The children sitting around a circle are playing the game as follows. At first the teacher gives each child an even number of candies (bigger than $ 0$, may be equal, maybe different). A certain child gives half of his candies to his neighbor on the right. Then the child who has just received candies does the same if he has an even number of candies, otherwise he gets one candy from the teacher and then does the job; and so on. Prove that after several steps there will be a child who will be able, giving the teacher half of his candies, to make the numbers of candies of all the children equal.

There are six ports on a lake. Is it possible to organize a series of routes satisfying the following conditions ? [i](i)[/i] Every route includes exactly three ports; [i](ii)[/i] No two routes contain the same three ports; [i](iii)[/i] The series offers exactly two routes to each tourist who desires to visit two different arbitrary ports.

In the plane we consider rectangles whose sides are parallel to the coordinate axes and have positive length. Such a rectangle will be called a [i]box[/i]. Two boxes [i]intersect[/i] if they have a common point in their interior or on their boundary. Find the largest $ n$ for which there exist $ n$ boxes $ B_1$, $ \ldots$, $ B_n$ such that $ B_i$ and $ B_j$ intersect if and only if $ i\not\equiv j\pm 1\pmod n$. [i]Proposed by Gerhard Woeginger, Netherlands[/i]

A set $S$ of positive integers is $\textit{sum-complete}$ if there are positive integers $m$ and $n$ such that an integer $a$ is the sum of the elements of some nonempty subset of $S$ if and only if $m \le a \le n$. Let $S$ be a sum-complete set such that $\{1, 3\} \subset S$ and $|S| = 8$. Find the greatest possible value of the sum of the elements of $S$. [i]Proposed by Michael Tang[/i]

The natural numbers $y>x$ are written on the board. Vassya decides to write the reminder of one number on the board to some other non-zero number in each step. Prove that Vassya can find a natural number $k$ such that if $y>k$ then the number distinct numbers on the board after arbitrary number of steps does not exceed $\frac{y}{1000000}.$

We consider the graph with vertices $A_1,A_2,\dots A_{2015}$ , $B_1,B_2,\dots B_{2015}$ and edges $A_iA _{i+1}, A_iB_i, B_iB_{i+17} $, taken cyclicaly. Is it true that 4 cops can catch a robber on this graph for every initial position?( First the 4 cops make a move, then the robber makes a move, then the cops make a move etc. A move consists of jumping from the vertex you stay on an adiacent vertex or by staying on your current vertex. Everyone knows the position of everyone everytime. The cops can coordinate their moves. The robber is caught when he shares the same vertex with a cop.) Tuymaada 2017 Q8 Juniors

For each positive integer $m$, we define $L_m$ as the figure that is obtained by overlapping two $1 \times m$ and $m \times 1$ rectangles in such a way that they coincide at the $1 \times 1$ square at their ends, as shown in the figure. [asy] pair h = (1, 0), v = (0, 1), o = (0, 0); for(int i = 1; i < 5; ++i) { o = (i*i/2 + i, 0); draw(o -- o + i*v -- o + i*v + h -- o + h + v -- o + i*h + v -- o + i*h -- cycle); string s = "$L_" + (string)(i) + "$"; label(s, o + ((i / 2), -1)); for(int j = 1; j < i; ++j) { draw(o + j*v -- o + j*v + h); draw(o + j*h -- o + j*h + v); } } label("...", (18, 0.5)); [/asy] Using some figures $L_{m_1}, L_{m_2}, \dots, L_{m_k}$, we cover an $n \times n$ board completely, in such a way that the edges of the figure coincide with lines in the board. Among all possible coverings of the board, find the minimal possible value of $m_1 + m_2 + \dots + m_k$. Note: In covering the board, the figures may be rotated or reflected, and they may overlap or not be completely contained within the board.

Let $n$ be a positive integer. Determine the minimum number of lines that can be drawn on the plane so that they intersect in exactly $n$ distinct points.

Given any positive real number $\varepsilon$, prove that, for all but finitely many positive integers $v$, any graph on $v$ vertices with at least $(1+\varepsilon)v$ edges has two distinct simple cycles of equal lengths. (Recall that the notion of a simple cycle does not allow repetition of vertices in a cycle.) [i]Fedor Petrov, Russia[/i]

The $64$ cells of an $8 \times 8$ chessboard have $64$ different colours. A Knight stays in one cell. In each move, the Knight jumps from one cell to another cell (the $2$ cells on the diagonal of an $2 \times 3$ board) also the colours of the $2$ cells interchange. In the end, the Knight goes to a cell having common side with the cell it stays at first. Can it happen that: there are exactly $3$ cells having the colours different from the original colours?

Let $ p,q,n$ be three positive integers with $ p \plus{} q < n$. Let $ (x_{0},x_{1},\cdots ,x_{n})$ be an $ (n \plus{} 1)$-tuple of integers satisfying the following conditions : (a) $ x_{0} \equal{} x_{n} \equal{} 0$, and (b) For each $ i$ with $ 1\leq i\leq n$, either $ x_{i} \minus{} x_{i \minus{} 1} \equal{} p$ or $ x_{i} \minus{} x_{i \minus{} 1} \equal{} \minus{} q$. Show that there exist indices $ i < j$ with $ (i,j)\neq (0,n)$, such that $ x_{i} \equal{} x_{j}$.

You are given a checkered square, the side of which is $n – 1$ long and contains $n \geq 10$ nodes. A non-return path is a path along edges, the intersection of which with any horizontal or vertical line is a segment, point or empty set, and which does not pass along any edge more than once. What is the smallest number of non-return paths that can cover all the edges? (An edge is a unit segment between adjacent nodes.)

Alice and Bob take turns alternatively on a $2020\times2020$ board with Alice starting the game. In every move each person colours a cell that have not been coloured yet and will be rewarded with as many points as the coloured cells in the same row and column. When the table is coloured completely, the points determine the winner. Who has the wining strategy and what is the maximum difference he/she can grantees? [i]Proposed by Seyed Reza Hosseini[/i]

$25$ chess players are going to participate in a chess tournament. All are on distinct skill levels, and of the two players the one who plays better always wins. What is the least number of games needed to select the two best players?

Let $n > 1$ be a positive integer. A 2-dimensional grid, infinite in all directions, is given. Each 1 by 1 square in a given $n$ by $n$ square has a counter on it. A [i]move[/i] consists of taking $n$ adjacent counters in a row or column and sliding them each by one space along that row or column. A [i]returning sequence[/i] is a finite sequence of moves such that all counters again fill the original $n$ by $n$ square at the end of the sequence. [list] [*] Assume that all counters are distinguishable except two, which are indistinguishable from each other. Prove that any distinguishable arrangement of counters in the $n$ by $n$ square can be reached by a returning sequence. [*] Assume all counters are distinguishable. Prove that there is no returning sequence that switches two counters and returns the rest to their original positions.[/list] [i]Mitchell Lee and Benjamin Gunby.[/i]

A $101\times 101$ square grid is given with rows and columns numbered in order from $1$ to $101$. Each square that is contained in both an even-numbered row and an even-numbered column is cut out. A small section of the grid is shown below, with the cut-out squares in black. Compute the maximum number of $L$-triominoes (pictured below) that can be placed in the grid so that each $L$-triomino lies entirely inside the grid and no two overlap. Each $L$-triomino may be placed in the orientation pictured below, or rotated by $90^o$, $180^o$, or $270^o$. [img]https://cdn.artofproblemsolving.com/attachments/2/5/016d4e823e3df4b83556a49f7e612d40e3deba.png[/img]

[b]p1.[/b] An evil galactic empire is attacking the planet Naboo with numerous automatic drones. The fleet defending the planet consists of $101$ ships. By the decision of the commander of the fleet, some of these ships will be used as destroyers equipped with one rocket each or as rocket carriers that will supply destroyers with rockets. Destroyers can shoot rockets so that every rocket destroys one drone. During the attack each carrier will have enough time to provide each destroyer with one rocket but not more. How many destroyers and how many carriers should the commander assign to destroy the maximal number of drones and what is the maximal number of drones that the fleet can destroy? [b]p2.[/b] Solve the inequality: $\sqrt{x^2-3x+2} \le \sqrt{x+7}$ [b]p3.[/b] Find all positive real numbers $x$ and $y$ that satisfy the following system of equations: $$x^y = y^{x-y}$$ $$x^x = y^{12y}$$ [b]p4.[/b] A convex quadrilateral $ABCD$ with sides $AB = 2$, $BC = 8$, $CD = 6$, and $DA = 7$ is divided by a diagonal $AC$ into two triangles. A circle is inscribed in each of the obtained two triangles. These circles touch the diagonal at points $E$ and $F$. Find the distance between the points $E$ and $F$. [b]p5.[/b] Find all positive integer solutions $n$ and $k$ of the following equation: $$\underbrace{11... 1}_{n} \underbrace{00... 0}_{2n+3} + \underbrace{77...7}_{n+1} \underbrace{00...0}_{n+1}+\underbrace{11...1}_{n+2} = 3k^3.$$ [b]p6.[/b] The Royal Council of the planet Naboo consists of $12$ members. Some of these members mutually dislike each other. However, each member of the Council dislikes less than half of the members. The Council holds meetings around the round table. Queen Amidala knows about the relationship between the members so she tries to arrange their seats so that the members that dislike each other are not seated next to each other. But she does not know whether it is possible. Can you help the Queen in arranging the seats? Justify your answer. PS. You should use hide for answers.

On a long enough highway, a passenger in a bus observes the traffic. He notes that, during an hour, the bus going with a constant velocity overpasses $a$ cars and gets overpassed by $b$ cars, while $c$ cars pass in the opposite direction. Assuming that the traffic is the same in both directions, is it possible to determine the number of cars that pass along the highway per hour? (You may assume that the velocity of a car can take only two values.)

What is the greatest number of rays in space beginning at one point and forming pairwise obtuse angles?

Let $m, n$ be positive integers. There are $n$ piles of gold coins, so that pile $i$ has $a_i > 0$ coins in it $(i = 1, ..., n)$. Consider the following game: Step 1. Nadech picks sets $B_1, B_2, ... , B_n$, where each $B_i$ is a nonempty subset of $\{1, 2, . . . , m\}$, and gives them to Yaya. Step 2. Yaya picks a set $S$ which is also a nonempty subset of $\{1, 2, . . . , m\}$. Step 3. For each $i = 1, 2, . . . , n$, Nadech wins the coins in pile $i$ if $B_i \cap S$ has an even number of elements, and Yaya wins the coins in pile $i$ if $B_i \cap S$ has an odd number of elements. Show that, no matter how Nadech picks the sets $B_1, B_2, . . . , B_n$, Yaya can always pick $S$ so that she ends up with more gold coins than Nadech

a) Given a quartet of positive numbers $(a,b,c,d)$. It is transformed to the new one according to the rule: $a'=ab, b' =bc, c'=cd,d'=da$. The second one is transformed to the third according to the same rule and so on. Prove that if at least one initial number does not equal 1, than You can never obtain the initial set. b) Given a set of $2^k$ ($k$-th power of two) numbers, equal either to $1$ or to $-1$. It is transformed as that was in the a) problem (each one is multiplied by the next, and the last -- by the first. Prove that You will always finally obtain the set of positive units.

On the VI-th International Festival of Young Mathematicians in Sozopol $n$ teams were participating, each of which was with $k$ participants ($n>k>1$). The organizers of the competition separated the $nk$ participants into $n$ groups, each with $k$ people, in such way that no two teammates are in the same group. Prove that there can be found $n$ participants no two of which are in the same team or group.

Let $n, k, l$ be positive integers satisfying $n \ge 3$, $l \le n - 2, l - k \le \frac{n-3}{2}$. Suppose that $a_1, a_2, \dots, a_k$ are integers chosen from $\{1, 2, \dots, n\}$ such that the set of remainders of the subset sums over all subsets of $a_i$ when divided by $n$ is exactly $\{1, 2, \dots, l\}$. Show that \[ a_1 + a_2 + \dots + a_k = l. \]

In a Cartesian coordinate plane, call a rectangle $standard$ if all of its sides are parallel to the $x$- and $y$- axes, and call a set of points $nice$ if no two of them have the same $x$- or $y$- coordinate. First, Bert chooses a nice set $B$ of $2016$ points in the coordinate plane. To mess with Bert, Ernie then chooses a set $E$ of $n$ points in the coordinate plane such that $B\cup E$ is a nice set with $2016+n$ points. Bert returns and then miraculously notices that there does not exist a standard rectangle that contains at least two points in $B$ and no points in $E$ in its interior. For a given nice set $B$ that Bert chooses, define $f(B)$ as the smallest positive integer $n$ such that Ernie can find a nice set $E$ of size $n$ with the aforementioned properties. Help Bert determine the minimum and maximum possible values of $f(B)$. [i]Yannick Yao[/i]

Ana chose some unit squares of a $50 \times 50$ board and placed a chip on each of them. Prove that Beto can always choose at most $99$ empty unit squares and place a chip on each so that each row and each column of the board contains an even number of chips.