For which values of $n$ is it possible to construct an $n$ by $n$ by $n$ cube with $n^3$ unit cubes, each of which is black or white, such that each cube shares a common face with exactly three cubes of the opposite colour? (S Tokarev)

Let $S=\{(x,y,z)\in \mathbb{Z}^3\}$ the set of points with integer coordinates in the space. Gugu has infinitely many solid spheres. All with radii $\ge (\frac{\pi}2)^3$. Is it possible for Gugu to cover all points of $S$ with his spheres?

Given a positive integer $n \ge 2$, let $\mathcal{P}$ and $\mathcal{Q}$ be two sets, each consisting of $n$ points in three-dimensional space. Suppose that these $2n$ points are distinct. Show that it is possible to label the points of $\mathcal{P}$ as $P_1,P_2,\dots,P_n$ and the points of $\mathcal{Q}$ as $Q_1,Q_2,\dots,Q_n$ such that for any indices $i$ and $j$, the balls of diameters $P_iQ_i$ and $P_jQ_j$ have at least one common point.

Prove that for any natural number $ m $ there exists a natural number $ n $ such that any $ n $ distinct points on the plane can be partitioned into $ m $ non-empty sets whose [i]convex hulls[/i] have a common point. The [i] convex hull [/i] of a finite set $ X $ of points on the plane is the set of points lying inside or on the boundary of at least one convex polygon with vertices in $ X $, including degenerate ones, that is, the segment and the point are considered convex polygons. No three vertices of a convex polygon are collinear. The polygon contains its border.

On a large chessboard $2n$ of its $1 \times 1$ squares have been marked such thar the rook (which moves only horizontally or vertically) can visit all the marked squares without jumpin over any unmarked ones. Prove that the figure consisting of all the marked squares can be cut into rectangles. (A Shapovalov)

A number of unit discs are given inside a square of side $100$ such that (i) no two of the discs have a common interior point, and (ii) every segment of length $10$, lying entirely within the square, meets at least one disc. Prove that there are at least $400$ discs in the square.

We are given $n$ discs in a plane, possibly overlapping, whose union has the area $1$. Prove that we can choose some of them which are mutually disjoint and have the total area greater than $1/9$.

Prove that in space there is a sphere containing exactly $1994$ points with integer coordinates.

Determine all integers $n \geq 3$ for which there exists a conguration of $n$ points in the plane, no three collinear, that can be labelled $1$ through $n$ in two different ways, so that the following condition be satisfied: For every triple $(i,j,k), 1 \leq i < j < k \leq n$, the triangle $ijk$ in one labelling has the same orientation as the triangle labelled $ijk$ in the other, except for $(i,j,k) = (1,2,3)$.

Suppose that $2002$ numbers, each equal to $1$ or $-1$, are written around a circle. For every two adjacent numbers, their product is taken; it turns out that the sum of all $2002$ such products is negative. Prove that the sum of the original numbers has absolute value less than or equal to $1000$. (The absolute value of $x$ is usually denoted by $|x|$. It is equal to $x$ if $x \ge 0$, and to $-x$ if $x < 0$. For example, $|6| = 6, |0| = 0$, and $|-7| = 7$.)

Consider an equilateral triangle with every side divided by $n$ points into $n+1$ equal parts. We put a marker on every of the $3n$ division points. We draw lines parallel to the sides of the triangle through the division points, and this way divide the triangle into $(n+1)^2$ smaller ones. Consider the following game: if there is a small triangle with exactly one vertex unoccupied, we put a marker on it and simultaneously take markers from the two its occupied vertices. We repeat this operation as long as it is possible. (a) If $n\equiv1\pmod3$, show that we cannot manage that only one marker remains. (b) If $n\equiv0$ or $n\equiv2\pmod3$, prove that we can finish the game leaving exactly one marker on the triangle.

The points of this lattice $4\times 4 = 16$ points can be vertices of squares. [asy] unitsize(1 cm); int i, j; for (i = 0; i <= 3; ++i) { draw((i,0)--(i,3)); draw((0,i)--(3,i)); } draw((1,1)--(2,2)--(1,3)--(0,2)--cycle); for (i = 0; i <= 3; ++i) { for (j = 0; j <= 3; ++j) { dot((i,j)); }} [/asy] Calculate the number of different squares that can be formed in a lattice of $100\times 100$ points.

$N^3$ unit cubes are made into beads by drilling a hole through them along a diagonal, put on a string and binded. Thus the cubes can move freely in space as long as the vertices of two neighboring cubes (including the first and last one) are touching. For which $N$ is it possible to build a cube of edge $N$ using these cubes?

Let $P = \{(x, y) | x, y \in \{0, 1, 2,... , 2015\}\}$ be a set of points on the plane. Straight wires of unit length are placed to connect points in $P$ so that each piece of wire connects exactly two points in $P$, and each point in $P$ is an endpoint of exactly one wire. Prove that no matter how the wires are placed, it is always possible to draw a straight line parallel to either the horizontal or vertical axis passing through midpoints of at least $506$ pieces of wire.

Given are $5$ segments, such that from any three of them one can form a triangle. Prove that from some three of them one can form an acute-angled triangle.

Two straight lines divide a square of side length $1$ into four regions. Show that at least one of the regions has a perimeter greater than or equal to $2$. [i]Proposed by Dylan Toh[/i]

Show that there exists a tetrahedron which can be partitioned into eight congruent tetrahedra, each of which is similar to the original one.

Show that there exists an integer $N$ such that for all $n \ge N$ a square can be partitioned into $n$ smaller squares.

Let $n> 2$ be a natural. Given $2n$ points in the plane, no $3$ are collinear. What is the maximum number of lines that can be drawn between them, without forming a triangle? [hide=original wording]Sea n > 2 un natural. Dados 2n puntos en el plano, tres a tres no colineales, Cual es el numero maximo de trazos que pueden dibujarse entre ellos, sin formar un triangulo?[/hide]

We are given $2001$ lines in the plane, no two of which are parallel and no three of which are concurrent. These lines partition the plane into regions (not necessarily finite) bounded by segments of these lines. These segments are called [i]sides[/i], and the collection of the regions is called a [i]map[/i]. Intersection points of the lines are called [i]vertices[/i]. Two regions are [i]neighbors [/i]if they share a side, and two vertices are neighbors if they lie on the same side. A [i]legal coloring[/i] of the regions (resp. vertices) is a coloring in which each region (resp. vertex) receives one color, such that any two neighboring regions (vertices) have different colors. (a) What is the minimum number of colors required for a legal coloring of the regions? (b) What is the minimum number of colors required for a legal coloring of the vertices?

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.

In a plane we have $n$ lines, no two of which are parallel or perpendicular, and no three of which are concurrent. A cartesian system of coordinates is chosen for the plane with one of the lines as the $x$-axis. A point $P$ is located at the origin of the coordinate system and starts moving along the positive $x$-axis with constant velocity. Whenever $P$ reaches the intersection of two lines, it continues along the line it just reached in the direction that increases its $x$-coordinate. Show that it is possible to choose the system of coordinates in such a way that $P$ visits points from all $n$ lines.

There are given 2050 points in a unit cube. Show that there are 5 points lying in an (open) ball with the radius 1/9.

There are distinct points $O, A, B, K_1, . . . , K_n, L_1, . . . , L_n$ on a plane such that no three points are collinear. The open line segments $K_1L_1, . . . , K_nL_n$ are coloured red, other points on the plane are left uncoloured. An allowed path from point $O$ to point $X$ is a polygonal chain with first and last vertices at points $O$ and $X$, containing no red points. For example, for $n = 1$, and $K_1 = (-1, 0)$, $L_1 = (1, 0)$, $O = (0,-1)$, and $X = (0,1)$, $OK_1X$ and $OL_1X$ are examples of allowed paths from $O$ to $X$, there are no shorter allowed paths. Find the least positive integer n such that it is possible that the first vertex that is not $O$ on any shortest possible allowed path from $O$ to $A$ is closer to $B$ than to $A$, and the first vertex that is not $O$ on any shortest possible allowed path from $O$ to $B$ is closer to $A$ than 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)?