Find all positive integers $k$ such that there exists some permutation of $(1, 2,...,1000)$ namely $(a_1, a_2,..., a_{1000}) $ and satisfy $|a_i - i| = k$ for all $i = 1,1000$.

[b]p1.[/b] Farmer James goes to Kristy’s Krispy Chicken to order a crispy chicken sandwich. He can choose from $3$ types of buns, $2$ types of sauces, $4$ types of vegetables, and $4$ types of cheese. He can only choose one type of bun and cheese, but can choose any nonzero number of sauces, and the same with vegetables. How many different chicken sandwiches can Farmer James order? [b]p2.[/b] A line with slope $2$ and a line with slope $3$ intersect at the point $(m, n)$, where $m, n > 0$. These lines intersect the $x$ axis at points $A$ and $B$, and they intersect the y axis at points $C$ and $D$. If $AB = CD$, find $m/n$. [b]p3.[/b] A multi-set of $11$ positive integers has a median of $10$, a unique mode of $11$, and a mean of $ 12$. What is the largest possible number that can be in this multi-set? (A multi-set is a set that allows repeated elements.) [b]p4.[/b] Farmer James is swimming in the Eggs-Eater River, which flows at a constant rate of $5$ miles per hour, and is recording his time. He swims $ 1$ mile upstream, against the current, and then swims $1$ mile back to his starting point, along with the current. The time he recorded was double the time that he would have recorded if he had swum in still water the entire trip. To the nearest integer, how fast can Farmer James swim in still water, in miles per hour? [b]p5.[/b] $ABCD$ is a square with side length $60$. Point $E$ is on $AD$ and $F$ is on $CD$ such that $\angle BEF = 90^o$. Find the minimum possible length of $CF$. [b]p6.[/b] Farmer James makes a trianglomino by gluing together $5$ equilateral triangles of side length $ 1$, with adjacent triangles sharing an entire edge. Two trianglominoes are considered the same if they can be matched using only translations and rotations (but not reflections). How many distinct trianglominoes can Farmer James make? [b]p7.[/b] Two real numbers $x$ and $y$ satisfy $x^2 - y^2 = 2y - 2x$ , and $x + 6 = y^2 + 2y$. What is the sum of all possible values of$ y$? [b]p8.[/b] Let $N$ be a positive multiple of $840$. When $N$ is written in base $6$, it is of the form $\overline{abcdef}_6$ where $a, b, c, d, e, f$ are distinct base $6$ digits. What is the smallest possible value of $N$, when written in base $6$? [b]p9.[/b] For $S = \{1, 2,..., 12\}$, find the number of functions $f : S \to S$ that satisfy the following $3$ conditions: (a) If $n$ is divisible by $3$, $f(n)$ is not divisible by $3$, (b) If $n$ is not divisible by $3$, $f(n)$ is divisible by $3$, and (c) $f(f(n)) = n$ holds for exactly $8$ distinct values of $n$ in $S$. [b]p10.[/b] Regular pentagon $JAMES$ has area $ 1$. Let $O$ lie on line $EM$ and $N$ lie on line $MA$ so that $E, M, O$ and $M, A, N$ lie on their respective lines in that order. Given that $MO = AN$ and $NO = 11 \cdot ME$, find the area of $NOM$. [b]p11.[/b] Hen Hao is flipping a special coin, which lands on its sunny side and its rainy side each with probability $1/2$. Hen Hao flips her coin ten times. Given that the coin never landed with its rainy side up twice in a row, find the probability that Hen Hao’s last flip had its sunny side up. [b]p12.[/b] Find the product of all integer values of a such that the polynomial $x^4 + 8x^3 + ax^2 + 2x - 1$ can be factored into two non-constant polynomials with integer coefficients. [b]p13.[/b] Isosceles trapezoid $ABCD$ has $AB = CD$ and $AD = 6BC$. Point $X$ is the intersection of the diagonals $AC$ and $BD$. There exist a positive real number $k$ and a point $P$ inside $ABCD$ which satisfy $$[PBC] : [PCD] : [PDA] = 1 : k : 3,$$ where $[XYZ]$ denotes the area of triangle $XYZ$. If $PX \parallel AB$, find the value of $k$. [b]p14.[/b] How many positive integers $n < 1000$ are there such that in base $10$, every digit in $3n$ (that isn’t a leading zero) is greater than the corresponding place value digit (possibly a leading zero) in $n$? For example, $n = 56$, $3n = 168$ satisfies this property as $1 > 0$, $6 > 5$, and $8 > 6$. On the other hand, $n = 506$, $3n = 1518$ does not work because of the hundreds place. [b]p15.[/b] Find the greatest integer that is smaller than $$\frac{2018}{37^2}+\frac{2018}{39^2}+ ... +\frac{2018}{ 107^2}.$$ PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

There are $27$ cards, each has some amount of ($1$ or $2$ or $3$) shapes (a circle, a square or a triangle) with some color (white, grey or black) on them. We call a triple of cards a [i]match[/i] such that all of them have the same amount of shapes or distinct amount of shapes, have the same shape or distinct shapes and have the same color or distinct colors. For instance, three cards shown in the figure are a [i]match[/i] be cause they have distinct amount of shapes, distinct shapes but the same color of shapes. What is the maximum number of cards that we can choose such that non of the triples make a [i]match[/i]? [i]Proposed by Amin Bahjati[/i]

The Tokyo Metro system is one of the most efficient in the world. There is some odd positive integer $k$ such that each metro line passes through exactly $k$ stations, and each station is serviced by exactly $k$ metro lines. One can get from any station to any otherstation using only one metro line - but this connection is unique. Furthermore, any two metro lines must share exactly one station. David is planning an excursion for the IMO team, and wants to visit a set $S$ of $k$ stations. He remarks that no three of the stationsin $S$ are on a common metro line. Show that there is some station not in $S$, which is connected to every station in $S$ by a different metro line.

$N^2$ pieces are placed on an $N \times N$ chessboard. Is it possible to rearrange them in such a way that any two pieces which can capture each other (when considered to be knights) after the rearrangement are on adjacent squares (i.e. squares having at least one common boundary point)? Consider two cases: (a) $N = 3$. (b) $N = 8$ (S Stefanov)

In a table-tennis tournament of $10$ contestants, any $2$ contestants meet only once. We say that there is a winning triangle if the following situation occurs: $i$-th contestant defeated the $j$-th contestant, $j$-th contestant defeated the $k$-th contestant, and, $k$-th contestant defeated the $i$-th contestant. Let, $W_i$ and $L_i $ be respectively the number of games won and lost by the $i$-th contestant. Suppose, $L_i+W_j\geq 8$ whenever the $j$-th contestant defeats the $i$-th contestant. Prove that, there are exactly $40$ winning triangles in this tournament.

$\forall n\in \mathbb{Z_{+}}-\{1,2\}$ find the maximal length of a sequence with elements from a set $\{1,2,\ldots,n\}$, such that any two consecutive elements of this sequence are different and after removing all elements except for the four we do not receive a sequence in form $x,y,x,y$ ($x\neq y$).

Each person stands on a whole number on the number line from $0$ to $2022$ . In each turn, two people are selected by a distance of at least $2$. These go towards each other by $1$. When no more such moves are possible, the process ends. Show that this process always ends after a finite number of moves, and determine all possible configurations where people can end up standing. (whereby is for each configuration is only of interest how many people stand at each number.) [i](Birgit Vera Schmidt)[/i] [hide=original wording]Bei jeder ganzen Zahl auf dem Zahlenstrahl von 0 bis 2022 steht zu Beginn eine Person. In jedem Zug werden zwei Personen mit Abstand mindestens 2 ausgewählt. Diese gehen jeweils um 1 aufeinander zu. Wenn kein solcher Zug mehr möglich ist, endet der Vorgang. Man zeige, dass dieser Vorgang immer nach endlich vielen Zügen endet, und bestimme alle möglichen Konfigurationen, wo die Personen am Ende stehen können. (Dabei ist für jede Konfiguration nur von Interesse, wie viele Personen bei jeder Zahl stehen.)[/hide]

Let $ n\geq 3 $ be an integer and let $ p\geq 2n-3 $ be a prime number. For a set $ M $ of $ n $ points in the plane, no 3 collinear, let $ f: M\to \{0,1,\ldots, p-1\} $ be a function such that (i) exactly one point of $ M $ maps to 0, (ii) if a circle $ \mathcal{C} $ passes through 3 distinct points of $ A,B,C\in M $ then $ \sum_{P\in M\cap \mathcal{C}} f(P) \equiv 0 \pmod p $. Prove that all the points in $ M $ lie on a circle.

Ali has $6$ types of $2\times2$ squares with cells colored in white or black, and has presented them to Mohammad as forbidden tiles. $a)$ Prove that Mohammad can color the cells of the infinite table (from each $4$ sides.) in black or white such that there's no forbidden tiles in the table. $b)$ Prove that Ali can present $7$ forbidden tiles such that Mohammad cannot achieve his goal.

A [i]word[/i] is a sequence of 0 and 1 of length 8. Let $ x$ and $ y$ be two words differing in exactly three places. Prove that the number of words differing from each of $ x$ and $ y$ in at least five places is 188.

We are given $k$ colors and we have to assign a single color to every vertex. An edge is [u][b]satisfied[/b][/u] if the vertices on that edge, are of different colors. [list] [*]Prove that you can always find an algorithm which assigns colors to vertices so that at least $\frac{k - 1}{k}|E|$ edges are satisfied where \(|E|\) is the cardinality of the edges in the graph.[/*] [*]Prove that there is a poly time deterministic algorithm for this [/*] [/list]

Each of the lattice points $(x,y)$ (where $x$ and $y$ are integers) in the plane can be coloured black or white. A single strike by an $L$-shaped punch changes the colour of the four lattice points $(a,b)$, $(a+1,b)$, $(a,b+1)$ and $(a,b+2)$. All lattice points are initially coloured white. Prove that after any number of strikes, the number of black lattice points will be either zero or greater than or equal to four.

Pairwise distinct points $P_1,\dots,P_{1024}$, which lie on a circle, are marked by distinct reals $a_1,\dots,a_{1024}$. Let $P_i$ be $Q-$good for a $Q$ on the circle different than $P_1,\dots,P_{1024}$, if and only if $a_i$ is the greatest number on at least one of the two arcs $P_iQ$. Let the score of $Q$ be the number of $Q-$good points on the circle. Determine the greatest $k$ such that regardless of the values of $a_1,\dots,a_{1024}$, there exists a point $Q$ with score at least $k$.

A game consists in pushing a flat stone along a sequence of squares $S_0, S_1, S_2, . . .$ that are arranged in linear order. The stone is initially placed on square $S_0$. When the stone stops on a square $S_k$ it is pushed again in the same direction and so on until it reaches $S_{1987}$ or goes beyond it; then the game stops. Each time the stone is pushed, the probability that it will advance exactly $n$ squares is $\frac{1}{2^n}$. Determine the probability that the stone will stop exactly on square $S_{1987}.$

Does it exist a permutation of the numbers $1,2,\ldots,1992$ such that the arithmetic mean of arbitrary two of the numbers is not equal to any of the numbers which is placed between these two numbers in the permutation?

Positive integers $x_1, x_2, \dots, x_n$ ($n \ge 4$) are arranged in a circle such that each $x_i$ divides the sum of the neighbors; that is \[ \frac{x_{i-1}+x_{i+1}}{x_i} = k_i \] is an integer for each $i$, where $x_0 = x_n$, $x_{n+1} = x_1$. Prove that \[ 2n \le k_1 + k_2 + \dots + k_n < 3n. \]

Eight politicians stranded on a desert island on January 1st, 1991, decided to establish a parliament. They decided on the following rules of attendance: (a) There should always be at least one person present on each day. (b) On no two days should the same subset attend. (c) The members present on day $N$ should include for each $K<N$, $(K\ge 1)$ at least one member who was present on day $K$. For how many days can the parliament sit before one of the rules is broken?

The square table $n\times n$ is filled by integers. If the fields have common side, the difference of numbers in them doesn't exceed $1$. Prove that some number is encountered not less than a) not less than $[n/2]$ times ($[ ...]$ mean the whole part), b) not less than $n$ times.

An equilateral triangle is divided into $9000000$ congruent equilateral triangles by lines parallel to its sides. Each vertex of the small triangles is coloured in one of three colours. Prove that there exist three points of the same colour being the vertices of a triangle with its sides parallel to the lines of the original triangle.

Alice and Bob play a game on a connected graph with $2n$ vertices, where $n\in \mathbb{N}$ and $n>1$.. Alice and Bob have tokens named A and B respectively. They alternate their turns with Alice going first. Alice gets to decide the starting positions of A and B. Every move, the player with the turn moves their token to an adjacent vertex. Bob's goal is to catch Alice, and Alice's goal is to prevent this. Note that positions of A, B are visible to both Alice and Bob at every moment. Provided they both play optimally, what is the maximum possible number of edges in the graph if Alice is able to evade Bob indefinitely? [i]Proposed by Shashank Ingalagavi and Vighnesh Sangle[/i]

Find the smallest positive integer $n$ with the following property: in every sequence of $n$ positive integers such that the sum of the $n$ numbers is equal to $2013$, there are some consecutive terms whose sum is equal to $31$.

Let $n$ be a positive integer. How many integer solutions $(i, j, k, l) , \ 1 \leq i, j, k, l \leq n$, does the following system of inequalities have: \[1 \leq -j + k + l \leq n\]\[1 \leq i - k + l \leq n\]\[1 \leq i - j + l \leq n\]\[1 \leq i + j - k \leq n \ ?\]

[b]p1.[/b] What is the units digit of the product of the first seven primes? [b]p2. [/b]In triangle $ABC$, $\angle BAC$ is a right angle and $\angle ACB$ measures $34$ degrees. Let $D$ be a point on segment $ BC$ for which $AC = CD$, and let the angle bisector of $\angle CBA$ intersect line $AD$ at $E$. What is the measure of $\angle BED$? [b]p3.[/b] Chad numbers five paper cards on one side with each of the numbers from $ 1$ through $5$. The cards are then turned over and placed in a box. Jordan takes the five cards out in random order and again numbers them from $ 1$ through $5$ on the other side. When Chad returns to look at the cards, he deduces with great difficulty that the probability that exactly two of the cards have the same number on both sides is $p$. What is $p$? [b]p4.[/b] Only one real value of $x$ satisfies the equation $kx^2 + (k + 5)x + 5 = 0$. What is the product of all possible values of $k$? [b]p5.[/b] On the Exeter Space Station, where there is no effective gravity, Chad has a geometric model consisting of $125$ wood cubes measuring $ 1$ centimeter on each edge arranged in a $5$ by $5$ by $5$ cube. An aspiring carpenter, he practices his trade by drawing the projection of the model from three views: front, top, and side. Then, he removes some of the original $125$ cubes and redraws the three projections of the model. He observes that his three drawings after removing some cubes are identical to the initial three. What is the maximum number of cubes that he could have removed? (Keep in mind that the cubes could be suspended without support.) [b]p6.[/b] Eric, Meena, and Cameron are studying the famous equation $E = mc^2$. To memorize this formula, they decide to play a game. Eric and Meena each randomly think of an integer between $1$ and $50$, inclusively, and substitute their numbers for $E$ and $m$ in the equation. Then, Cameron solves for the absolute value of $c$. What is the probability that Cameron’s result is a rational number? [b]p7.[/b] Let $CDE$ be a triangle with side lengths $EC = 3$, $CD = 4$, and $DE = 5$. Suppose that points $ A$ and $B$ are on the perimeter of the triangle such that line $AB$ divides the triangle into two polygons of equal area and perimeter. What are all the possible values of the length of segment $AB$? [b]p8.[/b] Chad and Jordan are raising bacteria as pets. They start out with one bacterium in a Petri dish. Every minute, each existing bacterium turns into $0, 1, 2$ or $3$ bacteria, with equal probability for each of the four outcomes. What is the probability that the colony of bacteria will eventually die out? [b]p9.[/b] Let $a = w + x$, $b = w + y$, $c = x + y$, $d = w + z$, $e = x + z$, and $f = y + z$. Given that $af = be = cd$ and $$(x - y)(x - z)(x - w) + (y - x)(y - z)(y - w) + (z - x)(z - y)(z - w) + (w - x)(w - y)(w - z) = 1,$$ what is $$2(a^2 + b^2 + c^2 + d^2 + e^2 + f^2) - ab - ac - ad - ae - bc - bd - bf - ce - cf - de - df - ef ?$$ [b]p10.[/b] If $a$ and $b$ are integers at least $2$ for which $a^b - 1$ strictly divides $b^a - 1$, what is the minimum possible value of $ab$? Note: If $x$ and $y$ are integers, we say that $x$ strictly divides $y$ if $x$ divides $y$ and $|x| \ne |y|$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given a sequence of eight integers $x_{1},x_{2},...,x_{8}$ in a single operation one replaces these numbers with $|x_{1}-x_{2}|,|x_{2}-x_{3}|,...,|x_{8}-x_{1}|$. Find all the eight-term sequences of integers which reduce to a sequence with all the terms equal after finitely many single operations.