Let $n \geq 2$ be an integer. There are $n$ distinct circles in general position, that is, any two of them meet in two distinct points and there are no three of them meeting at one point. Those circles divide the plane in limited regions by circular edges, that meet at vertices (note that each circle have exactly $2n-2$ vertices). For each circle, temporarily color its vertices alternately black and white (note that, doing this, each vertex is colored twice, one for each circle passing through it). If the two temporarily colouring of a vertex coincide, this vertex is definitely colored with this common color; otherwise, it will be colored with gray. Show that if a circle has more than $n-2 + \sqrt{n-2}$ gray points, all vertices of some region are grey. Observation: In this problem, a region cannot contain vertices or circular edges on its interior. Also, the outer region of all circles also counts as a region.

For dessert, Melinda eats a spherical scoop of ice cream with diameter $2$ inches. She prefers to eat her ice cream in cube-like shapes, however. She has a special machine which, given a sphere placed in space, cuts it through the planes $x = n$, $y = n$, and $z = n$ for every integer $n$ (not necessarily positive). Melinda centers the scoop of ice cream uniformly at random inside the cube $0 \le x, y,z \le 1$, and then cuts it into pieces using her machine. What is the expected number of pieces she cuts the ice cream into?

Are there convex $n$-gonal ($n \ge 4$) and triangular pyramids such that the four trihedral angles of the $n$-gonal pyramid are equal trihedral angles of a triangular pyramid? [hide=original wording] Существуют ли выпуклая n-угольная (n>= 4) и треугольная пирамиды такие, что четыре трехгранных угла n-угольной пирамиды равны трехгранным углам треугольной пирамиды?[/hide]

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)?

Numbers from$ 1$ to $8$ were written at the vertices of the cube, and on each edge the absolute value of the difference between the numbers at its ends.. What is the smallest number of different numbers that can be written on the edges?

For which pairs $(m,n)$ does there exist an injective function $f:\mathbb{R}^2\to\mathbb{R}^2$ under which the image of every regular $m$-gon is a regular $n$-gon. (Note that $m,n\geq 3$, and that by a regular $N$-gon we mean the union of the boundary segments, not the closed polygonal region.) [i]Proposed by Sutanay Bhattacharya, Bishnupur, India[/i]

Let $n \geq 5$ be an integer. Consider $n$ squares with side lengths $1, 2, \dots , n$, respectively. The squares are arranged in the plane with their sides parallel to the $x$ and $y$ axes. Suppose that no two squares touch, except possibly at their vertices. Show that it is possible to arrange these squares in a way such that every square touches exactly two other squares.

Given any convex polygon, show that there are three consecutive vertices such that the polygon lies inside the circle through them.

A village is constructed in the form of a square, consisting of $9$ blocks , each of side length $\ell$, in a $3 \times 3$ formation . Each block is bounded by a bitumen road . If we commence at a corner of the village, what is the smallest distance we must travel along bitumen roads , if we are to pass along each section of bitumen road at least once and finish at the same corner? (Muscovite folklore)

Let $M$ be a finite subset of the plane such that for any two different points $A,B\in M$ there is a point $C\in M$ such that $ABC$ is equilateral. What is the maximal number of points in $M?$

Let $ K$ be a compact convex body in the $ n$-dimensional Euclidean space. Let $ P_1,P_2,...,P_{n\plus{}1}$ be the vertices of a simplex having maximal volume among all simplices inscribed in $ K$. Define the points $ P_{n\plus{}2},P_{n\plus{}3},...$ successively so that $ P_k \;(k>n\plus{}1)$ is a point of $ K$ for which the volume of the convex hull of $ P_1,...,P_k$ is maximal. Denote this volume by $ V_k$. Decide, for different values of $ n$, about the truth of the statement "the sequence $ V_{n\plus{}1},V_{n\plus{}2},...$ is concave." [i]L. Fejes- Toth, E. Makai[/i]

A plane is divided into $N$ regions by three families of parallel lines. No three lines pass through the same point. What is the smallest number of lines needed so that $N > 1999$?

One bird lives in each of $n$ bird-nests in a forest. The birds change nests, so that after the change there is again one bird in each nest. Also for any birds $A, B, C, D$ (not necessarily distinct), if the distance $AB < CD$ before the change, then $AB > CD$ after the change. Find all possible values of $n$.

Let $f(r)$ be the number of lattice points inside the circle radius $r$, center the origin. Show that $\lim_{r\to \infty} \frac{f(r)}{r^2}$ exists and find it. If the limit is $k$, put $g(r) = f(r) - kr^2$. Is it true that $\lim_{r\to \infty} \frac{g(r)}{r^h} = 0$ for any $h < 2$?

Given a sheet of tin $6\times 6$. It is allowed to bend it and to cut it but in such a way that it doesn’t fall to pieces. How to make a cube with edge $2$, divided by partitions into unit cubes?

a) There are $2n$ rays marked in a plane, with $n$ being a natural number. Given that no two marked rays have the same direction and no two marked rays have a common initial point, prove that there exists a line that passes through none of the initial points of the marked rays and intersects with exactly $n$ marked rays. (b) Would the claim still hold if the assumption that no two marked rays have a common initial point was dropped?

$(USS 5)$ Given $5$ points in the plane, no three of which are collinear, prove that we can choose $4$ points among them that form a convex quadrilateral.

Let $n\ge3$ be an integer. Prove that there is a set of $n$ points in the plane such that the distance between any two points is irrational and each set of three points determines a non-degenerate triangle with rational area.

Prove that every cube's cross-section, containing its centre, has the area not less then its face's area.

A number of $n$ lamps ($n\ge 3$) are put at $n$ vertices of a regular $n$-gon. Initially, all the lamps are off. In each step. Lisa will choose three lamps that are located at three vertices of an isosceles triangle and change their states (from off to on and vice versa). Her aim is to turn on all the lamps. At least how many steps are required to do so?

For each natural number $n\ge3$, let $m(n)$ be the maximum number of points inside or on the sides of a regular $n$-agon of side $1$ such that the distance between any two points is greater than $1$. Prove that $m(n)\ge n$ for $n>6$.

What greatest number of triples of points can be selected from $1955$ given points, so that each two triples have one common point?

Is it possible to tile space using a combination of regular tetrahedra and regular octahedra? (A Belov)

Consider six points in the interior of a square of side length $3$. Prove that among the six points, there are two whose distance is less than $2$.

We attempt to cover the plane with an infinite sequence of rectangles, overlapping allowed. (a) Is the task always possible if the area of the $n$th rectangle is $n^2$ for each $n$? (b) Is the task always possible if each rectangle is a square, and for any number $N$, there exist squares with total area greater than $N$?