many different ways can a [i]staircase of height[/i] $n$ can be divided into several rectangles whose sides go along the grid lines, but the areas are different in pairs? [img]https://cdn.artofproblemsolving.com/attachments/f/0/f66d7e9ada0978e8403fbbd8989dc1b201f2cd.png[/img]

Let $n$ be an even positive integer. We consider rectangles with integer side lengths $k$ and $k +1$, where $k$ is greater than $\frac{n}{2}$ and at most equal to $n$. Show that for all even positive integers $ n$ the sum of the areas of these rectangles equals $$\frac{n(n + 2)(7n + 4)}{24}.$$

A point $M (x, y)$ of the Cartesian plane of $xOy$ coordinates is called [i]lattice [/i] if it has integer coordinates. Each lattice point is colored red or blue. Prove that in the plan there is at least one rectangle with lattice vertices of the same color.

Suppose there are $997$ points given in a plane. If every two points are joined by a line segment with its midpoint coloured in red, show that there are at least $1991$ red points in the plane. Can you find a special case with exactly $1991$ red points?

A single segment contains several non-intersecting red segments, the total length of which is greater than $0.5$. Are there necessarily two red dots at the distance: a) $1/99$ b) $1/100$ ?

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

Let $\,S\,$ be a finite set of points in three-dimensional space. Let $\,S_{x},\,S_{y},\,S_{z}\,$ be the sets consisting of the orthogonal projections of the points of $\,S\,$ onto the $yz$-plane, $zx$-plane, $xy$-plane, respectively. Prove that \[ \vert S\vert^{2}\leq \vert S_{x} \vert \cdot \vert S_{y} \vert \cdot \vert S_{z} \vert, \] where $\vert A \vert$ denotes the number of elements in the finite set $A$. [hide="Note"] Note: The orthogonal projection of a point onto a plane is the foot of the perpendicular from that point to the plane. [/hide]

On the map, the Flower City has the form of a right triangle $ABC$ (see Fig.1). The length of each leg is $6$ meters. All the streets of the city run parallel to one of the legs at a distance of $1$ meter from each other. A river flows along the hypotenuse. From their houses that are located at points $V$ and $S$, at the same time get the Cog and Tab. Each short moves to rivers according to the following rule: tosses his coin, and if the [b]heads[/b] falls, he passes $1$ meter parallel to the leg $AB$ to the north (up), and if tails, then passes $1$ meter parallel to the leg $AC$ on east (right). If the Cog and the Tab meet at the same point, then they move together, tossing a coin. a) Which is more likely: Cog and Tab will meet on the way to the river, or will they come to different points on the shore? b) At what point near the river should the Stranger sit, if he wants the most did Gvintik and Shpuntik come to him together? [img]https://cdn.artofproblemsolving.com/attachments/d/c/5d6f75d039e8f2dd6a0ddfe6c4cb046b83f24c.png[/img] [hide=original wording] На мапi Квiткове мiсто має вигляд прямокутного трикутника ABC (див. рисунок 1). Довжина кожного катету – 6 метрiв. Всi вулицi мiста проходять паралельно одному за катетiв на вiдстанi 1 метра одна вiд одної. Вздовж гiпотенузи тече рiка. Зi своїх будиночкiв, що знаходяться в точках V та S, одночасно виходять Гвинтик та Шпунтик. Кожен коротулька рухається до рiчки за таким правилом: пiдкидає свою монетку, та якщо випадає Орел, вiн проходить 1 метр паралельно катету AB на пiвнiч (вгору), а якщо Решка, то проходить 1 метр паралельно катету AC на схiд (вправо). Якщо Гвинтик та Шпунтик зустрiчаються в однiй точцi, то далi вони рушають разом, пiдкидаючи монетку Гвинтика. 1. Що бiльш ймовiрно: Гвинтик та Шпунтик зустрiнуться на шляху до рiки, або вони прийдуть у рiзнi точки берега? 2. В якiй точцi бiля рiки має сидiти Незнайка, якщо вiн хоче, щоб найбiльш ймовiрно до нього прийшли Гвинтик та Шпунтик разом?[/hide]

There are $ n \geq4 $ points on the plane, the distance between any two of which is an integer. Prove that there are at least $ \frac {1} {6} $ distances, each of which is divisible by $3$.

In how many ways can the ”Nikolaus’ House” (see the picture) be drawn? Edges may not be erased nor duplicated, and no additional edges may be drawn. [img]https://cdn.artofproblemsolving.com/attachments/0/5/33795820e0335686b06255180af698e536a9be.png[/img]

In the plane there are $2014$ plotted points, such that no $3$ are collinear. For each pair of plotted points, draw the line that passes through them. prove that for every three of marked points there are always two that are separated by an amount odd number of lines.

Consider a polyhedron having $100$ edges. (a) Find the maximal possible number of its edges which can be intersected by a plane (not containing any vertices of the polyhedron) if the polyhedron is convex. (b) Prove that for a non-convex polyhedron this number i. can be as great as $96$, ii. cannot be as great as $100$. (A Andjans, Riga

The $2001 \times 2001$ trees in a park form a square grid. What is the largest number of trees that can be cut so that no tree stump can be seen from any other? (Each tree has zero width.)

Every vertex of a polygon has both integer coordinates; the length of each side of this polygon is a natural number. Prove that the perimeter of the polygon is an even number.

For which integers $N\ge 3$ can we find $N$ points on the plane such that no three are collinear, and for any triangle formed by three vertices of the points’ convex hull, there is exactly one point within that triangle?

On a $1000\times 1000$-board we put dominoes, in such a way that each domino covers exactly two squares on the board. Moreover, two dominoes are not allowed to be adjacent, but are allowed to touch in a vertex. Determine the maximum number of dominoes that we can put on the board in this way. [i]Attention: you have to really prove that a greater number of dominoes is impossible. [/i]