Each of the numbers in the set $N = \{1, 2, 3, \cdots, n - 1\}$, where $n \geq 3$, is colored with one of two colors, say red or black, so that: [i](i)[/i] $i$ and $n - i$ always receive the same color, and [i](ii)[/i] for some $j \in N$, relatively prime to $n$, $i$ and $|j - i|$ receive the same color for all $i \in N, i \neq j.$ Prove that all numbers in $N$ must receive the same color.

Let $p$ be a prime number and a table of size $(p^2+ p+1)\times (p^2+p + 1)$ which is divided into unit cells. The way to color some cells of this table is called nice if there are no four colored cells that form a rectangle (the sides of rectangle are parallel to the sides of given table). 1. Let $k$ be the number of colored cells in some nice coloring way. Prove that $k \le (p + 1)(p^2 + p + 1)$. Denote this number as $k_{max}$. 2. Prove that all ordered tuples $(a, b, c)$ with $0 \le a, b, c < p$ and $a + b + c > 0$ can be partitioned into $p^2 + p + 1$ sets $S_1, S_2, .. . S_{p^2+p+1}$ such that two tuples $(a_1, b_1, c_1)$ and $(a_2, b_2, c_2)$ belong to the same set if and only if $a_1 \equiv ka_2, b_1 \equiv kb_2, c_1 \equiv kc_2$ (mod $p$) for some $k \in \{1,2, 3, ... , p - 1\}$. 3. For $1 \le i, j \le p^2+p+1$, if there exist $(a_1, b_1, c_1) \in S_i$ and $(a_2, b_2, c_2) \in S_j$ such that $a_1a_2 + b_1b_2 + c_1c_2 \equiv 0$ (mod $p$), we color the cell $(i, j)$ of the given table. Prove that this coloring way is nice with $k_{max}$ colored cells

A balance has a left pan, a right pan, and a pointer that moves along a graduated ruler. Like many other grocer balances, this one works as follows: An object of weight $ L$ is placed in the left pan and another of weight $ R$ in the right pan, the pointer stops at the number $ R \minus{} L$ on the graduated ruler. There are $ n, (n \geq 2)$ bags of coins, each containing $ \frac{n(n\minus{}1)}{2} \plus{} 1$ coins. All coins look the same (shape, color, and so on). $ n\minus{}1$ bags contain real coins, all with the same weight. The other bag (we don’t know which one it is) contains false coins. All false coins have the same weight, and this weight is different from the weight of the real coins. A legal weighing consists of placing a certain number of coins in one of the pans, putting a certain number of coins in the other pan, and reading the number given by the pointer in the graduated ruler. With just two legal weighings it is possible to identify the bag containing false coins. Find a way to do this and explain it.

Some cells of a $2007\times 2007$ table are colored. The table is [i]charrua[/i] if none of the rows and none of the columns are completely colored.[list](a) What is the maximum number $k$ of colored cells that a charrua table can have? (b) For such $k$, calculate the number of distinct charrua tables that exist.[/list]

Let $ n$ be a positive integer, and let $ M\equal{}\left\{ 1,2,...,n\right\}$. Two players take turns at the following game: Each player, at his turn, has to select an element of $ M$ and remove all divisors of this element (including this element itself) from the set $ M$. [b]a)[/b] Assume that the player who cannot move anymore (because the set $ M$ is empty when it's his move) wins. For which values of $ n$ does the first player have a winning strategy? [b]b)[/b] Assume that the player who cannot move anymore (because the set $ M$ is empty when it's his move) loses. For which values of $ n$ does the first player have a winning strategy?

There are $999$ black points marked on a circle, dividing it into $999$ arcs of length $1$. We need to place $d$ arcs of lengths $1, 2, \dots, d$ such that each arc starts and ends at two black points, and none of the $d$ arcs is contained within another. Find the maximum value of $d$ for which this construction is possible. [b]Note:[/b] Two arcs can have one or more black points in common.

An $7\times 7$ array is divided in $49$ unit squares. Find all integers $n \in N^*$ for which $n$ checkers can be placed on the unit squares so that each row and each line have an even number of checkers. ($0$ is an even number, so there may exist empty rows or columns. A square may be occupied by at most $1$ checker).

Addison and Emerson are playing a card game with three rounds. Addison has the cards $1, 3$, and $5$, and Emerson has the cards $2, 4$, and $6$. In advance of the game, both designate each one of their cards to be played for either round one, two, or three. Cards cannot be played for multiple rounds. In each round, both show each other their designated card for that round, and the person with the higher-numbered card wins the round. The person who wins the most rounds wins the game. Let $m/n$ be the probability that Emerson wins, where $m$ and $n$ are relatively prime positive integers. Find $m +n$. [i]Proposed by Ada Tsui[/i]

Let there be $3399$ numbers arbitrarily chosen among the first $6798$ integers $1, 2, \ldots , 6798$ in such a way that none of them divides another. Prove that there are exactly $1982$ numbers in $\{1, 2, \ldots, 6798\}$ that must end up being chosen.

[b]p1.[/b] Two players play the following game. On the lowest left square of an $8 \times 8$ chessboard there is a rook (castle). The first player is allowed to move the rook up or to the right by an arbitrary number of squares. The second layer is also allowed to move the rook up or to the right by an arbitrary number of squares. Then the first player is allowed to do this again, and so on. The one who moves the rook to the upper right square wins. Who has a winning strategy? [b]p2.[/b] Find the smallest positive whole number that ends with $17$, is divisible by $17$, and the sum of its digits is $17$. [b]p3.[/b] Three consecutive $2$-digit numbers are written next to each other. It turns out that the resulting $6$-digit number is divisible by $17$. Find all such numbers. [b]p4.[/b] Let $ABCD$ be a convex quadrilateral (a quadrilateral $ABCD$ is called convex if the diagonals $AC$ and $BD$ intersect). Suppose that $\angle CBD = \angle CAB$ and $\angle ACD = \angle BDA$ . Prove that $\angle ABC = \angle ADC$. [b]p5.[/b] A circle of radius $1$ is cut into four equal arcs, which are then arranged to make the shape shown on the picture. What is its area? [img]https://cdn.artofproblemsolving.com/attachments/f/3/49c3fe8b218ab0a5378ecc635b797a912723f9.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For any $n$-tuple $a=(a_1,a_2,\ldots,a_n)\in\mathbb{N}_0^n$ of nonnegative integers, let $d_a$ denote the number of pairs of indices $(i,j)$ such that $a_i-a_j=1.$ Determine the maximum possible value of $d_a$ as $a$ ranges over all elements of $\mathbb{N}_0^n.$

Several zeros, ones and twos are written on the blackboard. An anonymous clean in turn pairs of different numbers, writing, instead of cleaned, the number not equal to each. ($0$ instead of pair $\{1,2\}, 1$ instead of $\{0,2\}, 2$ instead of $\{0,1\}$). Prove that if there remains one number only, it does not depend on the processing order.

There is an even number of cards on a table; a positive integer is written on each card. Let $a_k$ be the number of cards having $k$ written on them. It is known that \[a_n-a_{n-1}+a_{n-2}- \cdots \ge 0 \] for each positive integer $n$. Prove that the cards can be partitioned into pairs so that the numbers in each pair differ by $1$. [i](A. Golovanov)[/i]

The following $100$ numbers are written on the board: $$2^1 - 1, 2^2 - 1, 2^3 - 1, \dots, 2^{100} - 1.$$ Alice chooses two numbers $a,b,$ erases them and writes the number $\dfrac{ab - 1}{a+b+2}$ on the board. She keeps doing this until a single number remains on the board. If the sum of all possible numbers she can end up with is $\dfrac{p}{q}$ where $p, q$ are coprime, then what is the value of $\log_{2}(p+q)$?

In a football league, whenever a player is transferred from a team $X$ with $x$ players to a team $Y$ with $y$ players, the federation is paid $y-x$ billions liras by $Y$ if $y \geq x$, while the federation pays $x-y$ billions liras to $X$ if $x > y$. A player is allowed to change as many teams as he wishes during a season. Suppose that a season started with $18$ teams of $20$ players each. At the end of the season, $12$ of the teams turn out to have again $20$ players, while the remaining $6$ teams end up with $16,16, 21, 22, 22, 23$ players, respectively. What is the maximal amount the federation may have won during the season?

The integers $ 1,2,\dots,20$ are written on the blackboard. Consider the following operation as one step: [i]choose two integers $ a$ and $ b$ such that $ a\minus{}b \ge 2$ and replace them with $ a\minus{}1$ and $ b\plus{}1$[/i]. Please, determine the maximum number of steps that can be done. [i]Yudi Satria, Jakarta[/i]

Assume that a set $M$ of real numbers is the union of finitely many disjoint intervals with the total length greater than $1$. Prove that $M$ contains a pair of distinct numbers whose difference is an integer.

In fields of a $19 \times 19$ table are written integers so that any two lying on neighboring fields differ at most by $2$ (two fields are neighboring if they share a side). Find the greatest possible number of mutually different integers in such a table.

Consider a graph $G$ with $n$ vertices and at least $n^2/10$ edges. Suppose that each edge is colored in one of $c$ colors such that no two incident edges have the same color. Assume further that no cycles of size $10$ have the same set of colors. Prove that there is a constant $k$ such that $c$ is at least $kn^\frac{8}{5}$ for any $n$. [i]David Yang.[/i]

Given points $A, M, M_1$ and rational number $\lambda \neq -1$. Construct the triangle $ABC$, such that $M$ lies on $BC$ and $M_1$ lies on $B_1C_1$ ($B_1, C_1$ are the projections of $B, C$ on $AC, AB$ respectively), and $\frac{BM}{MC}=\frac{B_1M_1}{M_1C_1}=\lambda.$

In the Cartesian plane $Oxy$ we consider the points ${A_1}\left( {40,1} \right), {A_2}\left( {40,2} \right), \ldots , {A_{40}}\left( {40,40} \right)$ as well as the segments $O{A_1},O{A_2},\ldots,O{A_{40}}$. A point of the Cartesian plane $Oxy$ is called "good", if its coordinates are integers and it is internal of one segment $O{A_i}, i=1,2,3,\ldots,40$. Additionally, one of the segments $O{A_1},O{A_2},\ldots,O{A_{40}}$ is called "good" if it contains a "good" point. Find the number of "good" segments and "good" points.

We say that a polygon is rectangular when all of its angles are $90^\circ$ or $270^\circ$. Is it true that each rectangular polygon, which sides are with length equal to odd numbers only, [u]can't[/u] be covered with 2x1 domino tiles?

(a) A square is cut into right triangles with legs of lengths $3$ and $4$. Prove that the total number of the triangles is even. (b) A rectangle is cut into right triangles with legs of lengths $1$ and $2$. Prove that the total number of the triangles is even. (A Shapovalov)

Let $n\geqslant 3$ be a positiv integer. Ana chooses the positive integers $a_1,a_2,\ldots,a_n$ and for any non-empty subset $A\subseteq\{1,2,\ldots,n\}$ she computes the sum \[s_A=\sum_{k \in A}a_k.\]She orders these sums $s_1\leqslant s_2\leqslant\cdots\leqslant s_{2^n-1}.$ Prove that there exists a subset $B\subseteq\{1,2,\ldots,2^n-1\}$ with $2^{n-2}+1$ elements such that, regardless of the integers $a_1,a_2,\ldots,a_n$ chosen by Ana, these can be determined by only knowing the sums $s_i$ with $i\in B.$

The audience chooses two of twenty-nine cards, numbered from $1$ to $29$ respectively. The assistant of a magician chooses two of the remaining twenty-seven cards, and asks a member of the audience to take them to the magician, who is in another room. The two cards are presented to the magician in an arbitrary order. By an arrangement with the assistant beforehand, the magician is able to deduce which two cards the audience has chosen only from the two cards he receives. Explain how this may be done.