Show that there is a subset of $A$ from $\{1,2, 3,... , 2014\}$ such that : (i) $|A| = 12$ (ii) for each coloring number in $A$ with red or white , we can always find some numbers colored in $A$ whose sum is $2015$.

Let $n \geq 2$ be a positive integer. There are $2n$ cities $M_1, M_2, \ldots, M_{2n}$ in the country of Mathlandia. Currently there roads only between $M_1$ and $M_2, M_3, \ldots, M_n$ and the king wants to build more roads so that it is possible to reach any city from every other city. The cost to build a road between $M_i$ and $M_j$ is $k_{i, j}>0$. Let $$K=\sum_{j=n+1}^{2n} k_{1,j}+\sum_{2 \leq i<j \leq 2n} k_{i, j}.$$Prove that the king can fulfill his plan at cost no more than $\frac{2K}{3n-1}$.

Is it possible to find a set of $100$ (or $200$) points on the boundary of a cube such that this set remains fixed under all rotations which leave the cube fixed ?

There are $n$ blocks placed on the unit squares of a $n \times n$ chessboard such that there is exactly one block in each row and each column. Find the maximum value $k$, in terms of $n$, such that however the blocks are arranged, we can place $k$ rooks on the board without any two of them threatening each other. (Two rooks are not threatening each other if there is a block lying between them.)

(a) Prove that $x^4+3x^3+6x^2+9x+12$ cannot be expressed as product of two polynomials of degree 2 with integers coefficients. (b) $2n+1$ segments are marked on a line. Each of these segments intersects at least $n$ other segments. Prove that one of these segments intersects all other segments.

Let $n \ge 2$ be an integer. In each cell of a $4n \times 4n$ table we write the sum of the cell row index and the cell column index. Initially, no cell is colored. A move consists of choosing two cells which are not colored and coloring one of them in red and one of them in blue. Show that, however Alex perfors $n^2$ moves, Jane can afterwards perform a number of moves (eventually none) after which the sum of the numbers written in the red cells is the same as the sum of the numbers written in the blue ones.

Consider the following table where initially all squares contain zeros: $ \begin{tabular}{ | l | c | r| } \hline 0 & 0 & 0 \\ \hline 0 & 0 & 0 \\ \hline 0 & 0 & 0 \\ \hline \end{tabular} $ To change the table, the following operation is allowed: a $2 \times 2$ square formed by adjacent squares is chosen, and a unit is added to all its numbers. Complete the following table, knowing that it was obtained by a sequence of permitted operations $ \begin{tabular}{ | l | c | r| } \hline 14 & & \\ \hline 19 & 36 & \\ \hline & 16 & \\ \hline \end{tabular} $

Assume $a_1,a_2,...,a_{14}$ are positive integers such that $\sum_{i=1}^{14}3^{a_i} = 6558$. Prove that the numbers $a_1,a_2,...,a_{14}$ consist of the numbers $1,...,7$, each taken twice.

Define $M_n = \{1,2,...,n\}$, for any $n\in N$. A collection of $3$-element subsets of $M_n$ is said to be [i]fine [/i] if for any coloring of elements of $M_n$ in two colors there is a subset of the collection all three elements of which are of the same color. For any $n\ge 5$ find the minimal possible number of the $3$-element subsets of $M_n$ in the fine collection. (E. Barabanov, S. Mazanik, I. Voronovich)

All faces of a polyhedron are triangles. Each of the vertices of this polyhedron is assigned independently one of three colors : green, white or black. We say that a face is [i]Extremadura[/i] if its three vertices are of different colors, one green, one white and one black. Is it true that regardless of how the vertices's color, the number of [i]Extremadura[/i] faces of this polyhedron is always even?

Given a $9\times 9$ grid, we assign either $+1$ or $-1$ to each square on the grid. We define an [i]adjustment[/i] as follow: for each square on the grid, we make a product of all numbers of those squares which share a common side with the square (excluding itself).Then we have $81$ products. Next we replace all the squares’ values with their corresponding products. Determine if we can make all values in the grid equal to $1$ through finite [i]adjustments[/i].

$n > 4$ planes are in general position (so every $3$ planes have just one common point, and no point belongs to more than $3$ planes). Show that there are at least $\frac{2n-3}{ 4}$ tetrahedra among the regions formed by the planes.

A finite set $M$ of unit squares on the plane is considered. The sides of the squares are parallel to the coordinate axes and the squares are allowed to intersect. It is known that the distance between the centres of any pair of squares is no greater than $2$. Prove that there exists a unit square (not necessarily belonging to $M$) with sides parallel to the coordinate axes and which has at least one common point with each of the squares in $M$. (A Andjans, Riga)

A plane is divided into unit squares by two collections of parallel lines. For any $n\times n$ square with sides on the division lines, we define its frame as the set of those unit squares which internally touch the boundary of the $n\times n$ square. Prove that there exists only one way of covering a given $100\times 100$ square whose sides are on the division lines with frames of $50$ squares (not necessarily contained in the $100\times 100$ square). (A. Perlin)

A square grid $100 \times 100$ is tiled in two ways - only with dominoes and only with squares $2 \times 2$. What is the least number of dominoes that are entirely inside some square $2 \times 2$?

In a tournament, $n$ players participate. Each player plays each other exactly once, with no ties. A player $A$ is said to be [i]champion [/i] if, for every other player $B$, one of the following two situations occurs: (a) $A$ beat $B$; (b) $A$ beat a player $C$ who in turn beat $B$. Prove that in such a tournament there cannot be exactly two champions.

There are $998$ cities in a country. Some pairs of cities are connected by two-way flights. According to the law, between any pair cities should be no more than one flight. Another law requires that for any group of cities there will be no more than $5k+10$ flights connecting two cities from this group, where $k$ is the number number of cities in the group. Prove that several new flights can be introduced so that laws still hold and the total number of flights in the country is equal to $5000$.

Consider $N$ points in the plane such that the area of a triangle formed by any three of the points does not exceed $1$. Prove that there is a triangle of area not more than $4$ that contains all $N$ points.

Given a finite set of points in the plane, each with integer coordinates, is it always possible to color the points red or white so that for any straight line $L$ parallel to one of the coordinate axes the difference (in absolute value) between the numbers of white and red points on $L$ is not greater than $1$?

There is a graph with 30 vertices. If any of 26 of its vertices with their outgoiing edges are deleted, then the remained graph is a connected graph with 4 vertices. What is the smallest number of the edges in the initial graph with 30 vertices?

A class contains $80$ boys and $80$ girls. On each weekday (Monday to Friday) of the week before final exams, the teacher has $16$ books for the students to borrow, where a book can only be borrowed for one day at a time, and each student can only borrow once during the entire week. Show that there are two days and two books such that one of the following two statements is true: (i) Both books were not borrowed on both days (ii) Both books were borrowed on both days, and the four students who borrowed the books on these days are either all boys or all girls.

The plan of a picture gallery is a chequered figure where each square is a room, and every room can be reached from each other by moving to adjacent rooms. A custodian in a room can watch all the rooms that can be reached from this room by one move of a chess queen (without leaving the gallery). What minimum number of custodians is sufficient to watch all the rooms in every gallery of $n$ rooms ($n > 2$)?

[b][i](a) [/i][/b] Show that the set $\mathbb N$ of all positive integers can be partitioned into three disjoint subsets $A, B$, and $C$ satisfying the following conditions: \[A^2 = A, B^2 = C, C^2 = B,\] \[AB = B, AC = C, BC = A,\] where $HK$ stands for $\{hk | h \in H, k \in K\}$ for any two subsets $H, K$ of $\mathbb N$, and $H^2$ denotes $HH.$ [b][i](b)[/i][/b] Show that for every such partition of $\mathbb N$, $\min\{n \in N | n \in A \text{ and } n + 1 \in A\}$ is less than or equal to $77.$

[b]p1.[/b] The total cost of $1$ football, $3$ tennis balls and $7$ golf balls is $\$14$ , while that of $1$ football, $4$ tennis balls and $10$ golf balls is $\$17$.If one has $\$20$ to spend, is this sufficient to buy a) $3$ footballs and $2$ tennis balls? b) $2$ footballs and $3$ tennis balls? [b]p2.[/b] Let $\overline{AB}$ and $\overline{CD}$ be two chords in a circle intersecting at a point $P$ (inside the circle). a) Prove that $AP \cdot PB = CP\cdot PD$. b) If $\overline{AB}$ is perpendicular to $\overline{CD}$ and the length of $\overline{AP}$ is $2$, the length of $\overline{PB}$ is $6$, and the length of $\overline{PD}$ is $3$, find the radius of the circle. [b]p3.[/b] A polynomial $P(x)$ of degree greater than one has the remainder $2$ when divided by $x-2$ and the remainder $3$ when divided by $x-3$. Find the remainder when $P(x)$ is divided by $x^2-5x+6$. [b]p4.[/b] Let $x_1= 2$ and $x_{n+1}=x_n+ (3n+2)$ for all $n$ greater than or equal to one. a) Find a formula expressing $x_n$ as a function of$ n$. b) Prove your result. [b]p5.[/b] The point $M$ is the midpoint of side $\overline{BC}$ of a triangle $ABC$. a) Prove that $AM \le \frac12 AB + \frac12 AC$. b) A fly takes off from a certain point and flies a total distance of $4$ meters, returning to the starting point. Explain why the fly never gets outside of some sphere with a radius of one meter. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In each square of a garden shaped like a $2022 \times 2022$ board, there is initially a tree of height $0$. A gardener and a lumberjack alternate turns playing the following game, with the gardener taking the first turn: [list] [*] The gardener chooses a square in the garden. Each tree on that square and all the surrounding squares (of which there are at most eight) then becomes one unit taller. [*] The lumberjack then chooses four different squares on the board. Each tree of positive height on those squares then becomes one unit shorter. [/list] We say that a tree is [i]majestic[/i] if its height is at least $10^6$. Determine the largest $K$ such that the gardener can ensure there are eventually $K$ majestic trees on the board, no matter how the lumberjack plays.