In each unit square of square $100*100$ write any natural number. Called rectangle with sides parallel sides of square $good$ if sum of number inside rectangle divided by $17$. We can painted all unit squares in $good$ rectangle. One unit square cannot painted twice or more. Find maximum $d$ for which we can guaranteed paint at least $d$ points.

Two permutations of $1,\ldots, n$ are written on the board: $a_1,\ldots,a_n$ $b_1,\ldots,b_n$ A move consists of one of the following two operations: 1) Change the first row to $b_{a_1},\ldots,b_{a_n}$ 2) Change the second row to $a_{b_1},\ldots,a_{b_n}$ The starting position is: $2134\ldots n$ $234\ldots n1$ Is it possible by finitely many moves to get: $2314\ldots n$ $234 \ldots n1$? [i]D. Zmiaikou[/i]

Each of the points $G$ and $H$ lying from different sides of the plane of hexagon $ABCDEF$ is connected with all vertices of the hexagon. Is it possible to mark 18 segments thus formed by the numbers $1, 2, 3, \ldots, 18$ and arrange some real numbers at points $A, B, C, D, E, F, G, H$ so that each segment is marked with the difference of the numbers at its ends? [i]Proposed by A. Golovanov[/i]

Let $n\geq 3$ be a positive integer. Alice and Bob are playing a game in which they take turns colouring the vertices of a regular $n$-gon. Alice plays the first move. Initially, no vertex is coloured. Both players start the game with $0$ points. In their turn, a player colours a vertex $V$ which has not been coloured and gains $k$ points where $k$ is the number of already coloured neighbouring vertices of $V$. (Thus, $k$ is either $0$, $1$ or $2$.) The game ends when all vertices have been coloured and the player with more points wins; if they have the same number of points, no one wins. Determine all $n\geq 3$ for which Alice has a winning strategy and all $n\geq 3$ for which Bob has a winning strategy.

Let $G$ is a graph and $x$ is a vertex of $G$. Define the transformation $\varphi_{x}$ over $G$ as deleting all incident edges with respect of $x$ and drawing the edges $xy$ such that $y\in G$ and $y$ is not connected with $x$ with edge in the beginning of the transformation. A graph $H$ is called $G-$[i]attainable[/i] if there exists a sequece of such transformations which transforms $G$ in $H.$ Let $n\in\mathbb{N}$ and $4|n.$ Prove that for each graph $G$ with $4n$ vertices and $n$ edges there exists $G-$[i]attainable[/i] graph with at least $9n^{2}/4$ triangles.

In a sports tournament of $n$ players, each pair of players plays against each other exactly one match and there are no draws.Show that the players can be arranged in an order $P_1,P_2, .... , P_n$ such that $P_i$ defeats $P_{i+1}$ for all $1 \le i \le n-1$.

Prove that it is possible to color each positive integers with one of three colors so that the following conditions are satisfied: $i)$ For each $n\in\mathbb{N}_{0}$ all positive integers $x$ such that $2^n\le x<2^{n+1}$ have the same color. $ii)$ There are no positive integers $x,y,z$ of the same color (except $x=y=z=2$) such that $x+y=z^2.$

Let $a_1<a_2<...<a_t$ be $t$ given positive integers where no three form an arithmetic progression. For $k=t,t+1,...$ define $a_{k+1}$ to be the smallest positive integer larger than $a_k$ satisfying the condition that no three of $a_1,a_2,...,a_{k+1}$ form an arithmetic progression. For any $x\in\mathbb{R}^+$ define $A(x)$ to be the number of terms in $\{a_i\}_{i\ge 1}$ that are at most $x$. Show that there exist $c>1$ and $K>0$ such that $A(x)\ge c\sqrt{x}$ for any $x>K$.

In an $m\times n$ rectangular chessboard,there is a stone in the lower leftmost square. Two persons A,B move the stone alternately. In each step one can move the stone upward or rightward any number of squares. The one who moves it into the upper rightmost square wins. Find all $(m,n)$ such that the first person has a winning strategy.

Kolya and Vasya each have $8$ cards with numbers from $1$ to $8$ (each has all the numbers from $1$ to $8$). Kolya put $4$ cards on the table, and Vasya put a card with a larger number on each of them. Now Vasya puts his remaining $4$ cards on the table. a) Can Kolya always put his own card with a larger number on each of Vasya’s cards? b) Can Kolya always put on each of Vasya’s cards his own card with a number no less than on Vasya’s card?

Given an initial integer $ n_0 > 1$, two players, $ {\mathcal A}$ and $ {\mathcal B}$, choose integers $ n_1$, $ n_2$, $ n_3$, $ \ldots$ alternately according to the following rules : [b]I.)[/b] Knowing $ n_{2k}$, $ {\mathcal A}$ chooses any integer $ n_{2k \plus{} 1}$ such that \[ n_{2k} \leq n_{2k \plus{} 1} \leq n_{2k}^2. \] [b]II.)[/b] Knowing $ n_{2k \plus{} 1}$, $ {\mathcal B}$ chooses any integer $ n_{2k \plus{} 2}$ such that \[ \frac {n_{2k \plus{} 1}}{n_{2k \plus{} 2}} \] is a prime raised to a positive integer power. Player $ {\mathcal A}$ wins the game by choosing the number 1990; player $ {\mathcal B}$ wins by choosing the number 1. For which $ n_0$ does : [b]a.)[/b] $ {\mathcal A}$ have a winning strategy? [b]b.)[/b] $ {\mathcal B}$ have a winning strategy? [b]c.)[/b] Neither player have a winning strategy?

Find all positive integers $n$ for which all positive divisors of $n$ can be put into the cells of a rectangular table under the following constraints: [list] [*]each cell contains a distinct divisor; [*]the sums of all rows are equal; and [*]the sums of all columns are equal. [/list]

A subset $S \in \{0, 1, 2, \cdots , 2000\}$ satisfies $|S|=401$. Prove that there exists a positive integer $n$ such that there are at least $70$ positive integers $x$ such that $x, x+n \in S$

Ruby writes the numbers $1, 2, 3, . . . , 10$ on the whiteboard. In each move, she selects two distinct numbers, $a$ and $b$, erases them, and replaces them with $a+b-1$. She repeats this process until only one number, $x$, remains. What are all the possible values of $x$?

The country Dreamland consists of $2016$ cities. The airline Starways wants to establish some one-way flights between pairs of cities in such a way that each city has exactly one flight out of it. Find the smallest positive integer $k$ such that no matter how Starways establishes its flights, the cities can always be partitioned into $k$ groups so that from any city it is not possible to reach another city in the same group by using at most $28$ flights. [i]Warut Suksompong, Thailand[/i]

An equilateral triangle $ A_1 A_2 A_3$ is divided into four smaller equilateral triangles by joining the midpoints $ A_4,A_5,A_6$ of its sides. Let $ A_7,...,A_{15}$ be the midpoints of the sides of these smaller triangles. The $ 15$ points $ A_1,...,A_{15}$ are colored either green or blue. Show that with any such colouring there are always three mutually equidistant points $ A_i,A_j,A_k$ having the same color.

Choose arbitrarily $n$ vertices of a regular $2n-$gon and colour them red. The remaining vertices are coloured blue. We arrange all red-red distances into a nondecreasing sequence and do the same with the blue-blue distances. Prove that the two sequences thus obtained are identical.

Turki has divided a square into finitely many white and green rectangles, each with sides parallel to the sides of the square. Within each white rectangle, he writes down its width divided by its height. Within each green rectangle, he writes down its height divided by its width. Finally, he calculates $S$, the sum of these numbers. If the total area of white rectangles equals the total area of green rectangles, determine the minimum possible value of $S$.

Consider an array of white unit squares arranged in a rectangular grid with $59$ rows of unit squares and $c$ columns of unit squares, for some positive integer $c$. What is the smallest possible value of $c$ such that, if we shade exactly $25$ unit squares in each column black, then there must necessarily be some row with at least $18$ black unit squares?

Let $n\geq 2$ be a positive integer. We call a [i]vertex[/i] every point in the coordinate plane, whose $x$ and $y$ coordinates both are from the set $\{1,2,3,...,n\}$. We call a segment between two vertices an [i]edge[/i], if its length if $1$. We've colored some edges red, such that between any two vertices, there is a unique path of red edges (a path may contain each edge at most once). The red edge $f$ is [i]vital[/i] for an edge $e$, if the path of red edges connecting the two endpoints of $e$ contain $f$. Prove that there is a red edge, such that it is vital for at least $n$ edges.

$k$ marbles are placed onto the cells of a $2024 \times 2024$ grid such that each cell has at most one marble and there are no two marbles are placed onto two neighboring cells (neighboring cells are defined as cells having an edge in common). a) Assume that $k=2024$. Find a way to place the marbles satisfying the conditions above, such that moving any placed marble to any of its neighboring cells will give an arrangement that does not satisfy both the conditions. b) Determine the largest value of $k$ such that for all arrangements of $k$ marbles satisfying the conditions above, we can move one of the placed marble onto one of its neighboring cells and the new arrangement satisfies the conditions above.

Three standard six-sided dice are rolled. What is the probability that the product of the values on the top faces of the three dice is a perfect cube?

Let $N$ be the number of convex $27$-gons up to rotation there are such that each side has length $ 1$ and each angle is a multiple of $2\pi/81$. Find the remainder when $N$ is divided by $23$.

Given a set $S$ that consists of $n \geq 3$ positive integers. It is known that if for some (not necessarily distinct) numbers $a,b,c,d$ from $S$ the equality $a-b=2(c-d)$ holds, then $a=b$ and $c=d$. Let $M$ be the biggest element in $S$. a) Prove that $M > \frac{n^2}{3}$. b) For $n=1024$ find the biggest possible value of $M$. [i]M. Zorka, Y. Sheshukou[/i]

[b]p1.[/b] A band of pirates unloaded some number of treasure chests from their ship. The number of pirates was between $60$ and $69$ (inclusive). Each pirate handled exactly $11$ treasure chests, and each treasure chest was handled by exactly $7$ pirates. Exactly how many treasure chests were there? Show that your answer is the only solution. [b]p2.[/b] Let $a$ and $b$ be the lengths of the legs of a right triangle, let $c$ be the length of the hypotenuse, and let $h$ be the length of the altitude drawn from the vertex of the right angle to the hypotenuse. Prove that $c+h>a+b$. [b]p3.[/b] Prove that $$\frac{1}{70}< \frac{1}{2} \frac{3}{4} \frac{5}{6} ... \frac{2001}{2002} < \frac{1}{40}$$ [b]p4.[/b] Given a positive integer $a_1$ we form a sequence $a_1 , a_2 , a _3,...$ as follows: $a_2$ is obtained from $a_1$ by adding together the digits of $a_1$ raised to the $2001$-st power; $a_3$ is obtained from $a_2$ using the same rule, and so on. For example, if $a_1 =25$, then $a_2 =2^{2001}+5^{2001}$, which is a $1399$-digit number containing $106$ $0$'s, $150$ $1$'s, 4124$ 42$'s, $157$ $3$'s, $148$ $4$'s, $141$ $5$'s, $128$ $6$'s, $1504 47$'s, $152$ $8$'s, $143$ $9$'s. So $a_3 = 106 \times 0^{2001}+ 150 \times 1^{2001}+ 124 \times 2^{2001}+ 157 \times 3^{2001}+ ...+ 143 \times 9^{2001}$ which is a $1912$-digit number, and so forth. Prove that if any positive integer $a_1$ is chosen to start the sequence, then there is a positive integer $M$ (which depends on $a_1$ ) that is so large that $a_n < M$ for all $n=1,2,3,...$ [b]p5.[/b] Let $P(x)$ be a polynomial with integer coefficients. Suppose that there are integers $a$, $b$, and $c$ such that $P(a)=0$, $P(b)=1$, and $P(c)=2$. Prove that there is at most one integer $n$ such that $P(n)=4$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].