Take a $n \times n$ chess page.Determine the $n$ such that we can put the numbers $1,2,3, \ldots ,n$ in the squares of the page such that we know the following two conditions are true: a) for each row we know all the numbers $1,2,3, \ldots ,n$ have appeared on it and the numbers that are in the black squares of that row have the same sum as the sum of the numbers in the white squares of that row. b) for each column we know all the numbers $1,2,3, \ldots ,n$ have appeared on it and the numbers that are in the black squares in that column have the same sum as the sum of the numbers in the white squares of that column.

A [i]zigzag [/i] is a polyline on a plane formed from two parallel rays and a segment connecting the origins of these rays. What is the maximum number of parts a plane can be split into using $ n $ zigzags?

You are responsible for arranging a banquet for an agency. In the agency, some pairs of agents are enemies. A group of agents are called [i]avengers[/i], if and only if the number of agents in the group is odd and at least $3$, and it is possible to arrange all of them around a round table so that every two neighbors are enemies. You figure out a way to assign all agents to $11$ tables so that any two agents on the same tables are not enemies, and that’s the minimum number of tables you can get. Prove that there are at least $2^{10}-11$ avengers in the agency. This problem is adapted from 2015 IMO Shortlist C7.

On a straight line lie $100$ points and another point outside the line. Which is the biggest the number of isosceles triangles can be formed from the vertices of these $101$ points?

The numbers $1, 2, 3, ... , 64$ are written in the cells of an $8 \times 8$ board (in some order, one per cell). Show that at least four $2 \times 2$ squares have sum greater than $100$.

A club of $11$ people has a committee. At every meeting of the committee a new committee is formed which differs by $1$ person from its predecessor (either one new member is included or one member is removed). The committee must always have at least three members and , according to the club rules, the committee membership at any stage must differ from its membership at every previous stage. Is it possible that after some time all possible compositions of the committee will have already occurred? (S. Fomin , Leningrad)

Let $n$ be a positive integer. Find the number of permutations $\sigma$ of the set $\{1, 2, ..., n\}$ such that $\sigma(j) \geq j$ holds for exactly two values of $j$.

Let $b\geq 3$ be a positive integer and let $\sigma$ be a nonidentity permutation of the set $\{0,1,\ldots,b-1\}$ such that $\sigma(0)=0.$ The [i]substitution cipher[/i] $C_\sigma$ encrypts every positive integer $n$ by replacing each digit $a$ in the representation of $n$ in base $b$ with $\sigma(a).$ Let $d$ be any positive integer such that $b$ does not divide $d.$ We say that $C_\sigma$ [i]complies[/i] with $d$ if $C_\sigma$ maps every multiple of $d$ onto a multiple of $d,$ and we say that $d$ is [i]cryptic[/i] if there exists some $C_\sigma$ such that $C_\sigma$ complies with $d.$ Let $k$ be any positive integer, and let $p=2^k+1.$ a) Find the greatest power of $2$ that is cryptic in base $2p,$ and prove that there is only one substitution cipher complying with it. b) Find the greatest power of $p$ that is cryptic in base $2p,$ and prove that there is only one substitution cipher complying with it. c) Suppose, furthermore, that $p$ is a prime number. Find the greatest cryptic positive integer in base $2p$ and prove that there is only one substitution cipher that complies with it. [i]Proposed by Nikolai Beluhov, Bulgaria[/i]

Let $X$ be an $n$-element set and let $A_1,\ldots ,A_m$ be subsets of $X$ such that i) $|A_i|=3$ for each $i=1,\ldots ,m$. ii) $|A_i\cap A_j|\le 1$ for any two distinct indices $i,j$. Show that there exists a subset of $X$ with at least $\lfloor\sqrt{2n}\rfloor$ elements which does not contain any of the $A_i$’s.

Set $T\subset\{1,2,\dots,n\}^3$ has the property that for any two triplets $(a,b,c)$ and $(x,y,z)$ in $T$, we have $a<b<c$, and also, we know that at most one of the equalities $a=x$, $b=y$, $c=z$ holds. Maximize $|T|$.

The points $S(i, j)$ with integer Cartesian coordinates $0 < i \leq n, 0 < j \leq m, m \leq n$, form a lattice. Find the number of: [b](a)[/b] rectangles with vertices on the lattice and sides parallel to the coordinate axes; [b](b)[/b] squares with vertices on the lattice and sides parallel to the coordinate axes; [b](c)[/b] squares in total, with vertices on the lattice.

$n$ players participated in a competition. Any two players have played exactly one game, and there was no tie game. For a set of $k(\le n)$ players, if it is able to line the players up so that each player won every player at the back, we call the set [i]ranked[/i]. For each player who participated in the competition, the set of players who lost to the player is ranked. Prove that the whole set of players can be split into three or less ranked sets.

For a permutation $\pi$ of $\{1,2,3,\ldots,n\}$, let $\text{Inv}(\pi)$ be the number of pairs $(i,j)$ with $1 \leq i < j \leq n$ and $\pi(i) > \pi(j)$. [list=1] [*] Given $n$, what is $\sum \text{Inv}(\pi)$ where the sum ranges over all permutations $\pi$ of $\{1,2,3,\ldots,n\}$? [*] Given $n$, what is $\sum \left(\text{Inv}(\pi)\right)^2$ where the sum ranges over all permutations $\pi$ of $\{1,2,3,\ldots,n\}$?[/list] [i]Brian Hamrick.[/i]

$n \geq 5$ football teams participate in a round-robin tournament. For every game played, the winner receives 3 points, the loser receives 0 points, and in the event of a draw, both teams receive 1 point. The third-from-bottom team has fewer points than all the teams ranked before it, and more points than the last 2 teams; it won more games than all the teams before it, but fewer games than the 2 teams behind it. Find the smallest possible $n$.

The number selection game is the following single-player game. Originally, on the table there were positive integers $1, 2,...,22$ (All positive integers not exceeding $22$ appear exactly once). In each move, the player chooses the three numbers $a, b, c$ that are on the table, then the selected numbers $a, b, c$ disappear but a new number $a + b + c$ appears; At the same time, the player's score is added $(a + b)(b+c)(c + a)$. The initial score was $0$. The game ends after $10$ moves (when there are only two numbers left on the board). Call $M, m$ respectively the highest and the lowest possible score of a game. Determine the value of $\dfrac{M}{m}$.

We wish to write $n$ distinct real numbers $(n\geq3)$ on the circumference of a circle in such a way that each number is equal to the product of its immediate neighbors to the left and right. Determine all of the values of $n$ such that this is possible.

[b]p1.[/b] Consider the equation $$x_1x_2 + x_2x_3 + x_3x_4 + · · · + x_{n-1}x_n + x_nx_1 = 0$$ where $x_i \in \{1,-1\}$ for $i = 1, 2, . . . , n$. (a) Show that if the equation has a solution, then $n$ is even. (b) Suppose $n$ is divisible by $4$. Show that the equation has a solution. (c) Show that if the equation has a solution, then $n$ is divisible by $4$. [b]p2.[/b] (a) Find a polynomial $f(x)$ with integer coefficients and two distinct integers $a$ and $b$ such that $f(a) = b$ and $f(b) = a$. (b) Let $f(x)$ be a polynomial with integer coefficients and $a$, $b$, and $c$ be three integers. Suppose $f(a) = b$, $f(b) = c$, and $f(c) = a$. Show that $a = b = c$. [b]p3.[/b] (a) Consider the triangle with vertices $M$ $(0, 2n + 1)$, $S$ $(1, 0)$, and $U \left(0, \frac{1}{2n^2}\right)$, where $n$ is a positive integer. If $\theta = \angle MSU$, prove that $\tan \theta = 2n - 1$. (b) Find positive integers $a$ and $b$ that satisfy the following equation. $$arctan \frac18 = arctan \,\,a - arctan \,\, b$$ (c) Determine the exact value of the following infinite sum. $$arctan \frac12 + arctan \frac18 + arctan \frac{1}{18} + arctan \frac{1}{32}+ ... + arctan \frac{1}{2n^2}+ ...$$ [b]p4.[/b] (a) Prove: $(55 + 12\sqrt{21})^{1/3} +(55 - 12\sqrt{21})^{1/3}= 5$. (b) Completely factor $x^8 + x^6 + x^4 + x^2 + 1$ into polynomials with integer coefficients, and explain why your factorization is complete. [b]p5.[/b] In this problem, we simulate a hula hoop as it gyrates about your waist. We model this situation by representing the hoop with a rotating a circle of radius $2$ initially centered at $(-1, 0)$, and representing your waist with a fixed circle of radius $1$ centered at the origin. Suppose we mark the point on the hoop that initially touches the fixed circle with a black dot (see the left figure). As the circle of radius $2$ rotates, this dot will trace out a curve in the plane (see the right figure). Let $\theta$ be the angle between the positive x-axis and the ray that starts at the origin and goes through the point where the fixed circle and circle of radius $2$ touch. Determine formulas for the coordinates of the position of the dot, as functions $x(\theta)$ and $y(\theta)$. The left figure shows the situation when $\theta = 0$ and the right figure shows the situation when $\theta = 2pi/3$. [img]https://cdn.artofproblemsolving.com/attachments/8/6/d15136872118b8e14c8f382bc21b41a8c90c66.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In a contest, there are $m$ candidates and $n$ judges, where $n\geq 3$ is an odd integer. Each candidate is evaluated by each judge as either pass or fail. Suppose that each pair of judges agrees on at most $k$ candidates. Prove that \[{\frac{k}{m}} \geq {\frac{n-1}{2n}}. \]

Two players $A$ and $B$ take stones one after the other from a heap with $n \ge 2$ stones. $A$ begins the game and takes at least one stone, but no more than $n -1$ stones. Thereafter, a player on turn takes at least one, but no more than the other player has taken before him. The player who takes the last stone wins. Who of the players has a winning strategy?

Let $n$ be positive integer and $S$= {$0,1,…,n$}, Define set of point in the plane. $$A = \{(x,y) \in S \times S \mid -1 \leq x-y \leq 1 \} $$, We want to place a electricity post on a point in $A$ such that each electricity post can shine in radius 1.01 unit. Define minimum number of electricity post such that every point in $A$ is in shine area

There are $n$ cities in a country, one of which is the capital. An airline operates bi-directional flights between some pairs of cities such that one can reach any city from any other city. The airline wants to close down some (possibly zero) number of flights, such that the number of flights needed to reach any particular city from the capital does not increase. Suppose that there are an odd number of ways that the airline can do this. Prove that the set of cities can be divided into two groups, such that there is no flight between two cities of the same group. [i]Proposed by Pranjal Srivastava[/i]

Find the sum of the values of $x$ for which $\binom{x}{0}-\binom{x}{1}+\binom{x}{2}-...+\binom{x}{2008}=0$

Let $n \geq 2$ be a positive integer and let $A, B$ be two finite sets of integers such that $|A| \leq n$. Let $C$ be a subset of the set $\{(a, b) | a \in A, b \in B\}$. Achilles writes on a board all possible distinct differences $a-b$ for $(a, b) \in C$ and suppose that their count is $d$. He writes on another board all triplets $(k, l, m)$, where $(k, l), (k, m) \in C$ and suppose that their count is $p$. Show that $np \geq d^2.$

Given an alfabet of $n$ letters. A sequence of letters such that between any 2 identical letters there are no 2 identical letters is called a [i]word[/i]. a) Find the maximal possible length of a [i]word[/i]. b) Find the number of the [i]words[/i] of maximal length.

Is it possible to tile an $8\times8$ board with dominoes ($2\times1$ tiles) so that no two dominoes form a $2\times2$ square?