For every positive integer $n$, let $p(n)$ denote the number of sets $\{x_1, x_2, \dots, x_k\}$ of integers with $x_1 > x_2 > \dots > x_k > 0$ and $n = x_1 + x_3 + x_5 + \dots$ (the right hand side here means the sum of all odd-indexed elements). As an example, $p(6) = 11$ because all satisfying sets are as follows: $$\{6\}, \{6, 5\}, \{6, 4\}, \{6, 3\}, \{6, 2\}, \{6, 1\}, \{5, 4, 1\}, \{5, 3, 1\}, \{5, 2, 1\}, \{4, 3, 2\}, \{4, 3, 2, 1\}.$$ Show that $p(n)$ equals to the number of partitions of $n$ for every positive integer $n$.

$p, q, r$ are natural numbers greater than 1. There are $pq$ balls placed on a circle, and one number among $0, 1, 2, \cdots , pr-1$ is written on each ball, satisfying following conditions. (1) If $i$ and $j$ is written on two adjacent balls, $|i-j|=1$ or $|i-j|=pr-1$. (2) $i$ is written on a ball $A$. If we skip $q-1$ balls clockwise from $A$ and see $q^{th}$ ball, $i+r$ or $i-(p-1)r$ is written on it. (This condition is satisfied for every ball.) If $p$ is even, prove that the number of pairs of two adjacent balls with $1$ and $2$ written on it is odd.

All of the numbers $1, 2,3,...,1000000$ are initially colored black. On each move it is possible to choose the number $x$ (among the colored numbers) and change the color of $x$ and of all of the numbers that are not co-prime with $x$ (black into white, white into black). Is it possible to color all of the numbers white?

Aidan, Boyan, Calvin, Derek, Ephram, and Fanalex are all chamber musicians at a concert. In each performance, any combination of musicians of them can perform for all the others to watch. What is the minimum number of performances necessary to ensure that each musician watches every other musician play?

There are two round tables with $n{}$ dwarves sitting at each table. Each dwarf has only two friends: his neighbours to the left and to the right. A good wizard wants to seat the dwarves at one round table so that each two neighbours are friends. His magic allows him to make any $2n$ pairs of dwarves into pairs of friends (the dwarves in a pair may be from the same or from different tables). However, he knows that an evil sorcerer will break $n{}$ of those new friendships. For which $n{}$ is the good wizard able to achieve his goal no matter what the evil sorcerer does? [i]Mikhail Svyatlovskiy[/i]

James has a red jar, a blue jar and a pile of $100$ pebbles. Initially both jars are empty. A move consists of moving a pebble from the pile into one of the jars or returning a pebble from one of the jars to the pile. The numbers of pebbles in the red and blue jars determine the state of the game. The followwing conditions must be satisfied: (a) The red jar may never contain fewer pebbles than the blue jar; (b) The game may never be returned to a previous state. What is the maximum number of moves that James can make?

Consider the starting position in a game of bughouse. Exhibit a sequence of moves on both boards, indicating the chronology, such that at the end: (a) The positions on both boards are the same as the original positions. (b) It is White to play on one board, but Black to play on the other. (c) All four players still have the right to castle subsequently (equivalently, the kings and rooks haven’t moved). for each of the following cases : (a) without moving any pawns. (b) without moving any queen.

Find the smallest positive integer $n$ such that we can write numbers $1,2,\dots ,n$ in a 18*18 board such that: i)each number appears at least once ii)In each row or column,there are no two numbers having difference 0 or 1

Let $n>1$ be an odd integer, and let $a_1$, $a_2$, $\dots$, $a_n$ be distinct real numbers. Let $M$ be the maximum of these numbers and $m$ the minimum. Show that it is possible to choose the signs of the expression $s=\pm a_1\pm a_2\pm\dots\pm a_n$ so that \[m<s<M\]

In a certain big city, all the streets go in one of two perpendicular directions. During a drive in the city, a car does not pass through any place twice, and returns to the parking place along a street from which it started. If it has made 100 left turns, how many right turns must it have made? [i](5 points)[/i]

[b]p1.[/b] In a given tetrahedron the sum of the measures of the three face angles at each of the vertices is $180$ degrees. Prove that all faces of the tetrahedron are congruent triangles. [img]https://cdn.artofproblemsolving.com/attachments/c/c/40f03324fd19f6a5e0a5e541153a2b38faac79.png[/img] [b]p2.[/b] The digital sum $D(n)$ of a positive integer $n$ is defined recursively by: $D(n) = n$ if $1 \le n \le 9$ $D(n) = D(a_0 + a_1 + ... + a_m)$ if $n>9$ where $a_0 , a_1 ,..,a_m$ are all the digits of $n$ expressed in base ten. (For example, $D(959) = D(26) = D(8) = 8$.) Prove that $D(n \times 1234)= D(n)$ fcr all positive integers $n$ . [b]p3.[/b] A right triangle has area $A$ and perimeter $P$ . Find the largest possible value for the positive constant $k$ such that for every such triangle, $P^2 \ge kA$ . [b]p4.[/b] In the accompanying diagram, $\overline{AB}$ is tangent at $A$ to a circle of radius $1$ centered at $O$ . The segment $\overline{AP}$ is equal in length to the arc $AB$ . Let $C$ be the point of intersection of the lines $AO$ and $PB$ . Determine the length of segment $\overline{AC}$ in terms of $a$ , where $a$ is the measure of $\angle AOB$ in radians. [img]https://cdn.artofproblemsolving.com/attachments/e/0/596e269a89a896365b405af7bc6ca47a1f7c57.png[/img] [b]p5.[/b] Let $a_1 = a > 0$ and $a_2 = b >a$. Consider the sequence $\{a_1,a_2,a_3,...\}$ of positive numbers defined by: $a_3=\sqrt{a_1a_2}$, $a_4=\sqrt{a_2a_3}$, $...$ and in general, $a_n=\sqrt{a_{n-2}a_{n-1}}$, for $n\ge 3$ . Develop a formula $a_n$ expressing in terms of $a$, $b$ and $n$ , and determine $\lim_{n \to \infty} a_n$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

There are $k$ cities in Belarus and $k$ cities in Armenia, between some cities there are non-directed flights. From any Belarusian city there are exactly $n$ flights to Armenian cities, and for every pair of Armenian cities exactly two Belarusian cities have flights to both of the Armenian cities. a) Prove that from every Armenian city there are exactly $n$ flights to Belarusian cities. b) Prove that there exists a flight route in which every city is visited at most once and that consists of at least $\lfloor \frac{(n+1)^2}{4} \rfloor$ cities in each of the countries. [i]D. Gorovoy[/i]

An equilateral triangle of side length $ n$ is divided into unit triangles. Let $ f(n)$ be the number of paths from the triangle in the top row to the middle triangle in the bottom row, such that adjacent triangles in a path share a common edge and the path never travels up (from a lower row to a higher row) or revisits a triangle. An example is shown on the picture for $ n \equal{} 5$. Determine the value of $ f(2005)$.

1. The transformation $ n \to 2n \minus{} 1$ or $ n \to 3n \minus{} 1$, where $ n$ is a positive integer, is called the 'change' of $ n$. Numbers $ a$ and $ b$ are called 'similar', if there exists such positive integer, that can be got by finite number of 'changes' from both $ a$ and $ b$. Find all positive integers 'similar' to $ 2005$ and less than $ 2005$.

A room has $2017$ boxes in a circle. A set of boxes is [i]friendly[/i] if there are at least two boxes in the set, and for each boxes in the set, if we go clockwise starting from the box, we would pass either $0$ or odd number of boxes before encountering a new box in the set. $30$ students enter the room and picks a set of boxes so that the set is friendly, and each students puts a letter inside all of the boxes that he/she chose. If the set of the boxes which have $30$ letters inside is not friendly, show that there exists two students $A, B$ and boxes $a, b$ satisfying the following condition. (i). $A$ chose $a$ but not $b$, and $B$ chose $b$ but not $a$. (ii). Starting from $a$ and going clockwise to $b$, the number of boxes that we pass through, not including $a$ and $b$, is not an odd number, and none of $A$ or $B$ chose such boxes that we passed.

Let $B$ be a set of $k$ sequences each having $n$ terms equal to $1$ or $-1$. The product of two such sequences $(a_1, a_2, \ldots , a_n)$ and $(b_1, b_2, \ldots , b_n)$ is defined as $(a_1b_1, a_2b_2, \ldots , a_nb_n)$. Prove that there exists a sequence $(c_1, c_2, \ldots , c_n)$ such that the intersection of $B$ and the set containing all sequences from $B$ multiplied by $(c_1, c_2, \ldots , c_n)$ contains at most $\frac{k^2}{2^n}$ sequences.

Toward the end of a game of Fish, the $2$ through $7$ of spades, inclusive, remain in the hands of three distinguishable players: DBR, RB, and DB, such that each player has at least one card. If it is known that DBR either has more than one card or has an even-numbered spade, or both, in how many ways can the players’ hands be distributed?

Let $m$ and $n$ be positive integers greater than $1$. In each unit square of an $m\times n$ grid lies a coin with its tail side up. A [i]move[/i] consists of the following steps. [list=1] [*]select a $2\times 2$ square in the grid; [*]flip the coins in the top-left and bottom-right unit squares; [*]flip the coin in either the top-right or bottom-left unit square. [/list] Determine all pairs $(m,n)$ for which it is possible that every coin shows head-side up after a finite number of moves. [i]Thanasin Nampaisarn, Thailand[/i]

Consider a $100 \times 100$ table, and identify the cell in row $a$ and column $b$, $1 \leq a, b \leq 100$, with the ordered pair $(a, b)$. Let $k$ be an integer such that $51 \leq k \leq 99$. A $k$-knight is a piece that moves one cell vertically or horizontally and $k$ cells to the other direction; that is, it moves from $(a, b)$ to $(c, d)$ such that $(|a-c|, |b - d|)$ is either $(1, k)$ or $(k, 1)$. The $k$-knight starts at cell $(1, 1)$, and performs several moves. A sequence of moves is a sequence of cells $(x_0, y_0)= (1, 1)$, $(x_1, y_1), (x_2, y_2)$, $\ldots, (x_n, y_n)$ such that, for all $i = 1, 2, \ldots, n$, $1 \leq x_i , y_i \leq 100$ and the $k$-knight can move from $(x_{i-1}, y_{i-1})$ to $(x_i, y_i)$. In this case, each cell $(x_i, y_i)$ is said to be reachable. For each $k$, find $L(k)$, the number of reachable cells.

What is the largest number of cells that can be colored in a $7\times7$ table in such a way that any $2\times2$ subtable has at most 2 colored cells?

A game has two players. In the game there is a rectangular chocolate bar, with $60$ pieces, arranged in a $6 \times 1 0$ formation , which can be broken only along the lines dividing the pieces. The first player breaks the bar along one line, discarding one section . The second player then breaks the remaining section, discarding one section. The first player repeats this process with the remaining section , and so on. The game is won by the player who leaves a single piece. In a perfect game which player wins? {S. Fomin , Leningrad)

Written on a blackboard are $n$ nonnegative integers whose greatest common divisor is $1$. A [i]move[/i] consists of erasing two numbers $x$ and $y$, where $x\ge y$, on the blackboard and replacing them with the numbers $x-y$ and $2y$. Determine for which original $n$-tuples of numbers on the blackboard is it possible to reach a point, after some number of moves, where $n-1$ of the numbers of the blackboard are zeroes.

In a group of 2017 persons, any pair of persons has exactly one common friend (other than the pair of persons). Determine the smallest possible value of the difference between the numbers of friends of the person with the most friends and the person with the least friends in such a group.

Let $n$ be a positive integer. Consider $n$ points in the plane such that no three of them are collinear and no two of the distances between them are equal. One by one, we connect each point to the two points nearest to it by line segments (if there are already other line segments drawn to this point, we do not erase these). Prove that there is no point from which line segments will be drawn to more than $11$ points.

A salmon in a mountain river must overpass two waterfalls. In every minute, the probability of the salmon to overpass the first waterfall is $p > 0$, and the probability to overpass the second waterfall is $q > 0$. These two events are assumed to be independent. Compute the probability that the salmon did not overpass the first waterfall in $n$ minutes, assuming that it did not overpass both waterfalls in that time.