Let $S = \mathbb Z \times \mathbb Z$. A subset $P$ of $S$ is called [i]nice[/i] if [list] [*] $(a, b) \in P \implies (b, a) \in P$ [*] $(a, b)$, $(c, d)\in P \implies (a + c, b - d) \in P$ [/list] Find all $(p, q) \in S$ so that if $(p, q) \in P$ for some [i]nice[/i] set $P$ then $P = S$.

The numbers $1, 2, 3, \ldots , 10$ are written on the blackboard. In each step, Andrew chooses two numbers $a, b$ which are written on the blackboard such that $a\geqslant 2b$, he erases them, and in their place writes the number $a-2b$. Find all numbers $n$, such that after a sequence of steps as above, at the end only the number $n$ will remain on the blackboard.

People from n different countries sit at a round table. Assume that for every two members of the same country their neighbours sitting next to them on the right hand side are from different countries. Find the largest possible number of people sitting around the table?

Find the number of positive integers less than or equal to $2017$ that have at least one pair of adjacent digits that are both even. For example, count the numbers $24$, $1862$, and $2012$, but not $4$, $58$, or $1276$.

The vertices of the regular $n$-gon are marked with some colours (each vertex -- with one colour) in such a way, that the vertices of one colour represent the right polygon. Prove that there are two equal ones among the smaller polygons.

Write the number $1000$ as the sum of at least two consecutive positive integers. How many (different) ways are there to write it?

Around a circular table an even number of persons have a discussion. After a break they sit again around the circular table in a different order. Prove that there are at least two people such that the number of participants sitting between them before and after a break is the same.

On the plane, a closed curve with simple auto intersections is drawn continuously. In the plane a finite number is determined in this way from disjoint regions. Show that each of these regions can be completely painted either white or blue, so that every two regions that share a curve segment at its edges, they always have different colors. Clarification: a car intersection is simple if looking at a very small disk around from her, the curve looks like a junction $\times$.

Players A and B play a game. They are given a box with $n=>1$ candies. A starts first. On a move, if in the box there are $k$ candies, the player chooses positive integer $l$ so that $l<=k$ and $(l, k) =1$, and eats $l$ candies from the box. The player who eats the last candy wins. Who has winning strategy, in terms of $n$.

[u]Round 9[/u] [b]p25.[/b] Let $S$ be the set of the first $2017$ positive integers. Find the number of elements $n \in S$ such that $\sum^n_{i=1} \left\lfloor \frac{n}{i} \right\rfloor$ is even. [b]p26.[/b] Let $\{x_n\}_{n \ge 0}$ be a sequence with $x_0 = 0$,$x_1 = \frac{1}{20}$ ,$x_2 = \frac{1}{17}$ ,$x_3 = \frac{1}{10}$ , and $x_n = \frac12 ((x_{n-2} +x_{n-4})$ for $n\ge 4$. Compute $$ \left\lfloor \frac{1}{x_{2017!} -x_{2017!-1}} \right\rfloor.$$ [b]p27.[/b] Let $ABCDE$ be be a cyclic pentagon. Given that $\angle CEB = 17^o$, find $\angle CDE + \angle EAB$, in degrees. [u]Round 10[/u] [b]p28.[/b] Let $S = \{1,2,4, ... ,2^{2016},2^{2017}\}$. For each $0 \le i \le 2017$, let $x_i$ be chosen uniformly at random from the subset of $S$ consisting of the divisors of $2^i$ . What is the expected number of distinct values in the set $\{x_0,x_1,x_2,... ,x_{2016},x_{2017}\}$? [b]p29.[/b] For positive real numbers $a$ and $b$, the points $(a, 0)$, $(20,17)$ and $(0,b)$ are collinear. Find the minimum possible value of $a+b$. [b]p30.[/b] Find the sum of the distinct prime factors of $2^{36}-1$. [u]Round 11[/u] [b]p31.[/b] There exist two angle bisectors of the lines $y = 20x$ and $y = 17x$ with slopes $m_1$ and $m_2$. Find the unordered pair $(m_1,m_2)$. [b]p32.[/b] Triangle 4ABC has sidelengths $AB = 13$, $BC = 14$, $C A =15$ and orthocenter $H$. Let $\Omega_1$ be the circle through $B$ and $H$, tangent to $BC$, and let $\Omega_2$ be the circle through $C$ and $H$, tangent to $BC$. Finally, let $R \ne H$ denote the second intersection of $\Omega_1$ and $\Omega_2$. Find the length $AR$. [b]p33.[/b] For a positive integer $n$, let $S_n = \{1,2,3, ...,n\}$ be the set of positive integers less than or equal to $n$. Additionally, let $$f (n) = |\{x \in S_n : x^{2017}\equiv x \,\, (mod \,\, n)\}|.$$ Find $f (2016)- f (2015)+ f (2014)- f (2013)$. [u]Round 12[/u] [b]p34. [/b] Estimate the value of $\sum^{2017}_{n=1} \phi (n)$, where $\phi (n)$ is the number of numbers less than or equal $n$ that are relatively prime to n. If your estimate is $E$ and the correct answer is $A$, your score for this problem will be max $\max \left(0,\lfloor 15 - 75 \frac{|A-E|}{A} \rceil \right).$ [b]p35.[/b] An up-down permutation of order $n$ is a permutation $\sigma$ of $(1,2,3, ..., n)$ such that $\sigma(i ) <\sigma (i +1)$ if and only if $i$ is odd. Denote by $P_n$ the number of up-down permutations of order $n$. Estimate the value of $P_{20} +P_{17}$. If your estimate is $E$ and the correct answer is $A$, your score for this problem will be $\max \left(0, 16 -\lceil \max \left(\frac{E}{A}, 2- \frac{E}{A}\right) \rceil \right).$ [b]p36.[/b] For positive integers $n$, superfactorial of $n$, denoted $n\$ $, is defined as the product of the first $n$ factorials. In other words, we have $n\$ = \prod^n_{i=1}(i !)$. Estimate the number of digits in the product $(20\$)\cdot (17\$)$. If your estimate is $E$ and the correct answer is $A$, your score for this problem will be $\max \left(0, \lfloor 15 -\frac12 |A-E| \rfloor \right).$ PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3158491p28715220]here [/url] and 5-8 [url=https://artofproblemsolving.com/community/c3h3158514p28715373]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Three piles of stones are given. One may add to, or remove from one of the piles in one operation the number of stones in the other two piles. For example $[12,3,5]$ can become$ [12,20,5]$ by adding $17 = 12 + 5$ stones to pile 2 or $[4,3,5]$ by removing $8 = 3 + 5$ stones from pile $1$. Is it possible starting from the piles with $1993$, $199$ and $19$ stones to get one empty heap after several operations? (MN Gusarov)

$A_0A_1\ldots A_n$ is a regular polygon with $n+1$ vertices ($n&gt;2$). Initially $n$ stones are placed at vertex $A_0$. In each allowed operation, $2$ stones are moved simultaneously, at the player's choice: each stone is moved from the vertex where it is located to one of the adjacent $2$ vertices. Find all the values of $n$ for which it is possible to have, after a succession of permitted operations, a stone at each of the vertices $A_1,A_2,\ldots ,A_n$. Clarification: The two stones that move in an allowed operation can be at the same vertex or at different vertices.

Find for which natural numbers $n$ one can color the sides and diagonals of a regular $n$-gon with $n$ colors in such a way that for each triplet in pairs of different colors, a triangle can be found, the sides of which are sides or diagonals of $n$-gon and which is colored with exactly these three colors.

Let $n$ be a fixed positive integer. Initially, $n$ 1's are written on a blackboard. Every minute, David picks two numbers $x$ and $y$ written on the blackboard, erases them, and writes the number $(x+y)^4$ on the blackboard. Show that after $n-1$ minutes, the number written on the blackboard is at least $2^{\frac{4n^2-4}{3}}$. [i]Proposed by Calvin Deng[/i]

A ternary sequence is one whose terms all lie in the set $\{0, 1, 2\}$. Let $w$ be a length $n$ ternary sequence $(a_1,\ldots,a_n)$. Prove that $w$ can be extended leftwards and rightwards to a length $m=6n$ ternary sequence \[(d_1,\ldots,d_m) = (b_1,\ldots,b_p,a_1,\ldots,a_n,c_1,\ldots,c_q), \quad p,q\geqslant 0,\]containing no length $t > 2n$ palindromic subsequence. (A sequence is called palindromic if it reads the same rightwards and leftwards. A length $t$ subsequence of $(d_1,\ldots,d_m)$ is a sequence of the form $(d_{i_1},\ldots,d_{i_t})$, where $1\leqslant i_1<\cdots<i_t \leqslant m$.)

John and James wish to divide $25$ coins, of denominations $1, 2, 3, \ldots , 25$ kopeks. In each move, one of them chooses a coin, and the other player decides who must take this coin. John makes the initial choice of a coin, and in subsequent moves, the choice is made by the player having more kopeks at the time. In the event that there is a tie, the choice is made by the same player in the preceding move. After all the coins have been taken, the player with more kokeps wins. Which player has a winning strategy? [i](6 points)[/i]

(a) A radio lamp has a $7$-contact plug, with the contacts arranged in a circle. The plug is inserted into a socket with $7$ holes. Is it possible to number the contacts and the holes so that for any insertion at least one contact would match the hole with the same number? (b) A radio lamp has a $20$-contact plug, with the contacts arranged in a circle. The plug is inserted into a socket with $20$ holes. Let the contacts in the plug and the socket be already numbered. Is it always possible to insert the plug so that none of the contacts matches its socket?

Find the smallest positive integer $k$ with the following property: if each cell of a $100\times 100$ grid is dyed with one color and the number of cells of each color is not more than $104$, then there is a $k\times1$ or $1\times k$ rectangle that contains cells of at least three different colors.

Two types of tiles, depicted on the figure below, are given. [img]https://wiki-images.artofproblemsolving.com//2/23/Izrezak.PNG[/img] Find all positive integers $n$ such that an $n\times n$ board consisting of $n^2$ unit squares can be covered without gaps with these two types of tiles (rotations and reflections are allowed) so that no two tiles overlap and no part of any tile covers an area outside the $n\times n$ board. \\ [i]Proposed by Art Waeterschoot[/i]

Let $n>1$ be an integer. For each numbers $(x_1, x_2,\dots, x_n)$ with $x_1^2+x_2^2+x_3^2+\dots +x_n^2=1$, denote $m=\min\{|x_i-x_j|, 0<i<j<n+1\}$ Find the maximum value of $m$.

Eight problems were given to each of $30$ students. After the test was given, point values of the problems were determined as follows: a problem is worth $n$ points if it is not solved by exactly $n$ contestants (no partial credit is given, only zero marks or full marks). (a) Is it possible that the contestant having got more points that any other contestant had also solved less problems than any other contestant? (b) Is it possible that the contestant having got less points than any other contestant has solved more problems than any other contestant?

In a company of people some pairs are enemies. A group of people is called [i]unsociable[/i] if the number of members in the group is odd and at least $3$, and it is possible to arrange all its members around a round table so that every two neighbors are enemies. Given that there are at most $2015$ unsociable groups, prove that it is possible to partition the company into $11$ parts so that no two enemies are in the same part. [i]Proposed by Russia[/i]

Four sages stand around a non-transparent baobab. Each of the sages wears red, blue, or green hat. A sage sees only his two neighbors. Each of them at the same time must make a guess about the color of his hat. If at least one sage guesses correctly, the sages win. They could consult before the game started. How should they act to win?

Two persons play the following game with integers. The initial number is $2011^{2011}$. The players move in turns. Each move consists of subtraction of an integer between $1$ and $2010$ inclusive, or division by $2011$, rounding down to the closest integer when necessary. The player who first obtains a non-positive integer wins. Which player has a winning strategy?

Given a graph with $n \geq 4$ vertices. It is known that for any two of vertices there is a vertex connected with none of these two vertices. Find the greatest possible number of the edges in the graph.