Some of the faces of a convex polyhedron $M$ are painted in blue, others are painted in white and there are no two walls with a common edge. Prove that if the sum of surfaces of the blue walls is bigger than half surface of $M$ then it may be inscribed a sphere in the polyhedron given $(M)$. [i](H. Lesov)[/i]

Let $ z_{1},z_{2},z_{3}$ be three complex numbers of moduli less than or equal to $ 1$. $ w_{1},w_{2}$ are two roots of the equation $ (z \minus{} z_{1})(z \minus{} z_{2}) \plus{} (z \minus{} z_{2})(z \minus{} z_{3}) \plus{} (z \minus{} z_{3})(z \minus{} z_{1}) \equal{} 0$. Prove that, for $ j \equal{} 1,2,3$, $\min\{|z_{j} \minus{} w_{1}|,|z_{j} \minus{} w_{2}|\}\leq 1$ holds.

For what values of $n$ is it possible to mark $n$ points on the plane so that each point has at least three other points at distance $1$?

A cube is assembled with $27$ white cubes. The larger cube is then painted black on the outside and disassembled. A blind man reassembles it. What is the probability that the cube is now completely black on the outside? Give an approximation of the size of your answer.

The corners of a fixed convex (but not necessarily regular) $n$-gon are labeled with distinct letters. If an observer stands at a point in the plane of the polygon, but outside the polygon, they see the letters in some order from left to right, and they spell a "word" (that is, a string of letters; it doesn't need to be a word in any language). For example, in the diagram below (where $n=4$), an observer at point $X$ would read "$BAMO$," while an observer at point $Y$ would read "$MOAB$." [center]Diagram to be added soon[/center] Determine, as a formula in terms of $n$, the maximum number of distinct $n$-letter words which may be read in this manner from a single $n$-gon. Do not count words in which some letter is missing because it is directly behind another letter from the viewer's position.

A square with the dimension $ 1 \times1 $ has been removed from a square board $ 3 ^n \times 3 ^n $ ($ n \in \mathbb {N}, $ $ n> 1 $). a) Prove that any defective board with the dimension $ 3 ^ n \times3 ^ n $ can be covered with shaped figures of shape 1 (the 3 squares' one) and of shape 2 (the 5 squares' one). Figures covering the board must not overlap each other and must not cross the edge of the board. Also the squares removed from the board must not be covered. (b) How many small figures in shape 2 must be used to cover the board? [img]https://cdn.artofproblemsolving.com/attachments/4/7/e970fadd7acc7fd6f5897f1766a84787f37acc.png[/img]

We consider an $n \times n$ table, with $n\ge1$. Aya wishes to color $k$ cells of this table so that that there is a unique way to place $n$ tokens on colored squares without two tokens are not in the same row or column. What is the maximum value of $k$ for which Aya's wish is achievable?

In a chess tournament $ 2n\plus{}3$ players take part. Every two play exactly one match. The schedule is such that no two matches are played at the same time, and each player, after taking part in a match, is free in at least $ n$ next (consecutive) matches. Prove that one of the players who play in the opening match will also play in the closing match.

A regular $n$-gon is inscribed in a circle radius $1$. Let $X$ be the set of all arcs $PQ$, where $P, Q$ are distinct vertices of the $n$-gon. $5$ elements $L_1, L_2, ... , L_5$ of $X$ are chosen at random (so two or more of the $L_i$ can be the same). Show that the expected length of $L_1 \cap L_2 \cap L_3 \cap L_4 \cap L_5$ is independent of $n$.

The vertices of the regular $n$-gon are marked with some colours (each vertex -- with one colour) in such a way, that the vertices of one colour represent the right polygon. Prove that there are two equal ones among the smaller polygons.

There are $n$ ($n \ge 4$) straight lines on the plane. For two straight lines $a$ and $b$, if there are at least two straight lines among the remaining $n-2$ lines that intersect both straight lines $a$ and $b$, then $a$ and $b$ are called a [i]congruent [/i] pair of staight lines, otherwise it is called a [i]separated[/i] pair of straight lines. If the number of [i]congruent [/i] pairs of straight line among $n$ straight lines is $2012$ more than the number of [i]separated[/i] pairs of straight line , find the smallest possible value of $n$ (the order of the two straight lines in a pair is not counted).

What is the largest number of parts into which $n$ planes can divide space? We assume that the set of planes is non-degenerate in the sense that any three planes intersect in one point and no four planes have a common point (and for n=2 it is necessary to require that the planes are not parallel).

The sides of a convex regular polygon of $L + M + N$ sides are to be given draw in three colors: $L$ of them with a red stroke, $M$ with a yellow stroke, and $N$ with a blue. Express, through inequalities, the necessary and sufficient conditions so that there is a solution (several, in general) to the problem of doing it without leaving two adjacent sides drawn with the same color.

A regular $1985$-gon is given in the plane. Let's pass a straight line through each side of it. Determine the number of parts into which these lines divide the plane.

Let $m$, $n$ be positive integers. On an $m\times{n}$ checkerboard, divided into $1\times1$ squares, we consider all paths that go from upper right vertex to the lower left vertex, travelling exclusively on the grid lines by going down or to the left. We define the area of a path as the number of squares on the checkerboard that are below this path. Let $p$ be a prime such that $r_{p}(m)+r_{p}(n)\geq{p}$, where $r_{p}(m)$ denotes the remainder when $m$ is divided by $p$ and $r_{p}(n)$ denotes the remainder when $n$ is divided by $p$. How many paths have an area that is a multiple of $p$?

Each square of a chessboard is painted either blue or red . Prove that the squares of one colour possess the property that the chess queen can perform a tour of all of them. The rules are that the queen may visit the squares of this colour not necessarily only once each , and may not be placed on squares of the other colour, although she may pass over them ; the queen moves along any horizontal , vertical or diagonal file over any distance. (A . K . Tolpugo , Kiev)

Several points on the plane are given, no three of them lie on the same line. Some of these points are connected by line segments. Assume that any line that does not pass through any of these points intersects an even number of these segments. Prove that from each point exits an even number of the segments.

Given $n$ points in the plane, show that the largest distance determined by these points cannot occur more than $n$ times.

a) Fold a $10 \times 10$ square from a $1 \times 118$ rectangular strip. b) Fold a $10 \times 10$ square from a $1 \times (100+9\sqrt3)$ rectangular strip (approximately $1\times 115.58$). The strip can be bent, but not torn.

In the coordinate plane, consider the set of all segments of integer lengths whose endpoints have integer coordinates. Prove that no two of these segments form an angle of $45^o$. Are there such segments in coordinate space?

A cube with edge length $2n+ 1$ is dissected into small cubes of size $1\times 1\times 1$ and bars of size $2\times 2\times 1$. Find the least possible number of cubes in such a dissection.

Two walkers are at the same altitude in a range of mountains. The path joining them is piecewise linear with all its vertices above the two walkers. Can they each walk along the path until they have changed places, so that at all times their altitudes are equal?

Krut and Vert go by car from point $A$ to point $B$. The car leaves $A$ in the direction of $B$, but every $3$ km of the road Krut turns $90^o$ to the left, and every $7$ km of the road Vert turns $90^o$ to the right ( if they try to turn at the same time, the car continues to go in the same direction). Will Krut and Vert be able to get to $B$ if the distance between $A$ and $B$ is $100$ km?

$m,n,p$ are positive integers which do not simultaneously equal to zero. $3$D Cartesian space is divided into unit cubes by planes each perpendicular to one of $3$ axes and cutting corresponding axis at integer coordinates. Each unit cube is filled with an integer from $1$ to $60$. A filling of integers is called [i]Dien Bien[/i] if, for each rectangular box of size $\{2m+1,2n+1,2p+1\}$, the number in the unit cube which has common centre with the rectangular box is the average of the $8$ numbers of the $8$ unit cubes at the $8$ corners of that rectangular box. How many [i]Dien Bien[/i] fillings are there? Two fillings are the same if one filling can be transformed to the other filling via a translation. [hide]translation from [url=http://artofproblemsolving.com/community/c6h592875p3515526]here[/url][/hide]

Two points are selected in a convex pentagon. Prove that you can choose a quadrilateral with vertices at the vertices of a pentagon so that both selected points fall into it.