Let $n$ and $k$ be relatively prime positive integers with $k<n$. Each number in the set $M=\{1,2,3,\ldots,n-1\}$ is colored either blue or white. For each $i$ in $M$, both $i$ and $n-i$ have the same color. For each $i\ne k$ in $M$ both $i$ and $|i-k|$ have the same color. Prove that all numbers in $M$ must have the same color.

For any finite sets $X$ and $Y$ of positive integers, denote by $f_X(k)$ the $k^{\text{th}}$ smallest positive integer not in $X$, and let $$X*Y=X\cup \{ f_X(y):y\in Y\}.$$Let $A$ be a set of $a>0$ positive integers and let $B$ be a set of $b>0$ positive integers. Prove that if $A*B=B*A$, then $$\underbrace{A*(A*\cdots (A*(A*A))\cdots )}_{\text{ A appears $b$ times}}=\underbrace{B*(B*\cdots (B*(B*B))\cdots )}_{\text{ B appears $a$ times}}.$$ [i]Proposed by Alex Zhai, United States[/i]

We call a matrix $\textsl{binary matrix}$ if all its entries equal to $0$ or $1$. A binary matrix is $\textsl{Good}$ if it simultaneously satisfies the following two conditions: (1) All the entries above the main diagonal (from left to right), not including the main diagonal, are equal. (2) All the entries below the main diagonal (from left to right), not including the main diagonal, are equal. Given positive integer $m$, prove that there exists a positive integer $M$, such that for any positive integer $n>M$ and a given $n \times n$ binary matrix $A_n$, we can select integers $1 \leq i_1 <i_2< \cdots < i_{n-m} \leq n$ and delete the $i_i$-th, $i_2$-th,$\cdots$, $i_{n-m}$-th rows and $i_i$-th, $i_2$-th,$\cdots$, $i_{n-m}$-th columns of $A_n$, then the resulting binary matrix $B_m$ is $\textsl{Good}$.

In a table-tennis tournament of $10$ contestants, any $2$ contestants meet only once. We say that there is a winning triangle if the following situation occurs: $i$-th contestant defeated the $j$-th contestant, $j$-th contestant defeated the $k$-th contestant, and, $k$-th contestant defeated the $i$-th contestant. Let, $W_i$ and $L_i $ be respectively the number of games won and lost by the $i$-th contestant. Suppose, $L_i+W_j\geq 8$ whenever the $j$-th contestant defeats the $i$-th contestant. Prove that, there are exactly $40$ winning triangles in this tournament.

There are $12$ delegates in a mathematical conference. It is known that every two delegates share a common friend. Prove that there is a delegate who has at least five friends in that conference. [i]Proposed by Nairy Sedrakyan[/i]

Ten gamblers started playing with the same amount of money. Each turn they cast (threw) five dice. At each stage the gambler who had thrown paid to each of his 9 opponents $\frac{1}{n}$ times the amount which that opponent owned at that moment. They threw and paid one after the other. At the 10th round (i.e. when each gambler has cast the five dice once), the dice showed a total of 12, and after payment it turned out that every player had exactly the same sum as he had at the beginning. Is it possible to determine the total shown by the dice at the nine former rounds ?

A piece of rectangular paper $20 \times 19$, divided into four units, is cut into several square pieces, the cuts being along the sides of the unit squares. Such a square piece is called odd square if the length of its side is an odd number. a) What is the minimum possible number of odd squares? b) What is the smallest value that the sum of the perimeters of the odd squares can take?

Given an integer $n > 2$, give an example of a set of $n$ mutually different numbers $a_1,...,a_n$ for which the set of their pairwise sums $a_i + a_j$ ($i \ne j$) contains as few different numbers as possible; also give an example of a set of n different numbers $b_1,...,b_n$ for which the set of their pairwise sums $b_i+b_j$ ($i \ne j$) contains as many different numbers as possible;

[b]p1.[/b] Suppose $5$ bales of hay are weighted two at a time in all possible ways. The weights obtained are $110$, $112$, $113$, $114$, $115$, $116$, $117$, $118$, $120$, $121$. What is the difference between the heaviest and the lightest bale? [b]p2.[/b] Paul and Paula are playing a game with dice. Each have an $8$-sided die, and they roll at the same time. If the number is the same they continue rolling; otherwise the one who rolled a higher number wins. What is the probability that the game lasts at most $3$ rounds? [b]p3[/b]. Find the unique positive integer $n$ such that $\frac{n^3+5}{n^2-1}$ is an integer. [b]p4.[/b] How many numbers have $6$ digits, some four of which are $2, 0, 1, 4$ (not necessarily consecutive or in that order) and have the sum of their digits equal to $9$? [b]p5.[/b] The Duke School has $N$ students, where $N$ is at most $500$. Every year the school has three sports competitions: one in basketball, one in volleyball, and one in soccer. Students may participate in all three competitions. A basketball team has $5$ spots, a volleyball team has $6$ spots, and a soccer team has $11$ spots on the team. All students are encouraged to play, but $16$ people choose not to play basketball, $9$ choose not to play volleyball and $5$ choose not to play soccer. Miraculously, other than that all of the students who wanted to play could be divided evenly into teams of the appropriate size. How many players are there in the school? [b]p6.[/b] Let $\{a_n\}_{n\ge 1}$ be a sequence of real numbers such that $a_1 = 0$ and $a_{n+1} =\frac{a_n-\sqrt3}{\sqrt3 a_n+1}$ . Find $a_1 + a_2 +.. + a_{2014}$. [b]p7.[/b] A soldier is fighting a three-headed dragon. At any minute, the soldier swings her sword, at which point there are three outcomes: either the soldier misses and the dragon grows a new head, the soldier chops off one head that instantaneously regrows, or the soldier chops off two heads and none grow back. If the dragon has at least two heads, the soldier is equally likely to miss or chop off two heads. The dragon dies when it has no heads left, and it overpowers the soldier if it has at least five heads. What is the probability that the soldier wins [b]p8.[/b] A rook moves alternating horizontally and vertically on an infinite chessboard. The rook moves one square horizontally (in either direction) at the first move, two squares vertically at the second, three horizontally at the third and so on. Let $S$ be the set of integers $n$ with the property that there exists a series of moves such that after the $n$-th move the rock is back where it started. Find the number of elements in the set $S \cap \{1, 2, ..., 2014\}$. [b]p9.[/b] Find the largest integer $n$ such that the number of positive integer divisors of $n$ (including $1$ and $n$) is at least $\sqrt{n}$. [b]p10.[/b] Suppose that $x, y$ are irrational numbers such that $xy$, $x^2 + y$, $y^2 + x$ are rational numbers. Find $x + y$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

There are $n$ students in a class, and some pairs of these students are friends. Among any six students, there are two of them that are not friends, and for any pair of students that are not friends there is a student among the remaining four that is friends with both of them. Find the maximum value of $n$.

Let $M = \{1,2,\cdots , 10\}$, and let $T$ be a set of 2-element subsets of $M$. For any two different elements $\{a,b\}, \{x,y\}$ in $T$, the integer $(ax+by)(ay+bx)$ is not divisible by 11. Find the maximum size of $T$.

$N$ unit squares on the infinite sheet of cross-lined paper are painted with black colour. Prove that you can cut out the finite number of square pieces and satisfy two conditions all the black squares are contained in those pieces the area of black squares is not less than $1/5$ and not greater than $4/5$ of every piece area.

Consider the set $S = \{1,2,3, ...,1441\}$. 1. Nora counts thoses subsets of $S$ having exactly two elements, tbe sum of which is even. Rania counts those subsets of $S$ having exactly two elements, the sum of which is odd. Determine the numbers counted by Nora and Rania. 2. Let $t$ be the number of subsets of $S$ which have at least two elements and the product of the elements is even. Determine the greatest power of $2$ which divides $t$. 3. Ahmad counts the subsets of $S$ having $77$ elements such that in each subset the sum of the elements is even. Bushra counts the subsets of $S$ having $77$ elements such that in each subset the sum of the elements is odd. Whose number is bigger? Determine the difference between the numbers found by Ahmad and Bushra.

In a faraway place in the Universe, a villain has a medal with special powers and wants to hide it so that no one else can use it. For this, the villain hides it in a vertex of a regular polygon with $2019$ sides. Olcoman, the savior of the Olcomita people, wants to get the medal to restore peace in the Universe, for which you have to pay $1000$ olcolones for each time he makes the following move: on each turn he chooses a vertex of the polygon, which turns green if the medal is on it or in one of the four vertices closest to it, or otherwise red. Find the fewest olcolones Olcoman needs to determine with certainty the position of the medal.

In a tournament each player played exactly one game against each of the other players. In each game the winner was awarded 1 point, the loser got 0 points, and each of the two players earned 1/2 point if the game was a tie. After the completion of the tournament, it was found that exactly half of the points earned by each player were earned against the ten players with the least number of points. (In particular, each of the ten lowest scoring players earned half of her/his points against the other nine of the ten). What was the total number of players in the tournament?

A square-shaped pizza with side length $30$ cm is cut into pieces (not necessarily rectangular). All cuts are parallel to the sides, and the total length of the cuts is $240$ cm. Show that there is a piece whose area is at least $36$ cm$^2$

Sunn and Tacks play a game alternately choosing a word among the following (German) words: ”bad”, ”binse”, ”kafig”, ”kosewort”, ”maitag”, ”name”, ”pol”, ”parade”, ”wolf”. Two words are said to compatible if they have exactly one consonant in common. In the first round, Sunn selects a word for herself and one for Tacks. In every consequent round, each player selects a word that is compatible with the one they chose in the previous round. Tacks wins the game if the two players successively select the same word. (a) Prove that Tacks can always win. How many rounds are necessary for that? (b) Upon Sunn’s desire, the word ”kafig” was replaced with the word ”feige”. Prove that Sunn can prevent Tacks from winning.

In each square of a $10\times 10$ board, there is an arrow pointing upwards, downwards, left, or right. Prove that it is possible to remove $50$ arrows from the board, such that no two remaining arrows point at each other.

A domino is a $1 \times 2$ (or 2 $\times 1$) rectangular piece; namely, made up of two squares. There is an $8 \times 8$ board such that each domino can be cover exactly two of its squares. John places $n$ dominoes on the board, so that each one covers exactly two squares of the board and it is no longer possible to place a piece more without overlapping with any of those already placed. Determine the smallest value of $n$ for which the described situation is possible.

A house has an even number of lamps distributed among its rooms in such a way that there are at least three lamps in every room. Each lamp shares a switch with exactly one other lamp, not necessarily from the same room. Each change in the switch shared by two lamps changes their states simultaneously. Prove that for every initial state of the lamps there exists a sequence of changes in some of the switches at the end of which each room contains lamps which are on as well as lamps which are off. [i]Proposed by Australia[/i]

Initially $A$ selects a graph with \( 2221 \) vertices such that each vertex is incident to at least one edge. Then $B$ deletes some of the edges (possibly none) from the chosen graph. Finally, $A$ pays $B$ one lev for each vertex that is incident to an odd number of edges. What is the maximum amount that $B$ can guarantee to earn?

We have a regular $n{}$-gon, with $n\geqslant 4$. We consider the arrangements of $n{}$ numbers on its vertices, each of which is equal to 1 or 2. For each such arrangement $K{}$, we find the number of odd sums among all sums of numbers in several consecutive vertices. This number is denoted by $\alpha(K)$. [list=a] [*]Find the largest possible value of $\alpha(K)$. [*]Find the number of arrangements for which $\alpha(K)$ takes this largest possible value. [/list] [i]Proposed by P. Kozhevnikov[/i]

Consider the following array: \[ 3, 5\\3, 8, 5\\3, 11, 13, 5\\3, 14, 24, 18, 5\\3, 17, 38, 42, 23, 5\\ \ldots \] Find the 5-th number on the $ n$-th row with $ n>5$.

Each of $n$ black squares and $n$ white squares can be obtained by a translation from each other. Every two squares of different colours have a common point. Prove that ther is a point belonging at least to $n$ squares. [i](V. Dolnikov)[/i]

In every vertex of a regular $n$ -gon exactly one chip is placed. At each $step$ one can exchange any two neighbouring chips. Find the least number of steps necessary to reach the arrangement where every chip is moved by $[\frac{n}{2}]$ positions clockwise from its initial position.