The plane is divided into squares by two sets of parallel lines, forming an infinite grid. Each unit square is coloured with one of $1201$ colours so that no rectangle with perimeter $100$ contains two squares of the same colour. Show that no rectangle of size $1\times1201$ or $1201\times1$ contains two squares of the same colour. [i]Note: Any rectangle is assumed here to have sides contained in the lines of the grid.[/i] [i](Bulgaria - Nikolay Beluhov)[/i]

A $9\times 7$ rectangle is tiled with tiles of the two types: L-shaped tiles composed by three unit squares (can be rotated repeatedly with $90^\circ$) and square tiles composed by four unit squares. Let $n\ge 0$ be the number of the $2 \times 2 $ tiles which can be used in such a tiling. Find all the values of $n$.

$N$ children no two of the same height stand in a line. The following two-step procedure is applied: first, the line is split into the least possible number of groups so that in each group all children are arranged from the left to the right in ascending order of their heights (a group may consist of a single child). Second, the order of children in each group is reversed, so now in each group the children stand in descending order of their heights. Prove that in result of applying this procedure $N - 1$ times the children in the line would stand from the left to the right in descending order of their heights. [i](12 points)[/i]

There are villager $0$, villager $1$, . . . , villager $2015$ i.e. $2016$ people in the village. You are villager $0$. Each villager is either honest or liar. You don’t know each villager is honest or liar, but you know you are honest and the number of liar is equal or smaller than integer $T$. The villagers became to write one letter without fail from one day. For integers $1 \le n \le 2015$, there are set integers $1 < k_n < 2015$. The letter villager $i$ wrote in day $n$ of the morning is delivered to villager $i + k_n$ if villager $i$ is honest, or villager $i - k_n$ if villager $i$ is liar in day $n$ of the evening. If $i - j$ is divisible by $2016$, villager $i$ and $j$ point same villager. Villagers don’t know $k_n$, but sender is told when letters are received. Villager can write anything on a letter, and each villager receives letters from any villagers a sufficient number of times after enough time. i.e. there are $n$ satisfying $k = k_n$ infinitely for each integer $1 \le k \le 2015$. You want to know the honest persons of this village. You can gather all villagers just once and instruct in one day of noon. The honest person obeys your instruction but the liar person not always obeys and he or she writes on a letter anything possible. One day from your instruction for a while, you could determine all honest persons of this village. Find the maximum value of $T$ such that it is possible to do this if you instruct appropriate regardless of the villagers who are honest or liar.

Two players, Aurelio and Bernardo, play the following game. Aurelio begins by writing the number $1$. Next it is Bernardo's turn, who writes number $2$. From then on, each player chooses whether to add $1$ to the number just written by the previous player, or whether multiply that number by $2$. Then write the result and it's the other player's turn. The first player to write a number greater than $ 2007$ loses the game. Determine if one of the players can ensure victory no matter what the other does.

A natural number is written in each square of an $ m \times n$ chess board. The allowed move is to add an integer $ k$ to each of two adjacent numbers in such a way that non-negative numbers are obtained. (Two squares are adjacent if they have a common side.) Find a necessary and sufficient condition for it to be possible for all the numbers to be zero after finitely many operations.

Let $n>2$ be an integer. A deck contains $\frac{n(n-1)}{2}$ cards,numbered \[1,2,3,\cdots , \frac{n(n-1)}{2}\] Two cards form a [i]magic pair[/i] if their numbers are consecutive , or if their numbers are $1$ and $\frac{n(n+1)}{2}$. For which $n$ is it possible to distribute the cards into $n$ stacks in such a manner that, among the cards in any two stacks , there is exactly one [i]magic pair[/i]?

Let $n$ be a positive integer. Two players $A$ and $B$ play a game in which they take turns choosing positive integers $k \le n$. The rules of the game are: (i) A player cannot choose a number that has been chosen by either player on any previous turn. (ii) A player cannot choose a number consecutive to any of those the player has already chosen on any previous turn. (iii) The game is a draw if all numbers have been chosen; otherwise the player who cannot choose a number anymore loses the game. The player $A$ takes the first turn. Determine the outcome of the game, assuming that both players use optimal strategies. [i]Proposed by Finland[/i]

What is the minimum number of planes determined by $6$ points in space which are not all coplanar, and among which no three are collinear?

Given a set of $2^{2016}$ cards with the numbers $1,2, ..., 2^{2016}$ written on them. We divide the set of cards into pairs arbitrarily, from each pair, we keep the card with larger number and discard the other. We now again divide the $2^{2015}$ remaining cards into pairs arbitrarily, from each pair, we keep the card with smaller number and discard the other. We now have $2^{2014}$ cards, and again divide these cards into pairs and keep the larger one in each pair. We keep doing this way, alternating between keeping the larger number and keeping the smaller number in each pair, until we have just one card left. Find all possible values of this final card.

A positive integer is called [i]vaivém[/i] when, considering its representation in base ten, the first digit from left to right is greater than the second, the second is less than the third, the third is bigger than the fourth and so on alternating bigger and smaller until the last digit. For example, $2021$ is [i]vaivém[/i], as $2 > 0$ and $0 < 2$ and $2 > 1$. The number $2023$ is not [i]vaivém[/i], as $2 > 0$ and $0 < 2$, but $2$ is not greater than $3$. a) How many [i]vaivém[/i] positive integers are there from $2000$ to $2100$? b) What is the largest [i]vaivém[/i] number without repeating digits? c) How many distinct $7$-digit numbers formed by all the digits $1, 2, 3, 4, 5, 6$ and $7$ are [i]vaivém[/i]?

Numbers $x,y,z$ are positive integers and satisfy the equation $x+y+z=2013$. (E) a) Find the number of the triplets $(x,y,z)$ that are solutions of the equation (E). b) Find the number of the solutions of the equation (E) for which $x=y$. c) Find the solution $(x,y,z)$ of the equation (E) for which the product $xyz$ becomes maximum.

[b]p1.[/b] Compute $$\left(\frac{1}{10}\right)^{\frac12}\left(\frac{1}{10^2}\right)^{\frac{1}{2^4}}\left(\frac{1}{10^3}\right)^{\frac{1}{2^3}} ...$$ [b]p2.[/b] Consider the sequence $1, 2, 2, 3, 3, 3, 4, 4, 4, 4,...,$ where the positive integer $m$ appears $m$ times. Let $d(n)$ denote the $n$th element of this sequence starting with $n = 1$. Find a closed-form formula for $d(n)$. [b]p3.[/b] Let $0 < \theta < \frac{\pi}{2}$, prove that $$ \left( \frac{\sin^2 \theta}{2}+\frac{2}{\cos^2 \theta} \right)^{\frac14}+ \left( \frac{\cos^2 \theta}{2}+\frac{2}{\sin^2 \theta} \right)^{\frac14} \ge (68)^{\frac14} $$ and determine the value of \theta when the inequality holds as equality. [b]p4.[/b] In $\vartriangle ABC$, parallel lines to $AB$ and $AC$ are drawn from a point $Q$ lying on side $BC$. If $a$ is used to represent the ratio of the area of parallelogram $ADQE$ to the area of the triangle $\vartriangle ABC$, (i) find the maximum value of $a$. (ii) find the ratio $\frac{BQ}{QC}$ when $a =\frac{24}{49}.$ [img]https://cdn.artofproblemsolving.com/attachments/5/8/eaa58df0d55e6e648855425e581a6ba0ad3ea6.png[/img] [b]p5.[/b] Prove the following inequality $$\frac{1}{2009} < \frac{1}{2} \cdot \frac{3}{4} \cdot \frac{5}{6} \cdot \frac{7}{8}...\frac{2007}{2008}<\frac{1}{40}$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].Thanks to gauss202 for sending the problems.

In how many ways can you rearrange the letters of ‘Alejandro’ such that it contains one of the words ‘ned’ or ‘den’?

In a kingdom, there are roads open between some cities with lanes both ways, in such a way, that you can come from one city to another using those roads. The roads are toll, and the price for taking each road is distinct. A minister made a list of all routes that go through each city exactly once. The king marked the most expensive road in each of the routes and said to close all the roads that he marked at least once. After that, it became impossible to go from city $A$ to city $B$, from city $B$ to city $C$, and from city $C$ to city $A$. Prove that the kings order was followed incorrectly.

In a box there are $31$, $41$ and $59$ stones coloured, respectively, red, green and blue. Three players, having t-shirts of these three colours, play the following game. They sequentially make one of two moves: (I) either remove three stones of one colour from the box, (II) or replace two stones of different colours by two stones of the third colour. The game ends when all the stones in the box have the same colour and the winner is the player whose t-shirt has this colour. Assuming that the players play optimally, is it possible to decide whether the game ends and who will win, depending on who the starting player is?

Let $n\ge 2$ be a positive integer, and let $P_1P_2\dots P_{2n}$ be a polygon with $2n$ sides such that no two sides are parallel. Denote $P_{2n+1}=P_1$. For some point $P$ and integer $i\in\{1,2,\ldots,2n\}$, we say that $i$ is a $P$-good index if $PP_{i}>PP_{i+1}$, and that $i$ is a $P$-bad index if $PP_{i}<PP_{i+1}$. Prove that there's a point $P$ such that the number of $P$-good indices is the same as the number of $P$-bad indices.

Amber and Brian are playing a game using $2010$ coins. Throughout the game, the coins are divided into a number of piles of at least 1 coin each. A move consists of choosing one or more piles and dividing each of them into two smaller piles. (So piles consisting of only $1$ coin cannot be chosen.) Initially, there is only one pile containing all $2010$ coins. Amber and Brian alternatingly take turns to make a move, starting with Amber. The winner is the one achieving the situation where all piles have only one coin. Show that Amber can win the game, no matter which moves Brian makes.

The Euclidian three-dimensional space has been partitioned into three nonempty sets $A_1,A_2,A_3$. Show that one of these sets contains, for each $d > 0$, a pair of points at mutual distance $d$.

A regular hexagon with a side of $50$ was divided to equilateral triangles with unit side, parallel to the sides of the hexagon. It is allowed to delete any three nodes of the resulting lattice defining a segment of length $2$. As a result of several such operations, exactly one node remains. How many ways is this possible?

Let $n\ge 2$ be a natural number. An $n \times n$ grid is drawn on a blackboard and each field with one of the numbers $-1$ or $+1$ labeled. Then the $n$ row and also the $n$ column sums calculated and the sum $S_n$ of all these $2n$ sums determined. (a) Show that for no odd number $n$ there is a label with $S_n = 0$. (b) Show that if $n$ is an even number, there are at least six different labels with $S_n = 0$.

Let $ M(n )\equal{}\{\minus{}1,\minus{}2,\ldots,\minus{}n\}$. For every non-empty subset of $ M(n )$ we consider the product of its elements. How big is the sum over all these products?

On a $300 \times 300$ board, several rooks are placed that beat the entire board. Within this case, each rook beats no more than one other rook. At what least $k$, it is possible to state that there is at least one rook in each $k\times k$ square ?

Positive integers are colored in black and white. We know that the sum of two numbers of different colors is always black, and that there are infinitely many numbers that are white. Prove that the sum and product of two white numbers are also white numbers.

In the plane are $10$ lines in general position, which means that no $2$ are parallel and no $3$ are concurrent. Where $2$ lines intersect, we measure the smaller of the two angles formed between them. What is the maximum value of the sum of the measures of these $45$ angles?