In the following $6\times 6$ matrix, one can choose any $k\times k$ submatrix, with $1<k\leq6 $ and add $1$ to all its entries. Is it possible to perform the operation a finite number of times so that all the entries in the $6\times 6$ matrix are multiples of $3$? $ \begin{pmatrix} 2 & 0 & 1 & 0 & 2 & 0 \\ 0 & 2 & 0 & 1 & 2 & 0 \\ 1 & 0 & 2 & 0 & 2 & 0 \\ 0 & 1 & 0 & 2 & 2 & 0 \\ 1 & 1 & 1 & 1 & 2 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix} $ Note: A $p\times q$ submatrix of a $m\times n$ matrix (with $p\leq m$, $q\leq n$) is a $p\times q$ matrix formed by taking a block of the entries of this size from the original matrix.

Let $ S$ be a set that contains $ n$ elements. Let $ A_{1},A_{2},\cdots,A_{k}$ be $ k$ distinct subsets of $ S$, where $ k\geq 2, |A_{i}| \equal{} a_{i}\geq 1 ( 1\leq i\leq k)$. Prove that the number of subsets of $ S$ that don't contain any $ A_{i} (1\leq i\leq k)$ is greater than or equal to $ 2^n\prod_{i \equal{} 1}^k(1 \minus{} \frac {1}{2^{a_{i}}}).$

For a natural number $n$, let $T(n)$ denote the number of ways we can place $n$ objects of weights $1,2,\cdots, n$ on a balance such that the sum of the weights in each pan is the same. Prove that $T(100) > T(99)$.

Let $a_1,a_2,...$ be a sequence of non-negative integers such that for any $m,n$ \[ \sum_{i=1}^{2m} a_{in} \leq m.\] Show that there exist $k,d$ such that \[ \sum_{i=1}^{2k} a_{id} = k-2014.\]

We have placed some red and blue points along a circle. The following operations are permitted: (a) we may add a red point somewhere and switch the color of its neighbors, (b) we may take off a red point from somewhere and switch the color of its neighbors (if there are at least $3$ points on the circle and there is a red one too). Initially, there are two blue points on the circle. Using a number of these operations, can we reach a state with exactly two red point?

Three distinct positive integers are chosen at random from $\{1,2,3...,12\}$. The probability that no two elements of the set have an absolute difference less than or equal to $2$ can be written as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a +b$.

Suppose that $S$ is a finite set of real numbers with the property that any two distinct elements of $S$ form an arithmetic progression with another element in $S$. Give an example of such a set with 5 elements and show that no such set exists with more than $5$ elements.

Let $A$ be the subset of $\{1, 2, ..., 16\}$ that has $6$ elements. Prove that there exist $2$ subsets of $A$ that are disjoint, and the sum of their elements are the same.

$110$ teams participate in a volleyball tournament. Every team has played every other team exactly once (there are no ties in volleyball). Turns out that in any set of $55$ teams, there is one which has lost to no more than $4$ of the remaining $54$ teams. Prove that in the entire tournament, there is a team that has lost to no more than $4$ of the remaining $109$ teams.

$3n$ lines are drawn on the plane ($n > 1$), such that no two of them are parallel and no three of them are concurrent. Prove that, if $2n$ of the lines are coloured red and the other $n$ lines blue, there are at least two regions of the plane such that all of their borders are red. Note: for each region, all of its borders are contained in the original set of lines, and no line passes through the region.

Let $ n(\ge2) $ be a positive integer. Find the minimum $ m $, so that there exists $x_{ij}(1\le i ,j\le n)$ satisfying: (1)For every $1\le i ,j\le n, x_{ij}=max\{x_{i1},x_{i2},...,x_{ij}\} $ or $ x_{ij}=max\{x_{1j},x_{2j},...,x_{ij}\}.$ (2)For every $1\le i \le n$, there are at most $m$ indices $k$ with $x_{ik}=max\{x_{i1},x_{i2},...,x_{ik}\}.$ (3)For every $1\le j \le n$, there are at most $m$ indices $k$ with $x_{kj}=max\{x_{1j},x_{2j},...,x_{kj}\}.$

One has marked $n$ points on a circle and has drawn a certain number of chords whose endpoints are the marked points. It turned out that the following property is satisfied: whenever any $2021$ drawn chords are removed one can join any two marked points by a broken line composed of some of the remaining drawn chords. Prove that one can remove some of the drawn chords so that at most $2022n$ chords remain and the property described above is preserved.

Let $n \geq 3$ be an integer. Two players play a game on an empty graph with $n + 1$ vertices, consisting of the vertices of a regular n-gon and its center. They alternately select a vertex of the n-gon and draw an edge (that has not been drawn) to an adjacent vertex on the n-gon or to the center of the n-gon. The player who first makes the graph connected wins. Between the player who goes first and the player who goes second, who has a winning strategy? [i]Note: an empty graph is a graph with no edges.[/i]

Each of the players in a tennis tournament played one match against each of the others. If every player won at least one match, show that there is a group A; B; C of three players for which A beat B, B beat C and C beat A.

In a tournament, every two of the $n$ players played exactly one match with each other (no draws). Prove that it is possible either (i) to partition the league in two groups $A$ and $B$ such that everybody in $A$ defeated everybody in $B$; or (ii) to arrange all the players in a chain $x_1, x_2, . . . , x_n, x_1$ in such a way that each player defeated his successor.

Bishops on a chessboard move along the diagonals ( that is , on lines parallel to the two main diagonals ) . Prove that the maximum number of non-attacking bishops on an $n*n$ chessboard is $2n-2$. (Two bishops are said to be attacking if they are on a common diagonal).

A not necessary nonplanar polygon in $\mathbb{R}^3$ is called [b]Grid Polygon[/b] if each of it's edges are parallel to one of the axes. (a) There's a right angle between each two neighbour sides of the grid polygon, the plane containing this angle could be parallel to either $xy$ plane, $yz$ plane, or $xz$ plane. Prove that parity of the number of the angles that the plane containing each of them is parallel to $xy$ plane is equal to parity of the number of the angles that the plane containing each of them is parallel to $yz$ plane and parity of the number of the angles that the plane containing each of them is parallel to $zx$ plane. (b) A grid polygon is called [b]Inscribed[/b] if there's a point in the space that has an equal distance from all of the vertices of the polygon. Prove that any nonplanar grid hexagon is inscribed. (c) Does there exist a grid 2014-gon without repeated vertices such that there exists a plane that intersects all of it's edges? (d) If $a,b,c \in \mathbb{N}-\{1\}$, prove that $a,b,c$ are sidelengths of a triangle iff there exists a grid polygon in which the number of it's edges that are parallel to $x$ axis is $a$, the number of it's edges that are parallel to $y$ axis is $b$ and the number of it's edges that are parallel to $z$ axis is $c$. Time allowed for this exam was 1 hour.

[b]8.1[/b] A $24 \times 60$ rectangle is divided by lines parallel to it sides, into unit squares. Draw another straight line so that after that the rectangle was divided into the largest possible number of parts. [b]8.2[/b] Engineers always tell the truth, but businessmen always lie. F and G are engineers. A declares that, B asserts that, C asserts that, D says that, E insists that, F denies that G is an businessman. C also announces that D is a businessman. If A is a businessman, then how much total businessmen in this company? [b]8.3 [/b]There is a straight road through the field. A tourist stands on the road at a point ?. It can walk along the road at a speed of 6 km/h and across the field at a speed of 3 km/h. Find the locus of the points where the tourist can get there within an hour's walk. [b]8.4 / 7.5 [/b] Let $ [A]$ denote the largest integer not greater than $A$. Solve the equation: $[(5 + 6x)/8] = (15x-7)/5$ . [b]8.5.[/b] In some state, every two cities are connected by a road. Each road is only allowed to move in one direction. Prove that there is a city from which you can travel around everything. state, having visited each city exactly once. [b]8.6[/b] Find all eights of prime numbers such that the sum of the squares of the numbers in the eight is 992 less than their quadruple product. [hide=original wording]Найдите все восьмерки простых чисел такие, что сумма квадратов чисел в восьмерке на 992 меньше, чем их учетверенное произведение.[/hide] PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988081_1965_leningrad_math_olympiad]here[/url].

A square board with a size of $2020 \times 2020$ is divided into $2020^2$ small squares of size $1 \times 1$. Each of these small squares will be coloured black or white. Determine the number of ways to colour the board such that for every $2\times 2$ square, which consists of $4$ small squares, contains $2$ black small squares and $2$ white small squares.

[b]p1.[/b] Compute $1 + 3 + 6 + 10 + 15 + 21 + 28 + 36 + 45 + 55$. [b]p2.[/b] Given that $a$, $b$, and $c$ are positive integers such that $a+b = 9$ and $bc = 30$, find the minimum possible value of $a + c$. [b]p3.[/b] Points $X$ and $Y$ lie outside regular pentagon $ABCDE$ such that $ABX$ and $DEY$ are equilateral triangles. Find the degree measure of $\angle XCY$ . [b]p4.[/b] Let $N$ be the product of the positive integer divisors of $8!$, including itself. The largest integer power of $2$ that divides $N$ is $2^k$. Compute $k$. [b]p5.[/b] Let $A=(-20, 22)$, $B = (k, 0)$, and $C = (202, 2)$ be points on the coordinate plane. Given that $\angle ABC = 90^o$, find the sum of all possible values of $k$. [b]p6.[/b] Tej is typing a string of $L$s and $O$s that consists of exactly $7$ $L$s and $4$ $O$s. How many different strings can he type that do not contain the substring ‘$LOL$’ anywhere? A substring is a sequence of consecutive letters contained within the original string. [b]p7.[/b] How many ordered triples of integers $(a, b, c)$ satisfy both $a+b-c = 12$ and $a^2+b^2-c^2 = 24$? [b]p8.[/b] For how many three-digit base-$7$ numbers $\overline{ABC}_7$ does $\overline{ABC}_7$ divide $\overline{ABC}_{10}$? (Note: $\overline{ABC}_D$ refers to the number whose digits in base $D$ are, from left to right, $A$, $B$, and $C$; for example, $\overline{123}_4$ equals $27$ in base ten). [b]p9.[/b] Natasha is sitting on one of the $35$ squares of a $5$-by-$7$ grid of squares. Wanda wants to walk through every square on the board exactly once except the one Natasha is on, starting and ending on any $2$ squares she chooses, such that from any square she can only go to an adjacent square (two squares are adjacent if they share an edge). How many squares can Natasha choose to sit on such that Wanda cannot go on her walk? [b]p10.[/b] In triangle $ABC$, $AB = 13$, $BC = 14$, and $CA = 15$. Point $P$ lies inside $ABC$ and points $D,E$, and $F$ lie on sides $BC$, $CA$, and $AB$, respectively, so that $PD \perp BC$, $PE \perp CA$, and $PF \perp AB$. Given that $PD$, $PE$, and $PF$ are all integers, find the sum of all possible distinct values of $PD \cdot PE \cdot PF$. [b]p11.[/b] A palindrome is a positive integer which is the same when read forwards or backwards. Find the sum of the two smallest palindromes that are multiples of $137$. [b]p12.[/b] Let $P(x) = x^2+px+q$ be a quadratic polynomial with positive integer coefficients. Compute the least possible value of p such that 220 divides p and the equation $P(x^3) = P(x)$ has at least four distinct integer solutions. [b]p13.[/b] Everyone at a math club is either a truth-teller, a liar, or a piggybacker. A truth-teller always tells the truth, a liar always lies, and a piggybacker will answer in the style of the previous person who spoke (i.e., if the person before told the truth, they will tell the truth, and if the person before lied, then they will lie). If a piggybacker is the first one to talk, they will randomly either tell the truth or lie. Four seniors in the math club were interviewed and here was their conversation: Neil: There are two liars among us. Lucy: Neil is a piggybacker. Kevin: Excluding me, there are more truth-tellers than liars here. Neil: Actually, there are more liars than truth-tellers if we exclude Kevin. Jacob: One plus one equals three. Define the base-$4$ number $M = \overline{NLKJ}_4$, where each digit is $1$ for a truth-teller, $2$ for a piggybacker, and $3$ for a liar ($N$ corresponds to Neil, $L$ to Lucy, $K$ corresponds to Kevin, and $J$ corresponds to Jacob). What is the sum of all possible values of $M$, expressed in base $10$? [b]p14.[/b] An equilateral triangle of side length $8$ is tiled by $64$ equilateral triangles of unit side length to form a triangular grid. Initially, each triangular cell is either living or dead. The grid evolves over time under the following rule: every minute, if a dead cell is edge-adjacent to at least two living cells, then that cell becomes living, and any living cell remains living. Given that every cell in the grid eventually evolves to be living, what is the minimum possible number of living cells in the initial grid? [b]p15.[/b] In triangle $ABC$, $AB = 7$, $BC = 11$, and $CA = 13$. Let $\Gamma$ be the circumcircle of $ABC$ and let $M$, $N$, and $P$ be the midpoints of minor arcs $BC$ , $CA$, and $AB$ of $\Gamma$, respectively. Given that $K$ denotes the area of $ABC$ and $L$ denotes the area of the intersection of $ABC$ and $MNP$, the ratio $L/K$ can be written as $a/b$ , where $a$ and $b$ are relatively prime positive integers. Compute $a + b$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $n$ be a positive integer. $n$ people take part in a certain party. For any pair of the participants, either the two are acquainted with each other or they are not. What is the maximum possible number of the pairs for which the two are not acquainted but have a common acquaintance among the participants?

Baron and Peter are playing a game. They are given a simple finite graph $G$ with $n\ge 3$ vertex and $k$ edges that connects the vertices. First Peter labels two vertices A and B, and places a counter at A. Baron starts first. A move for Baron is move the counter along an edge. Peter's move is to remove an edge from the graph. Baron wins if he reaches $B$, otherwise Peter wins. Given the value of $n$, what is the largest $k$ so that Peter can always win?

In a $2^n \times 2^n$ square with $n$ positive integer is covered with at least two non-overlapping rectangle pieces with integer dimensions and a power of two as surface. Prove that two rectangles of the covering have the same dimensions (Two rectangles have the same dimensions as they have the same width and the same height, wherein they, not allowed to be rotated.)

Every square on the infinite sheet of cross-lined paper contains some real number. Prove that some square contains a number that does not exceed at least four of eight neighbouring numbers.

There are $10{}$ apples, each with a with a weight which is no more than $100{}$ g. There is a weighing scale with two plates which shows the difference between the weights on the plates. Prove that 1) It is possible to put some (more than one) apples on the plates of the scale such that the difference between the weights on the plates will be less than $1$ g. 2) It is possible to put an equal amount (more than one) of apples on each plate of the scale such that the difference between the weights on the plates will be less than $2$ g.