Prove that for any positive integers $x, y, z$ with $xy-z^2 = 1$ one can find non-negative integers $a, b, c, d$ such that $x = a^2 + b^2, y = c^2 + d^2, z = ac + bd$. Set $z = (2q)!$ to deduce that for any prime number $p = 4q + 1$, $p$ can be represented as the sum of squares of two integers.

Let $a, b, c, d, e, f$ are nonzero digits such that the natural numbers $\overline{abc}, \overline{def}$ and $\overline{abcdef }$ are squares. a) Prove that $\overline{abcdef}$ can be represented in two different ways as a sum of three squares of natural numbers. b) Give an example of such a number.

Find all integers $x, y, z$ satisfy the $x^4-10y^4 + 3z^6 = 21$.

Let a sequence $\{a_n\}$, $n \in \mathbb{N}^{*}$ given, satisfying the condition \[0 < a_{n+1} - a_n \leq 2001\] for all $n \in \mathbb{N}^{*}$ Show that there are infinitely many pairs of positive integers $(p, q)$ such that $p < q$ and $a_p$ is divisor of $a_q$.

Prove that for every positive integer $n$, the number $A_n = 7^{2n} -48n - 1$ is a multiple of $9$.

A row of $2n$ real numbers is called [i]“Sozopolian”[/i], if for each $m$, such that $1\leq m\leq 2n$, the sum of the first $m$ members of the row is an integer or the sum of the last $m$ members of the row is an integer. What’s the least number of integers that a [i]Sozopolian[/i] row can have, if the number of its members is: a) 2016; b) 2017?

There is a row of $n$ chairs, numbered in order from left to right from $1$ to $n$. Additionally, the $n$ numbers from $1$ to $n$ are distributed on the backs of the chairs, one number per chair, such that the number on the back of a chair never matches the number of the chair itself. There is a child sitting on each chair. Every time the teacher claps, each child checks the number on the back of the chair they are sitting on and moves to the chair corresponding to that number. Prove that for any $m$ that is not a power of a prime, with $1 < m \leqslant n$, it is possible to distribute the numbers on the backrests such that, after the teacher claps $m$ times, for the first time, all the children are sitting in the chairs where they initially started. (During the process, it may happen that some children return to their original chairs, but they do not all do so simultaneously until the $m^{\text{th}}$ clap.)

Determine all pairs $(a, b)$ of integers which satisfy the equality $\frac{a + 2}{b + 1} +\frac{a + 1}{b + 2} = 1 +\frac{6}{a + b + 1}$

Let $a, b$ and $n$ be positive integers with $a>b$ such that all of the following hold: i. $a^{2021}$ divides $n$, ii. $b^{2021}$ divides $n$, iii. 2022 divides $a-b$. Prove that there is a subset $T$ of the set of positive divisors of the number $n$ such that the sum of the elements of $T$ is divisible by 2022 but not divisible by $2022^2$. [i]Proposed by Silouanos Brazitikos, Greece[/i]

Let $14$ integer numbers are given. Let Hamza writes on the paper the greatest common divisor for each pair of numbers. It occurs that the difference between the biggest and smallest numbers written on the paper is less than $91$. Prove that not all numbers on the paper are different.

Prove that for all $n \ge 3$ there are an infinite number of $n$-sided polygonal numbers which are also the sum of two other (not necessarily different) $n$-sided polygonal numbers! The first $n$-sided polygonal number is $1$. The kth n-sided polygonal number for $k \ge 2$ is the number of different points in a figure that consists of all of the regular $n$-sided polygons which have one common vertex, are oriented in the same direction from that vertex and their sides are $\ell$ cm long where $1 \le \ell \le k - 1$ cm and $\ell$ is an integer. [i]In this figure, what we call points are the vertices of the polygons and the points that break up the sides of the polygons into exactly $1$ cm long segments. For example, the first four pentagonal numbers are 1,5,12, and 22, like it is shown in the figure.[/i] [img]https://cdn.artofproblemsolving.com/attachments/1/4/290745d4be1888813678127e6d63b331adaa3d.png[/img]

If the sum of all positive divisors (including itself) of a positive integer $n$ is $2n$, then $n$ is called a perfect number. For example, the sum of the positive divisors of 6 is $1 + 2 + 3 + 6 = 2 \times 6$, hence 6 is a perfect number. Prove: There does not exist a perfect number of the form $p^a q^b r^c$, where $a, b, c$ are positive integers, and $p, q, r$ are odd primes.

Determine all positive integers $a$ for which there exist exactly $2014$ positive integers $b$ such that $\displaystyle2\leq\frac{a}{b}\leq5$.

A company of $n$ soldiers is such that (i) $n$ is a palindrome number (read equally in both directions); (ii) if the soldiers arrange in rows of $3, 4$ or $5$ soldiers, then the last row contains $2, 3$ and $5$ soldiers, respectively. Find the smallest $n$ satisfying these conditions and prove that there are infinitely many such numbers $n$.

Let $ n > 1$ be an integer, and $ n$ can divide $ 2^{\phi(n)} \plus{} 3^{\phi(n)} \plus{} \cdots \plus{} n^{\phi(n)},$ let $ p_{1},p_{2},\cdots,p_{k}$ be all distinct prime divisors of $ n$. Show that $ \frac {1}{p_{1}} \plus{} \frac {1}{p_{2}} \plus{} \cdots \plus{} \frac {1}{p_{k}} \plus{} \frac {1}{p_{1}p_{2}\cdots p_{k}}$ is an integer. ( where $ \phi(n)$ is defined as the number of positive integers $ \leq n$ that are relatively prime to $ n$.)

Let $m \leq n$ be positive integers and $p$ be a prime. Let $p-$expansions of $m$ and $n$ be \[m = a_0 + a_1p + \dots + a_rp^r\]\[n = b_0 + b_1p + \dots + b_sp^s\] respectively, where $a_r, b_s \neq 0$, for all $i \in \{0,1,\dots,r\}$ and for all $j \in \{0,1,\dots,s\}$, we have $0 \leq a_i, b_j \leq p-1$ . If $a_i \leq b_i$ for all $i \in \{0,1,\dots,r\}$, we write $ m \prec_p n$. Prove that \[p \nmid {{n}\choose{m}} \Leftrightarrow m \prec_p n\].

There are $100$ dwarfes with weight $1,2,...,100$. They sit on the left riverside. They can not swim, but they have one boat with capacity 100. River has strong river flow, so every dwarf has power only for one passage from right side to left as oarsman. On every passage can be only one oarsman. Can all dwarfes get to right riverside?

Find all positive integers $x$ and $n$ such that ${{x}^{3}}+3367={{2}^{n}}$.

Given a prime $p$, consider integers $0<a<b<c<d<p$ such that $a^4\equiv b^4\equiv c^4\equiv d^4\pmod{p}$. Show that \[a+b+c+d\mid a^{2013}+b^{2013}+c^{2013}+d^{2013}\]

Determine the least possible value of $f(1998),$ where $f:\Bbb{N}\to \Bbb{N}$ is a function such that for all $m,n\in {\Bbb N}$, \[f\left( n^{2}f(m)\right) =m\left( f(n)\right) ^{2}. \]

Let $n\in\mathbb{N}$ and $p$ is the odd prime number. Define the sequence $a_n$ such that $a_1=pn+1$ and $a_{k+1}=na_k+1$ for all $k \in \mathbb{N}$ . Prove that $a_{p-1}$ is compound number.

A student adds up rational fractions incorrectly: \[\frac{a}{b}+\frac{x}{y}=\frac{a+x}{b+y}\quad (\star) \] Despite that, he sometimes obtains correct results. For a given fraction $\frac{a}{b},a,b\in\mathbb{Z},b>0$, find all fractions $\frac{x}{y},x,y\in\mathbb{Z},y>0$ such that the result obtained by $(\star)$ is correct.

Let $n$ and $k$ be two positive integers such that $1\leq n \leq k$. Prove that, if $d^k+k$ is a prime number for each positive divisor $d$ of $n$, then $n+k$ is a prime number.

A sequence $P=\left \{ a_{n} \right \}$ is called a $ \text{Permutation}$ of natural numbers (positive integers) if for any natural number $m,$ there exists a unique natural number $n$ such that $a_n=m.$ We also define $S_k(P)$ as: $S_k(P)=a_{1}+a_{2}+\cdots +a_{k}$ (the sum of the first $k$ elements of the sequence). Prove that there exists infinitely many distinct $ \text{Permutations}$ of natural numbers like $P_1,P_2, \cdots$ such that$:$ $$\forall k, \forall i<j: S_k(P_i)|S_k(P_j)$$

Let $p_1,p_2,\ldots,p_n$ be distinct primes greater than $3$. Show that $2^{p_1p_2\cdots p_n}+1$ has at least $4^n$ divisors.