Find the maximum positive integer $k$ such that there exist $k$ positive integers which do not exceed $2017$ and have the property that every number among them cannot be a power of any of the remaining $k-1$ numbers.

Every cell of $100\times 100$ table is colored black or white. Every cell on table border is black. It is known, that in every $2\times 2$ square there are cells of two colors. Prove, that exist $2\times 2$ square that is colored in chess order.

In the Archery with an especial gun, the probability of goal is $90 \%.$ If we continue our work until we goal. [b]i)[/b] What is the probability which exactly $3$ balls consumed. [b]ii)[/b] What is the probability which at least $3$ balls consumed.

A $4\times 8$ grid is divided into $32$ unit squares. There are square tiles of sizes $1 \times 1$, $2 \times 2$, $3 \times 3$ and $4 \times 4$. The goal is to completely cover the grid using exactly $n$ of these tiles. [list=a] [*]Is it possible to do this if $n = 19$? [*]Is it possible to do this if $n = 14$? [*]Is it possible to do this if $n = 7$? [/list] [b]Note:[/b] The tiles cannot overlap or extend beyond the grid.

Each of $10$ identical jars contains some milk, up to $10$ percent of its capacity. At any time, we can tell the precise amount of milk in each jar. In a move, we may pour out an exact amount of milk from one jar into each of the other $9$ jars, the same amount in each case. Prove that we can have the same amount of milk in each jar after at most $10$ moves. [i](4 points)[/i]

Several points are given in the plane. Suppose that for any three of them, there exists an orthogonal coordinate system (determined by the two axes and the unit length) in which these three points have integer coordinates. Prove that there exists an orthogonal coordinate system in which all the given points have integer coordinates.

The [i]Olimpia[/i] country is formed by $n$ islands. The most populated one is called [i]Panacenter[/i], and every island has a different number of inhabitants. We want to build bridges between these islands, which we'll be able to travel in both directions, under the following conditions: a) No pair of islands is joined by more than one bridge. b) Using the bridges we can reach every island from Panacenter. c) If we want to travel from Panacenter to every other island, in such a way that we use each bridge at most once, the number of inhabitants of the islands we visit is strictly decreasing. Determine the number of ways we can build the bridges.

Let $n$ be a positive integer. In an $n\times n$ garden, a fountain is to be built with $1\times 1$ platforms covering the entire garden. Ana places all the platforms at a different height. Afterwards, Beto places water sources in some of the platforms. The water in each platform can flow to other platforms sharing a side only if they have a lower height. Beto wins if he fills all platforms with water. Find the least number of water sources that Beto needs to win no matter how Ana places the platforms.

Let $n\ge 4$ be a positive integer. Megavan and Minivan are playing a game, where Megavan secretly chooses a real number $x$ in $[0, 1]$. At the start of the game, the only information Minivan has about $x$ is $x$ in $[0, 1]$. He needs to now learn about $x$ based on the following protocols: at each turn of his, Minivan chooses a number $y$ and submits to Megavan, where Megavan replies immediately with one of $y > x$, $y < x$, or $y\simeq x$, subject to two rules: $\bullet$ The answers in the form of $y > x$ and $y < x$ must be truthful; $\bullet$ Define the score of a round, known only to Megavan, as follows: $0$ if the answer is in the form $y > x$ and $y < x$, and $|x - y|$ if in the form $y\simeq x$. Then for every positive integer $k$ and every $k$ consecutive rounds, at least one round has score no more than $\frac{1}{k + 1}$. Minivan's goal is to produce numbers $a, b$ such that $a\le x\le b$ and $b - a\le \frac 1n$. Let $f(n)$ be the minimum number of queries that Minivan needs in order to guarantee success, regardless of Megavan's strategy. Prove that $$n\le f(n) \le 4n$$ [i]Proposed by Anzo Teh Zhao Yang[/i]

Let $p$ be a prime number greater than $2$. Patricia wants $7$ not-necessarily different numbers from $\{1, 2, . . . , p\}$ to the black dots in the figure below, on such a way that the product of three numbers on a line or circle always has the same remainder when divided by $p$. [img]https://cdn.artofproblemsolving.com/attachments/3/1/ef0d63b8ff5341ffc340de0cc75b24c7229e23.png[/img] (a) Suppose Patricia uses the number $p$ at least once. How many times does she have the number $p$ then a minimum sum needed? (b) Suppose Patricia does not use the number $p$. In how many ways can she assign numbers? (Two ways are different if there is at least one black one dot different numbers are assigned. The figure is not rotated or mirrored.)

Let $k$ be a positive integer. Scrooge McDuck owns $k$ gold coins. He also owns infinitely many boxes $B_1, B_2, B_3, \ldots$ Initially, bow $B_1$ contains one coin, and the $k-1$ other coins are on McDuck's table, outside of every box. Then, Scrooge McDuck allows himself to do the following kind of operations, as many times as he likes: - if two consecutive boxes $B_i$ and $B_{i+1}$ both contain a coin, McDuck can remove the coin contained in box $B_{i+1}$ and put it on his table; - if a box $B_i$ contains a coin, the box $B_{i+1}$ is empty, and McDuck still has at least one coin on his table, he can take such a coin and put it in box $B_{i+1}$. As a function of $k$, which are the integers $n$ for which Scrooge McDuck can put a coin in box $B_n$?

If the set $S = \{1,2,3,…,16\}$ is partitioned into $n$ subsets, there must be a subset in which elements $a, b, c$ (can be the same) exist, satisfying $a+ b=c$. Find the maximum value of $n$.

Tracey baked a square cake whose surface is dissected in a $ 10 \times 10$ grid. In some of the fields she wants to put a strawberry such that for each four fields that compose a rectangle whose edges run in parallel to the edges of the cake boundary there is at least one strawberry. What is the minimum number of required strawberries?

Let $S = \lbrace A = (a_1, \ldots, a_s) \mid a_i = 0$ or $1, i = 1, \ldots, 8 \rbrace$. For any 2 elements of $S$, $A = \lbrace a_1, \ldots, a_8\rbrace$ and $B = \lbrace b_1, \ldots, b_8\rbrace$. Let $d(A,B) = \sum_{i=1}{8} |a_i - b_i|$. Call $d(A,B)$ the distance between $A$ and $B$. At most how many elements can $S$ have such that the distance between any 2 sets is at least 5?

Each of the squares on a $15$×$15$ board has a zero. At every step you choose a row or a column, we delete all the numbers from it and then we write the numbers from $1$ to $15$ in the empty cells, in an arbitrary order. find the sum possible maximum of the numbers on the board that can be achieved after a number finite number of steps.

Find the number of ordered triples of positive integers $(a, b, c)$, where $a, b,c$ is a strictly increasing arithmetic progression, $a + b + c = 2019$, and there is a triangle with side lengths $a, b$, and $c$.

In a badminton tournament there were 16 participants. Each pair of participants played at most one game and there were no draws. After the tournament it turned out that each participant has won a different number of games. Prove that each participant has lost a different number of games.

A set of (unit) squares of a $n\times n$ table is called [i]convenient[/i] if each row and each column of the table contains at least two squares belonging to the set. For each $n\geq 5$ determine the maximum $m$ for which there exists a [i]convenient [/i] set made of $m$ squares, which becomes in[i]convenient [/i] when any of its squares is removed.

Is it possible to place 100 consecutive numbers around a circle in some order such that the product of each pair of adjacent numbers is always a perfect square? (Recall that a number is a perfect square if it is the square of an integer.)

In a checkered square $101 \times 101$, each cell of the inner square $99 \times 99$ is painted in one of ten colors (cells adjacent to the border of the square, not painted). Could it turn out that in every in a $3\times 3$ square, is exactly one more cell painted the same color as the central cell?

Given a rectangle $\mathcal{R}$ of area $100000 $, Pancho must completely cover the rectangle $\mathcal{R}$ with a finite number of rectangles with sides parallel to the sides of $\mathcal{R}$ . Next, Martín colors some rectangles of Pancho's cover red so that no two red rectangles have interior points in common. If the red area is greater than $0.00001$, Martin wins. Otherwise, Pancho wins. Prove that Pancho can cover to ensure victory,

A $10 \times 10$ table consists of $100$ unit cells. A [i]block[/i] is a $2 \times 2$ square consisting of $4$ unit cells of the table. A set $C$ of $n$ blocks covers the table (i.e. each cell of the table is covered by some block of $C$ ) but no $n -1$ blocks of $C$ cover the table. Find the largest possible value of $n$.

The positive quadrant of a coordinate plane is divided into unit squares by lattice lines. Is it possible to color the squares in black and white so that: (i) In every square of side $n$ ($n \in N$) with a vertex at the origin and sides are parallel to the axes, there are more black than white squares; (ii) Every diagonal parallel to the line $y = x$ intersects only finitely many black squares?

Let $n$ be a positive integer. Dominoes are placed on a $2n \times 2n$ board in such a way that every cell of the board is adjacent to exactly one cell covered by a domino. For each $n$, determine the largest number of dominoes that can be placed in this way. (A domino is a tile of size $2 \times 1$ or $1 \times 2$. Dominoes are placed on the board in such a way that each domino covers exactly two cells of the board, and dominoes do not overlap. Two cells are said to be adjacent if they are different and share a common side.)

Let $n$ be a positive integer. Find the number of permutations $a_1$, $a_2$, $\dots a_n$ of the sequence $1$, $2$, $\dots$ , $n$ satisfying $$a_1 \le 2a_2\le 3a_3 \le \dots \le na_n$$. Proposed by United Kingdom