An integer $ n$ is called a good number if and only if $ |n|$ is not square of another intger. Find all integers $ m$ such that they can be written in infinitely many ways as sum of three different good numbers and product of these three numbers is square of an odd number.

Zachary tries to simplify the fraction $\frac{2020}{5050}$ by dividing the numerator and denominator by the same integer to get the fraction $\frac{m}{n}$ , where $m$ and $n$ are both positive integers. Find the sum of the (not necessarily distinct) prime factors of the sum of all the possible values of $m +n$

Find all integer solutions to \begin{align*} x^2+2y^2+3z^2&=36,\\ 3x^2+2y^2+z^2&=84,\\ xy+xz+yz&=-7. \end{align*}

Find all integes $a,b,c,d$ that form an arithmetic progression satisfying $d-c+1$ is prime number and $a+b^2+c^3=d^2b$

At the start of the day, the four numbers $(a_0,b_0,c_0,d_0)$ were written on the board. Every minute, Danny replaces the four numbers written on the board with new ones according to the following rule: if the numbers written on the board are $(a,b,c,d)$, then Danny first calculates the numbers \begin{align*} a'&=a+4b+16c+64d\\ b'&=b+4c+16d+64a\\ c'&=c+4d+16a+64b\\ d'&=d+4a+16b+64c \end{align*} and replaces the numbers $(a,b,c,d)$ with the numbers $(a'd',c'd',c'b',b'a')$. For which initial quadruples $(a_0,b_0,c_0,d_0)$, will Danny write at some point a quadruple of numbers all of which are divisible by $5780^{5780}$?

Let $p$ and $q$ be two positive integers that have no common divisor. The set of integers shall be partioned into three subsets $A$, $B$, $C$ such that for each integer $z$ in each of the sets $A$, $B$, $C$ there is exactly one of the numbers $z$, $z+p$ and $z+q$. a) Prove that such a decomposition is possible if and only if $p+q$ is divisible by $3$. b) In the case we omit the restriction that $p$, $q$ may not have a common divisor, prove that for $p \neq q$ the number $\frac{p+q}{\gcd(p,q)}$ is divisible by 3.

Let's call a natural number [i] interesting[/i] if any of its two digits consecutive forms a number that is a multiple of $19$ or $21$. For example, The number $7638$ is interesting, because $76$ is a multiple of $19$, $63$ is multiple of $21$, and $38$ is a multiple of $19$. How many interesting numbers of $2022$ digits exist?

Find all pairs of primes $(p, q)$ for which $p-q$ and $pq-q$ are both perfect squares.

For a positive integer $n$, let $d(n)$ denote the number of positive divisors of $n$. Determine all positive integers $n$ for which $d(n)$ is the second largest divisor of $n$.

Find the smallest positive integer $m$ such that $5m$ is an exact 5th power, $6m$ is an exact 6th power, and $7m$ is an exact 7th power.

A number is called polite if it can be written as $ m + (m+1)+...+ n$, for certain positive integers $ m <n$ . For example: $18$ is polite, since $18 =5 + 6 + 7$. A number is called a power of two if it can be written as $2^{\ell}$ for some integer $\ell \ge 0$. (a) Show that no number is both polite and a power of two. (b) Show that every positive integer is polite or a power of two.

Let $p$ be some prime number. a) Prove that there exist positive integers $a$ and $b$ such that $a^2 + b^2 + 2018$ is multiple of $p$. b) Find all $p$ for which the $a$ and $b$ from a) can be chosen in such way that both these numbers aren’t multiples of $p$.

Let $n$ be a $6$-digit number, perfect square and perfect cube, if $n -6$ is neither even nor multiple of $3$. Find $n$ .

Determine all positive integers $a$, $b$, $c$, $p$, where $p$ and $p+2$ are odd primes and \[2^ap^b=(p+2)^c-1.\]

Find all positive integers $n$ such that the set $\{n,n+1,n+2,n+3,n+4,n+5\}$ can be partitioned into two subsets so that the product of the numbers in each subset is equal.

In the sequence of integers $(a_n)$, the sum $a_m + a_n$ is divided by $m + n$ with any different $m$ and $n$. Prove that $a_n$ is a multiple of $n$ for any $n$.

$(GBR 1)$ The polynomial $P(x) = a_0x^k + a_1x^{k-1} + \cdots + a_k$, where $a_0,\cdots, a_k$ are integers, is said to be divisible by an integer $m$ if $P(x)$ is a multiple of $m$ for every integral value of $x$. Show that if $P(x)$ is divisible by $m$, then $a_0 \cdot k!$ is a multiple of $m$. Also prove that if $a, k,m$ are positive integers such that $ak!$ is a multiple of $m$, then a polynomial $P(x)$ with leading term $ax^k$can be found that is divisible by $m.$

Let $h(t)$ and $f(t)$ be polynomials such that $h(t)=t^2$ and $f_n(t)=h(h(h(h(h...h(t))))))-1$ where $h(t)$ occurs $n$ times. Prove that $f_n(t)$ is a factor of $f_N(t)$ whenever $n$ is a factor of $N$

Suppose that $n$ is s positive integer. Determine all the possible values of the first digit after the decimal point in the decimal expression of the number $\sqrt{n^3+2n^2+n}$

Let $A_1, A_2, ... , A_k$ be the subsets of $\left\{1,2,3,...,n\right\}$ such that for all $1\leq i,j\leq k$:$A_i\cap A_j \neq \varnothing$. Prove that there are $n$ distinct positive integers $x_1,x_2,...,x_n$ such that for each $1\leq j\leq k$: $$lcm_{i \in A_j}\left\{x_i\right\}>lcm_{i \notin A_j}\left\{x_i\right\}$$ [i]Proposed by Morteza Saghafian, Mahyar Sefidgaran[/i]

A [i]Benelux n-square[/i] (with $n\geq 2$) is an $n\times n$ grid consisting of $n^2$ cells, each of them containing a positive integer, satisfying the following conditions: $\bullet$ the $n^2$ positive integers are pairwise distinct. $\bullet$ if for each row and each column we compute the greatest common divisor of the $n$ numbers in that row/column, then we obtain $2n$ different outcomes. (a) Prove that, in each Benelux n-square (with $n \geq 2$), there exists a cell containing a number which is at least $2n^2.$ (b) Call a Benelux n-square [i]minimal[/i] if all $n^2$ numbers in the cells are at most $2n^2.$ Determine all $n\geq 2$ for which there exists a minimal Benelux n-square.

Prove that for any positive integer $n$ there exist coprime numbers $a$ and $b$ such that for all $1 \leq k \leq n$ numbers $a+k$ and $b+k$ are not coprime.

$\mathbb{N}_{10}$ is generalization of $\mathbb{N}$ that every hypernumber in $\mathbb{N}_{10}$ is something like: $\overline{...a_2a_1a_0}$ with $a_i \in {0,1..9}$ (Notice that $\overline {...000} \in \mathbb{N}_{10}$) Also we easily have $+,*$ in $\mathbb{N}_{10}$. first $k$ number of $a*b$= first $k$ nubmer of (first $k$ number of a * first $k$ number of b) first $k$ number of $a+b$= first $k$ nubmer of (first $k$ number of a + first $k$ number of b) Fore example $\overline {...999}+ \overline {...0001}= \overline {...000}$ Prove that every monic polynomial in $\mathbb{N}_{10}[x]$ with degree $d$ has at most $d^2$ roots.

A positive integer is called a repunit, if it is written only by ones. The repunit with $n$ digits will be denoted as $\underbrace{{11\cdots1}}_{n}$ . Prove that: α) the repunit $\underbrace{{11\cdots1}}_{n}$is divisible by $37$ if and only if $n$ is divisible by $3$ b) there exists a positive integer $k$ such that the repunit $\underbrace{{11\cdots1}}_{n}$ is divisible by $41$ if $n$ is divisible by $k$

For a positive integer $n$, let $\langle n \rangle$ denote the perfect square integer closest to $n$. For example, $\langle 74 \rangle = 81$, $\langle 18 \rangle = 16$. If $N$ is the smallest positive integer such that $$ \langle 91 \rangle \cdot \langle 120 \rangle \cdot \langle 143 \rangle \cdot \langle 180 \rangle \cdot \langle N \rangle = 91 \cdot 120 \cdot 143 \cdot 180 \cdot N $$ find the sum of the squares of the digits of $N$.