Given a table $m\times n$, in each cell of which a number $+1$ or $-1$ is written. It is known that initially exactly one $-1$ is in the table, all the other numbers being $+1$. During a move, it is allowed to chose any cell containing $-1$, replace this $-1$ by $0$, and simultaneously multiply all the numbers in the neighbouring cells by $-1$ (we say that two cells are neighbouring if they have a common side). Find all $(m,n)$ for which using such moves one can obtain the table containing zeros only, regardless of the cell in which the initial $-1$ stands.

A teacher wishes to separate her $12$ students into groups. Yesterday, the teacher put the students into $4$ groups of $3$. Today, the teacher decides to put the students into $4$ groups of $3$ again. However, she doesn’t want any pair of students to be in the same group on both days. Find how many ways she could formthe groups today.

Let $n$ and $k$ be positive integers and $G$ be a complete graph on $n$ vertices. Each edge of $G$ is colored one of $k$ colors such that every triangle consists of either three edges of the same color or three edges of three different colors. Furthermore, there exist two different-colored edges. Prove that $n\le(k-1)^2$. [i]Linus Tang[/i]

[b]p1.[/b] Prove that no matter what digits are placed in the four empty boxes, the eight-digit number $9999\Box\Box\Box\Box$ is not a perfect square. [b]p2.[/b] Prove that the number $m/3+m^2/2+m^3/6$ is integral for all integral values of $m$. [b]p3.[/b] An elevator in a $100$ store building has only two buttons: UP and DOWN. The UP button makes the elevator go $13$ floors up, and the DOWN button makes it go $8$ floors down. Is it possible to go from the $13$th floor to the $8$th floor? [b]p4.[/b] Cut the triangle shown in the picture into three pieces and rearrange them into a rectangle. (Pieces can not overlap.) [img]https://cdn.artofproblemsolving.com/attachments/4/b/ca707bf274ed54c1b22c4f65d3d0b0a5cfdc56.png[/img] [b]p5.[/b] Two players Tom and Sid play the following game. There are two piles of rocks, $7$ rocks in the first pile and $9$ rocks in the second pile. Each of the players in his turn can take either any amount of rocks from one pile or the same amount of rocks from both piles. The winner is the player who takes the last rock. Who does win in this game if Tom starts the game? [b]p6.[/b] In the next long multiplication example each letter encodes its own digit. Find these digits. $\begin{tabular}{ccccc} & & & a & b \\ * & & & c & d \\ \hline & & c & e & f \\ + & & a & b & \\ \hline & c & f & d & f \\ \end{tabular}$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

There are $15$ people, including Petruk, Gareng, and Bagong, which will be partitioned into $6$ groups, randomly, that consists of $3, 3, 3, 2, 2$, and $2$ people (orders are ignored). Determine the probability that Petruk, Gareng, and Bagong are in a group.

Suppose in the plane 10 pairwise nonparallel lines intersect one another. What is the maximum possible number of polygons (with finite areas) that can be formed?

Each cell of the checkered plane is colored in one of $n^2$ colors so that in any square of $n \times n$ cells all colors occur. It is known that in some line all the colors occur. Prove that there exists a column colored in exactly $n$ colors.

What is the number of ways in which one can choose $60$ unit squares from a $11 \times 11$ chessboard such that no two chosen squares have a side in common?

We denote with $p_n(k)$ the number of permutations of the numbers $1,2,...,n$ that have exactly $k$ fixed points. a) Prove that $\sum_{k=0}^n kp_n (k)=n!$. b) If $s$ is an arbitrary natural number, then: $\sum_{k=0}^n k^s p_n (k)=n!\sum_{i=1}^m R(s,i)$, where with $R(s,i)$ we denote the number of partitions of the set $\{1,2,...,s\}$ into $i$ non-empty non-intersecting subsets and $m=min(s,n)$.

There are three balls of distinct colors in a bag. We repeatedly draw out the balls one by one, the balls are put back into the bag after each drawing. What is the probability that, after $n$ drawings, (a) exactly one color occured? (b) exactly two oclors occured? (c) all three colors occured?

A rectangle $\mathcal{R}$ with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of $\mathcal{R}$ are either all odd or all even. [i]Proposed by Jeck Lim, Singapore[/i]

Let $M$ be a subset of $\mathbb{R}$ such that the following conditions are satisfied: a) For any $x \in M, n \in \mathbb{Z}$, one has that $x+n \in \mathbb{M}$. b) For any $x \in M$, one has that $-x \in M$. c) Both $M$ and $\mathbb{R}$ \ $M$ contain an interval of length larger than $0$. For any real $x$, let $M(x) = \{ n \in \mathbb{Z}^{+} | nx \in M \}$. Show that if $\alpha,\beta$ are reals such that $M(\alpha) = M(\beta)$, then we must have one of $\alpha + \beta$ and $\alpha - \beta$ to be rational.

There are $2008$ participants in a programming competition. In every round, all programmers are divided into two equal-sized teams. Find the minimal number of rounds after which there can be a situation in which every two programmers have been in different teams at least once.

A point starts at the origin of the coordinate plane. Every minute, it either moves one unit in the $x$-direction or is rotated $\theta$ degrees counterclockwise about the origin. (a) If $\theta = 90^o$, determine all locations where the point could end up. (b) If $\theta = 45^o$, prove that for every location $ L$ in the coordinate plane and every positive number $\varepsilon$, there is a sequence of moves after which the point has distance less than $\varepsilon$ from $L$. (c) Determine all rational numbers $\theta$ such that for every location $L$ in the coordinate plane and every positive number $\varepsilon$, there is a sequence of moves after which the point has distance less than $\varepsilon$ from $L$. (d) Prove that when $\theta$ is irrational, for every location $L$ in the coordinate plane and every positive number $\varepsilon$, there is a sequence of moves after which the point has distance less than $\varepsilon$ from $L.$

Twenty-five of the numbers $1, 2, \cdots , 50$ are chosen. Twenty-five of the numbers$ 51, 52, \cdots, 100$ are also chosen. No two chosen numbers differ by $0$ or $50$. Find the sum of all $50$ chosen numbers.

Let $ \left( a_i \right)_{1\le i\le n} $ and $ \left( b_i \right)_{1\le i\le n} $ be two sequences, the former being a decreasing sequence and the latter being an increasing sequence. All the terms of $ \left( a_i \right)_{1\le i\le n} $ and $ \left( b_i \right)_{1\le i\le n} $ form the set $ \{1,2,3,\ldots ,2n \} . $ Prove that: $$ \left| a_1-b_1 \right| +\left| a_2-b_2 \right| +\cdots +\left| a_n-b_n \right|=n^2 $$

Let $ N \equal{} \{1,2 \ldots, n\}, n \geq 2.$ A collection $ F \equal{} \{A_1, \ldots, A_t\}$ of subsets $ A_i \subseteq N,$ $ i \equal{} 1, \ldots, t,$ is said to be separating, if for every pair $ \{x,y\} \subseteq N,$ there is a set $ A_i \in F$ so that $ A_i \cap \{x,y\}$ contains just one element. $ F$ is said to be covering, if every element of $ N$ is contained in at least one set $ A_i \in F.$ What is the smallest value $ f(n)$ of $ t,$ so there is a set $ F \equal{} \{A_1, \ldots, A_t\}$ which is simultaneously separating and covering?

Given two integers $n\geq 1$ and $q\geq 2$, let $A=\{(a_1,\ldots ,a_n):a_i\in\{0,\ldots ,q-1\}, i=1,\ldots ,n\}$. If $a=(a_1,\ldots ,a_n)$ and $b=(b_1,\ldots ,b_n)$ are two elements of $A$, let $\delta(a,b)=\#\{i:a_i\neq b_i\}$. Let further $t$ be a non-negative integer and $B$ a non-empty subset of $A$ such that $\delta(a,b)\geq 2t+1$, whenever $a$ and $b$ are distinct elements of $B$. Prove that the two statements below are equivalent: a) For any $a\in A$, there is a unique $b\in B$, such that $\delta (a,b)\leq t$; b) $\displaystyle|B|\cdot \sum_{k=0}^t \binom{n}{k}(q-1)^k=q^n$

Set $X$ is called [i]fancy[/i] if it satisfies all of the following conditions: [list] [*]The number of elements of $X$ is $2022$. [*]Each element of $X$ is a closed interval contained in $[0, 1]$. [*]For any real number $r \in [0, 1]$, the number of elements of $X$ containing $r$ is less than or equal to $1011$. [/list] For [i]fancy[/i] sets $A, B$, and intervals $I \in A, J \in B$, denote by $n(A, B)$ the number of pairs $(I, J)$ such that $I \cap J \neq \emptyset$. Determine the maximum value of $n(A, B)$.

the code system of a new 'MO lock' is a regular $n$-gon,each vertex labelled a number $0$ or $1$ and coloured red or blue.it is known that for any two adjacent vertices,either their numbers or colours coincide. find the number of all possible codes(in terms of $n$).

Let $P$ be a given regular $100$-gon. Prove that if we take the union of two polygons that are congruent to $P,$ the ratio of the perimeter and area of the resulting shape cannot be more than the ratio of the perimeter and area of $P.$

Given a set $M$ of $1985$ positive integers, none of which has a prime divisor larger than $26$, prove that the set has four distinct elements whose geometric mean is an integer.

The vertices of the regular $n$-gon are marked. Two players play the following game: they, in turn, select a vertex and connect it by a segment to either the adjacent vertex or the center of the $n$-gon. The winner is a player if after his move it is possible to get any vertex from any other vertex moving along segments. For each integer $n\geqslant 3$ determine who has a winning strategy.

We say that a set $S$ of $3$ unit squares is \textit{commutable} if $S = \{s_1,s_2,s_3\}$ for some $s_1,s_2,s_3$ where $s_2$ shares a side with each of $s_1,s_3$. How many ways are there to partition a $3\times 3$ grid of unit squares into $3$ pairwise disjoint commutable sets? [i]Proposed by Srinivasan Sathiamurthy[/i]

[b](a)[/b] One black and one white pawn are placed on a chessboard. You may move the pawns in turn to the neighbouring empty squares of the chessboard using vertical and horizontal moves. Can you arrange the moves so that every possible position of the two pawns will appear on the chessboard exactly once? [b](b)[/b] Same question, but you don’t have to move the pawns in turn.