The $2013$ numbers $$\frac{1}{1\times 2}, \frac{1}{2\times 3},\frac{1}{3\times 4},...,\frac{1}{2013 \times 2014}$$ are arranged randomly on a circle. (a) Prove that there exist ten consecutive numbers on the circle whose sum is less than $\frac{1}{4000}$ . (b) Prove that there exist ten consecutive numbers on the circle whose sum is less than $\frac{1}{10000}$ .

$n$ positive numbers are given. Is it always possible to find a convex polygon with $n+3$ edges and a triangulation of it so that the length of the diameters used in the triangulation are the given $n$ numbers? [i]Proposed by Morteza Saghafian[/i]

Non-zero numbers are arranged in $n \times n$ square ($n>2$). Every number is exactly $k$ times less than the sum of all the other numbers in the same cross (i.e., $2n-2$ numbers written in the same row or column with this number). Find all possible $k$. [i]Proposed by D. Rostovsky, A. Khrabrov, S. Berlov [/i]

There are $n$ ($n \ge 10$) cards with numbers $1, 2, \ldots, n$ lying in a row on a table, face down, so that the numbers on any adjacent cards differ by at least $5.$ Is it always enough to turn at most $n-5$ cards to determine which of the cards has number $n$? (It is not necessary to turn the card with number $n$.)

Let $ N > M > 1$ be fixed integers. There are $ N$ people playing in a chess tournament; each pair of players plays each other once, with no draws. It turns out that for each sequence of $ M \plus{} 1$ distinct players $ P_0, P_1, \ldots P_M$ such that $ P_{i \minus{} 1}$ beat $ P_i$ for each $ i \equal{} 1, \ldots, M$, player $ P_0$ also beat $ P_M$. Prove that the players can be numbered $ 1,2, \ldots, N$ in such a way that, whenever $ a \geq b \plus{} M \minus{} 1$, player $ a$ beat player $ b$. [i]Gabriel Carroll.[/i]

Exactly $102$ country leaders arrived at the IMO. At the final session, the IMO chairperson wants to introduce some changes to the regulations, which the leaders must approve. To pass the changes, the chairperson must gather at least \(\frac{2}{3}\) of the votes "FOR" out of the total number of leaders. Some leaders do not attend such meetings, and it is known that there will be exactly $81$ leaders present. The chairperson must seat them in a square-shaped conference hall of size \(9 \times 9\), where each leader will be seated in a designated \(1 \times 1\) cell. It is known that exactly $28$ of these $81$ leaders will surely support the chairperson, i.e., they will always vote "FOR." All others will vote as follows: At the last second of voting, they will look at how their neighbors voted up to that moment — neighbors are defined as leaders seated in adjacent cells \(1 \times 1\) (sharing a side). If the majority of neighbors voted "FOR," they will also vote "FOR." If there is no such majority, they will vote "AGAINST." For example, a leader seated in a corner of the hall has exactly $2$ neighbors and will vote "FOR" only if both of their neighbors voted "FOR." (a) Can the IMO chairperson arrange their $28$ supporters so that they vote "FOR" in the first second of voting and thereby secure a "FOR" vote from at least \(\frac{2}{3}\) of all $102$ leaders? (b) What is the maximum number of "FOR" votes the chairperson can obtain by seating their 28 supporters appropriately? [i]Proposed by Bogdan Rublov[/i]

Let $f(k)$ be the number of integers $n$ satisfying the following conditions: (i) $0\leq n < 10^k$ so $n$ has exactly $k$ digits (in decimal notation), with leading zeroes allowed; (ii) the digits of $n$ can be permuted in such a way that they yield an integer divisible by $11$. Prove that $f(2m) = 10f(2m-1)$ for every positive integer $m$. [i]Proposed by Dirk Laurie, South Africa[/i]

In a table consisting of $n$ by $n$ small squares some squares are coloured black and the other squares are coloured white. For each pair of columns and each pair of rows the four squares on the intersections of these rows and columns must not all be of the same colour. What is the largest possible value of $n$?

Let $n$ be a positive integer. Lavi Dopes has two boards $n \times n$. On the first board, he writes an integer in each of his $n^2$ squares (the written numbers are not necessarily distinct). On the second board, he writes, on each square, the sum of the numbers corresponding, on the first board, to that square and to all its adjacent squares (that is, those that share a common vertex). For example, if $n = 3$ and if Lavi Dopes writes the numbers on the first board, as shown below, the second board will look like this. Next, Davi Lopes receives only the second board, and from it, he tries to discover the numbers written by Lavi Dopes on the first board. (a) If $n = 4$, is it possible that Davi Lopes always manages to find the numbers written by Lavi Dopes on the first board? (b) If $n = 5$, is it possible that Davi Lopes always manages to find the numbers written by Lavi Dopes on the first board?

On the board all the 20-digit numbers which have 10 ones and 10 twos in their decimal form are written. It is allowed to choose two different digits in any number and to reverse the order of digits in the interval between them. What is the maximal quantity of equal numbers which is possible to get on the board using such operations?

You are given a set of $n$ blocks, each weighing at least $1$; their total weight is $2n$. Prove that for every real number $r$ with $0 \leq r \leq 2n-2$ you can choose a subset of the blocks whose total weight is at least $r$ but at most $r + 2$.

Decide if it is possible to consider $2011$ points in a plane such that the distance between every two of these points is different from $1$ and each unit circle centered at one of these points leaves exactly $1005$ points outside the circle.

Find the number of ordered triples of sets $(T_1, T_2, T_3)$ such that 1. each of $T_1, T_2$, and $T_3$ is a subset of $\{1, 2, 3, 4\}$, 2. $T_1 \subseteq T_2 \cup T_3$, 3. $T_2 \subseteq T_1 \cup T_3$, and 4. $T_3\subseteq T_1 \cup T_2$.

A board consists of $2021 \times 2021$ squares all of which are white, except for one corner square which is black. Alma and Bertha play the following game. At the beginning, there is a piece on the black square. In each turn, the player must move the piece to a new square in the same row or column as the one in which the piece is currently. All squares that the piece moves across, including the ending square but excluding the starting square, must be white, and all squares that the piece moves across, including the ending square, become black by this move. Alma begins, and the first player unable to move loses. Which player may prepare a strategy which secures her the victory? [img]https://cdn.artofproblemsolving.com/attachments/a/7/270d82f37b729bfe661f8a92cea8be67e5625c.png[/img]

A circus act consists of $2024$ bamboo sticks of pairwise different heights placed in some order, with a monkey standing atop one of them. The circus master can then give commands to the monkey as follows: [color=#FFFFFF]___[/color]$\bullet$ Left! : When given this command, the monkey locates the closest bamboo stick to the left taller than the one it is currently atop, and jumps to it. If there is no such stick, the monkey stays put. [color=#FFFFFF]___[/color]$\bullet$ Right! : When given this command, the monkey locates the closest bamboo stick to the right taller than the one it is currently atop, and jumps to it. If there is no such stick, the monkey stays put. The circus master claims that given any two bamboo sticks, if the monkey is originally atop the shorter stick, then after giving at most $c$ commands he can reposition the monkey atop the taller stick. What is the smallest possible value of $c$? [i]Proposed by Archit Manas[/i]

Initially, there are 2018 distinct boxes on a table. In the first stage, Yazan and Bozan, starting with Yazan, take turns make $2016$ moves each, such that, in each move, the person whose turn selects a pair of boxes that is not written on the board, and writes the pair on the board. In the second stage, Bozan enumerates the $4032$ pairs with numbers from $1,2,\dots,4032$, in whichever order he wants, and puts $k$ balls in each boxes written contained in the $k^{th}$ pair. Is there a strategy for Bozan that guarantees that the number of balls in each box are distinct?

Give and Take divide $100$ coins between themselves as follows. In each step, Give chooses a handful of coins from the heap, and Take decides who gets this handful. This is repeated until all coins have been taken, or one of them has $9$ handfuls. In the latter case, the other gets all the remaining coins. What is the largest number of coins that Give can be sure of getting no matter what Take does? (A Shapovalov)

Toph wants to tile a rectangular $m\times n$ square grid with the $6$ types of tiles in the picture (moving the tiles is allowed, but rotating and reflecting is not). For which pairs $(m,n)$ is this possible?

Let $n \ge k \ge 3$ be integers. Show that for every integer sequence $1 \le a_1 < a_2 < . . . < a_k \le n$ one can choose non-negative integers $b_1, b_2, . . . , b_k$, satisfying the following conditions: [list=i] [*] $0 \le b_i \le n$ for each $1 \le i \le k$, [*] all the positive $b_i$ are distinct, [*] the sums $a_i + b_i$, $1 \le i \le k$, form a permutation of the first $k$ terms of a non-constant arithmetic progression. [/list]

[b]p1.[/b] Three camps are located in the vertices of an equilateral triangle. The roads connecting camps are along the sides of the triangle. Captain America is inside the triangle and he needs to know the distances between camps. Being able to see the roads he has found that the sum of the shortest distances from his location to the roads is 50 miles. Can you help Captain America to evaluate the distances between the camps? [b]p2.[/b] $N$ regions are located in the plane, every pair of them have a non-empty overlap. Each region is a connected set, that means every two points inside the region can be connected by a curve all points of which belong to the region. Iron Man has one charge remaining to make a laser shot. Is it possible for him to make the shot that goes through all $N$ regions? [b]p3.[/b] Money in Wonderland comes in $\$5$ and $\$7$ bills. (a) What is the smallest amount of money you need to buy a slice of pizza that costs $\$1$ and get back your change in full? (The pizza man has plenty of $\$5$ and $\$7$ bills.) For example, having $\$7$ won't do since the pizza man can only give you $\$5$ back. (b) Vending machines in Wonderland accept only exact payment (do not give back change). List all positive integer numbers which CANNOT be used as prices in such vending machines. (That is, find the sums of money that cannot be paid by exact change.) [b]p4.[/b] (a) Put $5$ points on the plane so that each $3$ of them are vertices of an isosceles triangle (i.e., a triangle with two equal sides), and no three points lie on the same line. (b) Do the same with $6$ points. [b]p5.[/b] Numbers $1,2,3,…,100$ are randomly divided in two groups $50$ numbers in each. In the first group the numbers are written in increasing order and denoted $a_1,a_2, ..., a_{50}$. In the second group the numberss are written in decreasing order and denoted $b_1,b_2, ..., b_{50}$. Thus $a_1<a_2<...<a_{50}$ and $ b_1>b_2>...>b_{50}$. Evaluate $|a_1-b_1|+|a_2-b_2|+...+|a_{50}-b_{50}|$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Originally, every square of $8 \times 8$ chessboard contains a rook. One by one, rooks which attack an odd number of others are removed. Find the maximal number of rooks that can be removed. (A rook attacks another rook if they are on the same row or column and there are no other rooks between them.) [i](6 points)[/i]

Is it possible to fill the cells of a table of size $2019\times2019$ with pairwise distinct positive integers in such a way that in each rectangle of size $1\times2$ or $2\times1$ the larger number is divisible by the smaller one, and the ratio of the largest number in the table to the smallest one is at most $2019?$

What is the maximum number of non-overlapping $ 2\times 1$ dominoes that can be placed on a $ 8\times 9$ checkerboard if six of them are placed as shown? Each domino must be placed horizontally or vertically so as to cover two adjacent squares of the board.

Does there exist a sphere passing through only one rational point? (A rational point is a point whose Cartesian coordinates are all rational numbers.) (A Rubin)

$2014$ points are placed on a circumference. On each of the segments with end points on two of the $2014$ points is written a non-negative real number. For any convex polygon with vertices on some of the $2014$ points, the sum of the numbers written on their sides is less or equal than $1$. Find the maximum possible value for the sum of all the written numbers.