Let $S$ be a set of three, not necessarily distinct, positive integers. Show that one can transform $S$ into a set containing $0$ by a finite number of applications of the following rule: Select two of the integers $x$ and $y$, where $x\leq y$ and replace them with $2x$ and $y-x.$

Suppose that every positve integer has been given one of the colors red, blue,arbitrarily. Prove that there exists an infinite sequence of positive integers $ a_{1} < a_{2} < a_{3} < \cdots < a_{n} < \cdots,$ such that inifinite sequence of positive integers $ a_{1},\frac {a_{1} \plus{} a_{2}}{2},a_{2},\frac {a_{2} \plus{} a_{3}}{2},a_{3},\frac {a_{3} \plus{} a_{4}}{2},\cdots$ has the same color.

For any natural numbers $n$, ( $n \geq 3$), let $f(n)$ denote the number of congruent integer-sided triangles with perimeter $n$. Show that (i) $f(1999) > f (1996)$; (ii) $f(2000) = f(1997)$.

There are several settlements on the bank of the Big Round Lake. Some of them are connected with the regular direct ship lines. Two settlements are connected if and only if two next (counterclockwise) to each ones are not connected. Prove that you can move from the arbitrary settlement to another arbitrary settlement, having used not more than three ships.

Let $n\ge 2$ be a given integer. Show that the number of strings of length $n$ consisting of $0'$s and $1'$s such that there are equal number of $00$ and $11$ blocks in each string is equal to \[2\binom{n-2}{\left \lfloor \frac{n-2}{2}\right \rfloor}\]

Every time Josh and Ron tap their screens, one of three emojis appears, each with equal probability: barbecue, bacon, or burger. Josh taps his screen until he gets a sequence of barbecue, bacon, and burger consecutively (in that specific order.) Ron taps his screen until he gets a sequence of three bacons in a row. Let $M$ and $N$ be the expected number of times Josh and Ron tap their screens, respectively. What is $|M-N|$?

Towns $A$, $B$ and $C$ are connected with a telecommunications cable. If you for example want to send a message from $A$ to $B$ is assigned to either a direct line between $A$ and $B$, or if necessary, a line via $C$. There are $43$ lines between $A$ and $B$, including those who go through $C$, and $29$ lines between $B$ and $C$, including those who go via $A$. How many lines, are there between $A$ and $C$ (including those who go via $B$)?

A $n \times n \times n$ cube is constructed using $1 \times 1 \times 1$ cubes, some of them black and others white, such that in each $n \times 1 \times 1$, $1 \times n \times 1$, and $1 \times 1 \times n$ subprism there are exactly two black cubes, and they are separated by an even number of white cubes (possibly 0). Show it is possible to replace half of the black cubes with white cubes such that each $n \times 1 \times 1$, $1 \times n \times 1$ and $1 \times 1 \times n$ subprism contains exactly one black cube.

Let $n$ be an integer greater than $1$. For a positive integer $m$, let $S_{m}= \{ 1,2,\ldots, mn\}$. Suppose that there exists a $2n$-element set $T$ such that (a) each element of $T$ is an $m$-element subset of $S_{m}$; (b) each pair of elements of $T$ shares at most one common element; and (c) each element of $S_{m}$ is contained in exactly two elements of $T$. Determine the maximum possible value of $m$ in terms of $n$.

On a table are given $1999$ red candies and $6661$ yellow candies. The candies are indistinguishable due to the same packing. A gourmet applies the following procedure as long as it is possible: (i) He picks any of the remaining candies, notes its color, eats it and goes to (ii). (ii) He picks any of the remaining candies, and notes its color: if it is the same as the color of the last eaten candy, eats it and goes to (ii); otherwise returns it upon repacking and goes to (i). Prove that all the candies will be eaten and find the probability that the last eaten candy will be red.

Can a square of side length $5$ be covered by three squares of side length $4$?

From a deck of playing cards, four [i]threes[/i], four [i]fours[/i] and four [i]fives[/i] are selected and put down on a table with the main side up. Players $A$ and $B$ alternately take the cards one by one and put them on the pile. Player $A$ begins. A player after whose move the sum of values of the cards on the pile is (a) greater than 34; (b) greater than 37; loses the game. Which player has a winning strategy?

A rectangle $\mathcal{R}$ with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of $\mathcal{R}$ are either all odd or all even. [i]Proposed by Jeck Lim, Singapore[/i]

[THE PROBLEM OF PAINTING THE THÁP RÙA (THE CENTRAL TOWER) MODEL] The following picture illustrates the model of the Tháp Rùa (the Central Tower) in Hanoi, which consists of $3$ levels. For the first and second levels, each has $10$ doorways among which $3$ doorways are located at the front, $3$ at the back, $2$ on the right side and $2$ on the left side. The top level of the tower model has no doorways. The front of the tower model is signified by a disk symbol on the top level. We paint the tower model with three colors: Blue, Yellow and Brown by fulfilling the following requirements: 1. The top level is painted with only one color. 2. In the second level, the $3$ doorways at the front are painted with the same color which is different from the one used for the center doorway at the back. Besides, any two adjacent doorways, including the pairs at the same corners, are painted with different colors. 3. For the first level, we apply the same rules as for the second level. [img]https://cdn.artofproblemsolving.com/attachments/2/3/18ee062b79693c4ccc26bf922a7f54e9f352ee.png[/img] (a) In how many ways the first level can be painted? (b) In how many ways the whole tower model can be painted?

Given a positive integer $N$. There are three squirrels that each have an integer. It is known that the largest integer and the least one differ by exactly $N$. Each time, the squirrel with the second largest integer looks at the squirrel with the largest integer. If the integers they have are different, then the squirrel with the second largest integer would be unhappy and attack the squirrel with the largest one, making its integer decrease by two times the difference between the two integers. If the second largest integer is the same as the least integer, only of the squirrels would attack the squirrel with the largest integer. The attack continues until the largest integer becomes the same as the second largest integer. What is the maximum total number of attacks these squirrels make? Proposed by USJL, ST.

In a multiple-choice test, there are 4 problems, each having 3 possible answers. In some group of examinees, it turned out that for every 3 of them, there was a question that each of them gave a different answer to. What is the maximal number of examinees in this group?

For every nonempty subset $R$ of $S = \{1,2,...,10\}$, we define the alternating sum $A(R)$ as follows: If $r_1,r_2,...,r_k$ are the elements of $R$ in the increasing order, then $A(R) = r_k -r_{k-1} +r_{k-2}- ... +(-1)^{k-1}r_1$. (a) Is it possible to partition $S$ into two sets having the same alternating sum? (b) Determine the sum $\sum_{R} A(R)$, where $R$ runs over all nonempty subsets of $S$.

Find the maximum number of Permutation of set {$1,2,3,...,2014$} such that for every 2 different number $a$ and $b$ in this set at last in one of the permutation $b$ comes exactly after $a$

We are given an angle $0^o < \phi \le 180^o$ and a circular disc. An ant begins its journey from an interior point of the disc, travelling in a straight line in a certain direction. When it reaches the edge of the disc, it does the following: it turns clockwise by the angle $\phi $, and if its new direction does not point towards the interior of the disc, it turns by the angle $\phi $ again, and repeats this until it faces the interior. Then it continues its journey in this new direction and turns as before every time when it reaches the edge. For what values of $\phi $ is it true that for any starting point and initial direction the ant eventually returns to its starting position?

On the blackboard, there are $K$ blanks. Alice decides $N$ values of blanks $(0-9)$ and then Bob determines the remaining digits. Find the largest possible integer $N$ such that Bob can guarantee to make the final number isn't a power of an integer.

Consider a checkered $3m\times 3m$ square, where $m$ is an integer greater than $1.$ A frog sits on the lower left corner cell $S$ and wants to get to the upper right corner cell $F.$ The frog can hop from any cell to either the next cell to the right or the next cell upwards. Some cells can be [i]sticky[/i], and the frog gets trapped once it hops on such a cell. A set $X$ of cells is called [i]blocking[/i] if the frog cannot reach $F$ from $S$ when all the cells of $X$ are sticky. A blocking set is [i] minimal[/i] if it does not contain a smaller blocking set.[list=a][*]Prove that there exists a minimal blocking set containing at least $3m^2-3m$ cells. [*]Prove that every minimal blocking set containing at most $3m^2$ cells.

Let $n\geqslant 2$ be a positive integer. Paul has a $1\times n^2$ rectangular strip consisting of $n^2$ unit squares, where the $i^{\text{th}}$ square is labelled with $i$ for all $1\leqslant i\leqslant n^2$. He wishes to cut the strip into several pieces, where each piece consists of a number of consecutive unit squares, and then [i]translate[/i] (without rotating or flipping) the pieces to obtain an $n\times n$ square satisfying the following property: if the unit square in the $i^{\text{th}}$ row and $j^{\text{th}}$ column is labelled with $a_{ij}$, then $a_{ij}-(i+j-1)$ is divisible by $n$. Determine the smallest number of pieces Paul needs to make in order to accomplish this.

A grid of cells is tiled with dominoes such that every cell is covered by exactly one domino. A subset $S$ of dominoes is chosen. Is it true that at least one of the following 2 statements is false? (1) There are $2022$ more horizontal dominoes than vertical dominoes in $S$. (2) The cells covered by the dominoes in $S$ can be tiled completely and exactly by $L$-shaped tetrominoes.

Let $n>d>0$ integers. Batman, Joker, Clark play the following game in an infinite checkered board. Initially, Batman and Joker are in cells with distance $n$ and a candy is in a cell with distance $d$ to Batman. Batman is blindfold, and can only see his cell. Clark and Joker can see the whole board. The following two moves go alternately. 1 - Batman goes to an adjacent cell. If he touches Joker, Batman loses. If he touches the candy, Batman wins. If the cell is empty, Clark chooses to say loudly one of the following two words [b]hot[/b] or [b]cold[/b]. 2 - Joker goes to an adjacent cell. If he touches Batman or candy, Joker wins. Otherwise, the game continues. Determine for each $d$, the least $n$, such that Batman, and Clark can plan an strategy to ensure the Batman's win, regardless of initial positions of the Joker and of the candy. Note: Two cells are adjacent if its have a common side. The distance between two cells $X$ and $Y$ is the least $p$ such that there exist cells $X=X_0,X_1,X_2,\dots, X_p=Y$ with $X_i$ adjacent to $X_{i-1}$ for all $i=1,2,\dots,p$.

Some students on the faculty speak several languages and some - Russian only. $50$ of them know English, $50$ -- French and $50$ -- Spanish. Prove that it is possible to divide them onto $5$ groups, not necessary equal, to get $10$ of them knowing English, $10$ -- French and $10$ -- Spanish in each group.