By number-theoretical functions, we will understand integer-valued functions defined on the set of all integers. Are there number-theoretical functions $ f_0(x),f_1(x),f_2(x),\dots$ such that every number theoretical function $ F(x)$ can be uniquely represented in the form $ F(x)\equal{}\sum_{k\equal{}0}^{\infty}a_kf_k(x)$, $ a_0,a_1,a_2,\dots$ being integers?

Find all four digit positive integers such that the sum of the squares of the digits equals twice the sum of the digits.

Let $a_0,a_1,a_2...$ be a sequence of natural numbers with the following property: $a_n^2$ divides $a_{n-1} a_{n+1}$ for $\forall$ $n\in \mathbb{N}$. Prove that, if for some natural $k\geq 2$ the numbers $a_1$ and $a_k$ are coprime, then $a_1$ divides $a_0$.

Find the smallest positive integer $n$ or show no such $n$ exists, with the following property: there are infinitely many distinct $n$-tuples of positive rational numbers $(a_1, a_2, \ldots, a_n)$ such that both $$a_1+a_2+\dots +a_n \quad \text{and} \quad \frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_n}$$ are integers.

Let $a \neq 1$ and be a real positive number and $n$ be an integer greater than $1.$ Demonstrate that $n^2 < \frac{(a^n + a^{-n} -2)}{(a + a^{-1} -2)}.$

Find all pair of integer numbers $(n,k)$ such that $n$ is not negative and $k$ is greater than $1$, and satisfying that the number: \[ A=17^{2006n}+4.17^{2n}+7.19^{5n} \] can be represented as the product of $k$ consecutive positive integers.

Jerry's favorite number is $97$. He knows all kinds of interesting facts about $97$: [list][*]$97$ is the largest two-digit prime. [*]Reversing the order of its digits results in another prime. [*]There is only one way in which $97$ can be written as a difference of two perfect squares. [*]There is only one way in which $97$ can be written as a sum of two perfect squares. [*]$\tfrac1{97}$ has exactly $96$ digits in the [smallest] repeating block of its decimal expansion. [*]Jerry blames the sock gnomes for the theft of exactly $97$ of his socks.[/list] A repunit is a natural number whose digits are all $1$. For instance, \begin{align*}&1,\\&11,\\&111,\\&1111,\\&\vdots\end{align*} are the four smallest repunits. How many digits are there in the smallest repunit that is divisible by $97?$

Show that a number $n(n+1)$ where $n$ is positive integer is the sum of 2 numbers $k(k+1)$ and $m(m+1)$ where $m$ and $k$ are positive integers if and only if the number $2n^2+2n+1$ is composite.

Let $q>3$ be a rational number, such that $q^2-4$ is a perfect square of a rational number. The sequence $a_0, a_1, \ldots$ is defined by $a_0=2, a_1=q, a_{i+1}=qa_i-a_{i-1}$ for all $i \geq 1$. Is it true that there exist a positive integer $n$ and nonzero integers $b_0, b_1, \ldots, b_n$ with sum zero, such that if $\sum_{i=0}^{n} a_ib_i=\frac{A} {B}$ for $(A, B)=1$, then $A$ is squarefree?

a) Find a pair(s) of integers $(x,y)$ such that: $y^2=x^3+2017$ b) Prove that there isn't integers $x$ and $y$, with $y$ not divisible by $3$, such that: $y^2=x^3-2017$

Each positive integer $a$ undergoes the following procedure in order to obtain the number $d = d\left(a\right)$: (i) move the last digit of $a$ to the first position to obtain the numb er $b$; (ii) square $b$ to obtain the number $c$; (iii) move the first digit of $c$ to the end to obtain the number $d$. (All the numbers in the problem are considered to be represented in base $10$.) For example, for $a=2003$, we get $b=3200$, $c=10240000$, and $d = 02400001 = 2400001 = d(2003)$.) Find all numbers $a$ for which $d\left( a\right) =a^2$. [i]Proposed by Zoran Sunic, USA[/i]

Let $a$ be a positive integer such that the last two digits of $a^2$ are both non-zero. When the last two digits of $a^2$ are deleted, the resulting number is still a perfect square. Find, with justification, all possible values of $a$.

Let $\mathbb{Z}^+$ be the set of positive integers. Determine all functions $f : \mathbb{Z}^+\to\mathbb{Z}^+$ such that $a^2+f(a)f(b)$ is divisible by $f(a)+b$ for all positive integers $a,b$.

Six decks of $n$ cards, numbered from $1$ to $n$, are given. Melanie arranges each of the decks in some order, such that for any distinct numbers $x$, $y$, and $z$ in $\{1, 2, . . . , n\}$, there is exactly one deck where card $x$ is above card $y$ and card $y$ is above card $z$. Show that there is some $n$ for which Melanie cannot arrange these six decks of cards with this property.

A sequence $ a_1, a_2, \ldots$ of non-negative integers is defined by the rule $ a_{n \plus{} 2} \equal{} |a_{n \plus{} 1} \minus{} a_n|$ for $ n\ge 1$. If $ a_1 \equal{} 999, a_2 < 999,$ and $ a_{2006} \equal{} 1$, how many different values of $ a_2$ are possible? $ \textbf{(A) } 165 \qquad \textbf{(B) } 324 \qquad \textbf{(C) } 495 \qquad \textbf{(D) } 499 \qquad \textbf{(E) } 660$

Some distinct positive integers were written on a blackboard such that the sum of any two integers is a power of $2$. What is the maximal possible number written on the blackboard?

Find all integers $m$ and $n$ such that the fifth power of $m$ minus the fifth power of $n$ is equal to $16mn$.

Find the number of positive divisors $d$ of $15!=15\cdot 14\cdot\cdots\cdot 2\cdot 1$ such that $\gcd(d,60)=5$.

Let $P$ be the set of prime numbers. Consider a subset $M$ of $P$ with at least three elements. We assume that, for each non empty and finite subset $A$ of $M$, with $A \neq M$, the prime divisors of the integer $( \prod_{p \in A} ) - 1$ belong to $M$. Prove that $M = P$.

Let $ a$ be the greatest positive root of the equation $ x^3 \minus{} 3 \cdot x^2 \plus{} 1 \equal{} 0.$ Show that $ \left[a^{1788} \right]$ and $ \left[a^{1988} \right]$ are both divisible by 17. Here $ [x]$ denotes the integer part of $ x.$

Let $P(x)$ be a non-constant polynomial with integer coefficients. Prove that there is no function $T$ from the set of integers into the set of integers such that the number of integers $x$ with $T^n(x)=x$ is equal to $P(n)$ for every $n\geq 1$, where $T^n$ denotes the $n$-fold application of $T$. [i]Proposed by Jozsef Pelikan, Hungary[/i]

Find minimum number $n$ that: 1) $80|n$ 2) we can permute 2 different numbers in $n$ to get $m$ and $80|m$

Find all functions $g:\mathbb{N}\rightarrow\mathbb{N}$ such that \[\left(g(m)+n\right)\left(g(n)+m\right)\] is a perfect square for all $m,n\in\mathbb{N}.$ [i]Proposed by Gabriel Carroll, USA[/i]

Find the number $\overline {523abc}$ divisible by $7, 8$ and $9$.

Given a square-free positive integer $n$. Show that there do not exist coprime positive integers $x,y$ such that $x^n+y^n$ is a multiple of $(x+y)^3$.