Two squids are forced to participate in a game. Before it begins, they will be informed of all the rules, and can discuss their strategies freely. Then, they will be locked in separate rooms, and be given distinct positive integers no larger than $2023$ as their IDs respectively. The two squids then take turns alternatively; on one's turn, the squid chooses one of the following: 1. announce a positive integer, which will be heard by the other squid; 2. declare which squid has the larger ID. If correct, they win and are released together; otherwise, they lose and are fried together. Find the smallest positive integer $N$ so that, no matter what IDs the squids have been given, they can always win in a finite number of turns, and the sum of the numbers announced during the game is no larger than $N$.

In every cell of an $n\times n$ board is written $1$ or $-1$. At each step we may choose any of the $4n-2$ diagonals of the board and change the signs of all the numbers on that diagonal. Determine the number of initial configurations from which, after a finite number of steps, we may arrive at a configuration where all products of numbers on rows and columns equal to $1$. [i]Proposed by Pavel Ciurea[/i]

How many knights you can put on chess table $5 \times 5$ such that every one of them attacks exactly two other knights ?

Consider a $25 \times 25$ grid of unit squares. Draw with a red pen contours of squares of any size on the grid. What is the minimal number of squares we must draw in order to colour all the lines of the grid?

There are $n$ sticks which have distinct integer length. Suppose that it's possible to form a non-degenerate triangle from any $3$ distinct sticks among them. It's also known that there are sticks of lengths $5$ and $12$ among them. What's the largest possible value of $n$ under such conditions? [i](Proposed by Bogdan Rublov)[/i]

Let $n$ be a positive integer. Given $n^2$ points in a unit square, prove that there exists a broken line of length $2n + 1$ that passes through all the points. [i]Allen Yuan[/i]

Let $S$ be a set with 2002 elements, and let $N$ be an integer with $0 \leq N \leq 2^{2002}$. Prove that it is possible to color every subset of $S$ either black or white so that the following conditions hold: (a) the union of any two white subsets is white; (b) the union of any two black subsets is black; (c) there are exactly $N$ white subsets.

There is a row of $100N$ sandwiches with ham. A boy and his cat play a game. In one action the boy eats the first sandwich from any end of the row. In one action the cat either eats the ham from one sandwich or does nothing. The boy performs 100 actions in each of his turns, and the cat makes only 1 action each turn; the boy starts first. The boy wins if the last sandwich he eats contains ham. Is it true that he can win for any positive integer $N{}$ no matter how the cat plays? [i]Ivan Mitrofanov[/i]

Alex places a rook on any square of an empty $8\times8$ chessboard. Then he places additional rooks one rook at a time, each attacking an odd number of rooks which are already on the board. A rook attacks to the left, to the right, above and below, and only the first rook in each direction. What is the maximum number of rooks Alex can place on the chessboard?

For two positive integers $n$ and $d$, let $S_n(d)$ be the set of all ordered $d$-tuples $(x_1,x_2,\dots ,x_d)$ that satisfy all of the following conditions: i. $x_i\in \{1,2,\dots ,n\}$ for every $i\in\{1,2,\dots ,d\}$; ii. $x_i\ne x_{i+1}$ for every $i\in\{1,2,\dots ,d-1\}$; iii. There does not exist $i,j,k,l\in\{1,2,\dots ,d\}$ such that $i<j<k<l$ and $x_i=x_k,\, x_j=x_l$; a. Compute $|S_3(5)|$ b. Prove that $|S_n(d)|>0$ if and only if $d\leq 2n-1$.

Can we put positive integers $1,2,3, \cdots 64$ into $8 \times 8$ grids such that the sum of the numbers in any $4$ grids that have the form like $T$ ( $3$ on top and $1$ under the middle one on the top, this can be rotate to any direction) can be divided by $5$?

[b]p1.[/b] Find the smallest whole number $n \ge 2$ such that the product $(2^2 - 1)(3^2 - 1) ... (n^2 - 1)$ is the square of a whole number. [b]p2.[/b] The figure below shows a $ 10 \times 10$ square with small $2 \times 2$ squares removed from the corners. What is the area of the shaded region? [img]https://cdn.artofproblemsolving.com/attachments/7/5/a829487cc5d937060e8965f6da3f4744ba5588.png[/img] [b]p3.[/b] Three cars are racing: a Ford $[F]$, a Toyota $[T]$, and a Honda $[H]$. They began the race with $F$ first, then $T$, and $H$ last. During the race, $F$ was passed a total of $3$ times, $T$ was passed $5$ times, and $H$ was passed $8$ times. In what order did the cars finish? [b]p4.[/b] There are $11$ big boxes. Each one is either empty or contains $8$ medium-sized boxes inside. Each medium box is either empty or contains $8$ small boxes inside. All small boxes are empty. Among all the boxes, there are a total of $102$ empty boxes. How many boxes are there altogether? [b]p5.[/b] Ann, Mary, Pete, and finally Vlad eat ice cream from a tub, in order, one after another. Each eats at a constant rate, each at his or her own rate. Each eats for exactly the period of time that it would take the three remaining people, eating together, to consume half of the tub. After Vlad eats his portion there is no more ice cream in the tube. How many times faster would it take them to consume the tub if they all ate together? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ n \geq 4$ be a fixed positive integer. Given a set $ S \equal{} \{P_1, P_2, \ldots, P_n\}$ of $ n$ points in the plane such that no three are collinear and no four concyclic, let $ a_t,$ $ 1 \leq t \leq n,$ be the number of circles $ P_iP_jP_k$ that contain $ P_t$ in their interior, and let \[m(S)=a_1+a_2+\cdots + a_n.\] Prove that there exists a positive integer $ f(n),$ depending only on $ n,$ such that the points of $ S$ are the vertices of a convex polygon if and only if $ m(S) = f(n).$

Let $n > 3$ be an odd positive integer not divisible by $3$. Determine if it is possible to form an $n \times n$ array of numbers such that [list] [*] [b](a)[/b] the set of the numbers in each row is a permutation of $0, 1, \dots , n - 1$; the set of the numbers in each column is a permutation of $0, 1, \dots , n-1$; [*] [b](b)[/b] the board is [i]totally non-symmetric[/i]: for $1 \le i < j \le n$ and $1 \le i' < j' \le n$, if $(i, j) \neq (i', j')$ then $(a_{i,j} , a_{j,i}) \neq (a_{i',j'} , a_{j',i'})$ where $a_{i,j}$ denotes the entry in the $i^\text{th}$ row and $j^\text{th}$ column.[/list]

A positive integer $n$ is fixed. Numbers $0$ and $1$ are placed in all cells (exactly one number in any cell) of a $k \times n$ table ($k$ is a number of the rows in the table, $n$ is the number of the columns in it). We call a table nice if the following property is fulfilled: for any partition of the set of the rows of the table into two nonempty subsets $R$[size=75]1[/size] and $R$[size=75]2[/size] there exists a nonempty set $S$ of the columns such that on the intersection of any row from $R$[size=75]1[/size] with the columns from $S$ there are even number of $1's$ while on the intersection of any row from $R$[size=75]2[/size] with the columns from $S$ there are odd number of $1's$. Find the greatest number of $k$ such that there exists at least one nice $k \times n$ table.

There are some real numbers on the board (at least two). In every step we choose two of them, for example $a$ and $b$, and then we replace them with $\frac{ab}{a+b}$. We continue until there is one number. Prove that the last number does not depend on which order we choose the numbers to erase.

In a $7\times 8$ chessboard, $56$ stones are placed in the squares. Now we have to remove some of the stones such that after the operation, there are no five adjacent stones horizontally, vertically or diagonally. Find the minimal number of stones that have to be removed.

A family of sets $F$ is called perfect if the following condition holds: For every triple of sets $X_1, X_2, X_3\in F$, at least one of the sets $$ (X_1\setminus X_2)\cap X_3,$$ $$(X_2\setminus X_1)\cap X_3$$ is empty. Show that if $F$ is a perfect family consisting of some subsets of a given finite set $U$, then $\left\lvert F\right\rvert\le\left\lvert U\right\rvert+1$. [i]Proposed by Michał Pilipczuk[/i]

A class consists of $7$ boys and $13$ girls. During the first three months of the school year, each boy has communicated with each girl at least once. Prove that there exist two boys and two girls such that both boys communicated with both girls first time in the same month.

Consider two disjoint finite sets of positive integers, $A$ and $B$, have $n$ and $m$ elements, respectively. It is knows that all $k$ belonging to $A \cup B$ satisfies at least one of the conditions $k + 17 \in A$ and $k - 31 \in B$. Prove that $17n = 31m$.

How many non-intersecting pairs of paths we have from (0,0) to (n,n) so that path can move two ways:top or right?

In a summer school, there are $n>4$ students. It is known that, among these students, i. If two ones are friends, then they don't have any common friends. ii If two ones are not friends, then they have exactly two common friends. 1. Prove that $8n-7$ must be a perfect square. 2. Determine the smallest possible value of $n$.

Prove that there are $2012$ points in the plane, none of which are three on one straight line and in pairs have integer distances .

Show that a convex polyhedron with an odd number of faces has at least one face with an even number of edges.

There are $2021$ people at a meeting. It is known that one person at the meeting doesn't have any friends there and another person has only one friend there. In addition, it is true that, given any $4$ people, at least $2$ of them are friends. Show that there are $2018$ people at the meeting that are all friends with each other. [i]Note. [/i]If $A$ is friend of $B$ then $B$ is a friend of $A$.