Find all $f : N \longmapsto N$ that there exist $k \in N$ and a prime $p$ that: $\forall n \geq k \ f(n+p)=f(n)$ and also if $m \mid n$ then $f(m+1) \mid f(n)+1$

Prove that among any ten consecutive positive integers at least one is relatively prime to the product of the others.

Given are $100$ positive integers whose sum equals their product. Determine the minimum number of $1$s that may occur among the $100$ numbers.

For some positive integer $n$, $n$ is considered a $\textbf{unique}$ number if for any other positive integer $m\neq n$, $\{\dfrac{2^n}{n^2}\}\neq\{\dfrac{2^m}{m^2}\}$ holds. Prove that there is an infinite list consisting of composite unique numbers whose elements are pairwise coprime.

[u]Set 1 [/u] [b]1.1[/b] Compute the number of real numbers x such that the sequence $x$, $x^2$, $x^3$,$ x^4$, $x^5$, $...$ eventually repeats. (To be clear, we say a sequence “eventually repeats” if there is some block of consecutive digits that repeats past some point—for instance, the sequence $1$, $2$, $3$, $4$, $5$, $6$, $5$, $6$, $5$, $6$, $...$ is eventually repeating with repeating block $5$, $6$.) [b]1.2[/b] Let $T$ be the answer to the previous problem. Nicole has a broken calculator which, when told to multiply $a$ by $b$, starts by multiplying $a$ by $b$, but then multiplies that product by b again, and then adds $b$ to the result. Nicole inputs the computation “$k \times k$” into the calculator for some real number $k$ and gets an answer of $10T$. If she instead used a working calculator, what answer should she have gotten? [b]1.3[/b] Let $T$ be the answer to the previous problem. Find the positive difference between the largest and smallest perfect squares that can be written as $x^2 + y^2$ for integers $x, y$ satisfying $\sqrt{T} \le x \le T$ and $\sqrt{T} \le y \le T$. PS. You should use hide for answers.

Let $c$ be a nonnegative integer, and define $a_n = n^2 + c$ (for $n \geq 1)$. Define $d_n$ as the greatest common divisor of $a_n$ and $a_{n + 1}$. (a) Suppose that $c = 0$. Show that $d_n = 1,\ \forall n \geq 1$. (b) Suppose that $c = 1$. Show that $d_n \in \{1,5\},\ \forall n \geq 1$. (c) Show that $d_n \leq 4c + 1,\ \forall n \geq 1$.

Let $a_1, a_2,..., a_n$ be positive integers and $a$ positive integer greater than $1$ which is a multiple of the product $a_1a_2...a_n$. Prove that $a^{n+1} + a - 1$ is not divisible by $(a + a_1 -1)(a + a_2 - 1) ... (a + a_n -1)$.

To unlock his cell phone, Joao slides his finger horizontally or vertically across a numerical box, similar to the one represented in the figure, describing a $7$-digit code, without ever passing through the same digit twice. For example, to indicate the code $1452369$, Joao follows the path indicated in the figure. [img]https://cdn.artofproblemsolving.com/attachments/8/a/511018ba4e43c2c6f0be350d57161eb5ea7c2b.png[/img] João forgot his code, but he remembers that it is divisible by $9$. How many codes are there under these conditions?

Let $n \ge 1$ be an integer. For each subset $S \subset \{1, 2, \ldots , 3n\}$, let $f(S)$ be the sum of the elements of $S$, with $f(\emptyset) = 0$. Determine, as a function of $n$, the sum $$\sum_{\mathclap{\substack{S \subset \{1,2,\ldots,3n\}\\ 3 \mid f(S)}}} f(S)$$ where $S$ runs through all subsets of $\{1, 2,\ldots, 3n\}$ such that $f(S)$ is a multiple of $3$.

A positive integer $N$ is called [i]balanced[/i], if $N=1$ or if $N$ can be written as a product of an even number of not necessarily distinct primes. Given positive integers $a$ and $b$, consider the polynomial $P$ defined by $P(x)=(x+a)(x+b)$. (a) Prove that there exist distinct positive integers $a$ and $b$ such that all the number $P(1)$, $P(2)$,$\ldots$, $P(50)$ are balanced. (b) Prove that if $P(n)$ is balanced for all positive integers $n$, then $a=b$. [i]Proposed by Jorge Tipe, Peru[/i]

Let $x = a_1a_2 \dots a_n$ be an n-digit number, where $a_1, a_2, \dots , an (a_1 \neq 0)$ are the digits. The $n$ numbers $ x_1 = x = a_1 a_2 ... a_n, $ $ x_2 = a_n a_1 ... a_{n-1}, $ $ x_3 = a_{n-1} a_n a _1 ... a_{n-2} $ , $ x_4 = a_{n-2} a_{n-1} a_n a_1 , ... a_{n-3} , $ $ ... , x_n = a_2 a_3 ... a_n a_1$ are said to be obtained from $x$ by the cyclic permutation of digits. [For example, if $n = 5$ and $x = 37001$, then the numbers are $x_1 = 37001, x_2 = 13700, $ $x_3 = 01370(= 1370), x_4 = 00137(= 137), $ $ x_5 = 70013.]$ Find, with proof, (i) the smallest natural number n for which there exists an n-digit number x such that the n numbers obtained from x by the cyclic permutation of digits are all divisible by 1989; and (ii) the smallest natural number x with this property.

Find all positive integers $k$ such that for the first $k$ prime numbers $2, 3, \ldots, p_k$ there exist positive integers $a$ and $n>1$, such that $2\cdot 3\cdot\ldots\cdot p_k - 1=a^n$. [i]V. Senderov[/i]

Find all primes $p$ such that $p+2$ and $p^2+2p-8$ are also primes.

In a quadratic table with $ 2007$ rows and $ 2007$ columns is an odd number written in each field. For $ 1\leq i\leq2007$ is $ Z_i$ the sum of the numbers in the $ i$-th row and for $ 1\leq j\leq2007$ is $ S_j$ the sum of the numbers in the $ j$-th column. $ A$ is the product of all $ Z_i$ and $ B$ the product of all $ S_j$. Show that $ A\plus{}B\neq0$

[u]Round 1 [/u] [b]p1.[/b] Pirate Jim had $8$ boxes with gun powder weighing $1, 2, 3, 4, 5, 6, 7$, and $8$ pounds (the weight is printed on top of every box). Pirate Bob hid a $1$-pound gold bar in one of these boxes. Pirate Jim has a balance scale that he can use, but he cannot open any of the boxes. Help him find the box with the gold bar using two weighings on the balance scale. [b]p2.[/b] James Bond will spend one day at Dr. Evil's mansion to try to determine the answers to two questions: a) Is Dr. Evil at home? b) Does Dr. Evil have an army of ninjas? The parlor in Dr. Evil's mansion has three windows. At noon, Mr. Bond will sneak into the parlor and use open or closed windows to signal his answers. When he enters the parlor, some windows may already be opened, and Mr. Bond will only have time to open or close one window (or leave them all as they are). Help Mr. Bond and Moneypenny design a code that will tell Moneypenny the answers to both questions when she drives by later that night and looks at the windows. Note that Moneypenny will not have any way to know which window Mr. Bond opened or closed. [b]p3.[/b] Suppose that you have a triangle in which all three side lengths and all three heights are integers. Prove that if these six lengths are all different, there cannot be four prime numbers among them. p4. Fred and George have designed the Amazing Maze, a $5\times 5$ grid of rooms, with Adorable Doors in each wall between rooms. If you pass through a door in one direction, you gain a gold coin. If you pass through the same door in the opposite direction, you lose a gold coin. The brothers designed the maze so that if you ever come back to the room in which you started, you will find that your money has not changed. Ron entered the northwest corner of the maze with no money. After walking through the maze for a while, he had $8$ shiny gold coins in his pocket, at which point he magically teleported himself out of the maze. Knowing this, can you determine whether you will gain or lose a coin when you leave the central room through the north door? [b]p5.[/b] Bill and Charlie are playing a game on an infinite strip of graph paper. On Bill’s turn, he marks two empty squares of his choice (not necessarily adjacent) with crosses. Charlie, on his turn, can erase any number of crosses, as long as they are all adjacent to each other. Bill wants to create a line of $2013$ crosses in a row. Can Charlie stop him? [u]Round 2 [/u] [b]p6.[/b] $1000$ non-zero numbers are written around a circle and every other number is underlined. It happens that each underlined number is equal to the sum of its two neighbors and that each non-underlined number is equal to the product of its two neighbors. What could the sum of all the numbers written on the circle be? [b]p7.[/b] A grasshopper is sitting at the edge of a circle of radius $3$ inches. He can hop exactly $4$ inches in any direction, as long as he stays within the circle. Which points inside the circle can the grasshopper reach if he can make as many jumps as he likes? [img]https://cdn.artofproblemsolving.com/attachments/1/d/39b34b2b4afe607c1232f4ce9dec040a34b0c8.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all triples $ (x,y,z)$ of integers which satisfy $ x(y \plus{} z) \equal{} y^2 \plus{} z^2 \minus{} 2$ $ y(z \plus{} x) \equal{} z^2 \plus{} x^2 \minus{} 2$ $ z(x \plus{} y) \equal{} x^2 \plus{} y^2 \minus{} 2$.

Find all primes $p$ and $q$ such that $3p^{q-1}+1$ divides $11^p+17^p$ Proposed by Stanislav Dimitrov,Bulgaria

Let $\mathbb{F}_{p}$ denote the $p$-element field for a prime $p>3$ and let $S$ be the set of functions from $\mathbb{F}_{p}$ to $\mathbb{F}_{p}$. Find all functions $D\colon S\to S$ satisfying \[D(f\circ g)=(D(f)\circ g)\cdot D(g)\] for all $f,g \in {S}$. Here, $\circ$ denotes the function composition, so $(f\circ g)(x)$ is the function $f(g(x))$, and $\cdot$ denotes multiplication, so $(f\cdot g)=f(x)g(x)$.

For which positive integers $n$ does there exist a positive integer $m$ such that among the numbers $m + n, 2m + (n - 1), \dots, nm + 1$, there are no two that share a common factor greater than $1$?

Define a [i]beautiful number[/i] to be an integer of the form $a^n$, where $a\in\{3,4,5,6\}$ and $n$ is a positive integer. Prove that each integer greater than $2$ can be expressed as the sum of pairwise distinct beautiful numbers. [i]Proposed by Matthew Babbitt[/i]

Find all strictly increasing sequences $\{a_n\}_{n=0}^\infty$ of positive integers such that for all positive integers $k,m,n$ $$\frac{a_{n+1} +a_{n+2} +\dots +a_{n+k}}{k+m}$$ is not an integer larger than $2020$.

There are $n$ distinct points on a circumference. Choose one of the points. Connect this point and the $m$th point from the chosen point counterclockwise with a segment. Connect this $m$th point and the $m$th point from this $m$th point counterclockwise with a segment. Repeat such steps until no new segment is constructed. From the intersections of the segments, let the number of the intersections - which are in the circle - be $I$. Answer the following questions ($m$ and $n$ are positive integers that are relatively prime and they satisfy $6 \leq 2m < n$). 1) When the $n$ points take different positions, express the maximum value of $I$ in terms of $m$ and $n$. 2) Prove that $I \geq n$. Prove that there is a case, which is $I=n$, when $m=3$ and $n$ is arbitrary even number that satisfies the condition.

Let $n \ge 2$ be a fixed positive integer. An integer will be called "$n$-free" if it is not a multiple of an $n$-th power of a prime. Let $M$ be an infinite set of rational numbers, such that the product of every $n$ elements of $M$ is an $n$-free integer. Prove that $M$ contains only integers.

$a, b, c, d$ are integers such that $ad + bc$ divides each of $a, b, c$ and $d$. Prove that $ad + bc =\pm 1$

Determine the number of all ordered triplets of positive integers $(a, b, c)$, which satisfy the equalities: $$[a, b] =1000, [b, c] = 2000, [c, a] =2000.$$ ([x, y]represents the least common multiple of positive integers x,y)