Find six distinct positive integers such that the product of any two of them is divisible by their sum. (D. Fomin, Leningrad)

A positive integer $\ell \ge 2$ is called [i]sweet[/i] if there exists a positive integer $n \ge 10$ such that when the leftmost nonzero decimal digit of $n$ is deleted, the resulting number $m$ satisfies $n = m\ell.$ Let $S$ denote the set of all sweet numbers $\ell.$ If the sum $\sum_{\ell \in S} \tfrac{1}{\ell-1}$ can be written as $\tfrac{A}{B}$ for relatively prime positive integers $A,B,$ find $A+B.$

Four positive integers $x,y,z$ and $t$ satisfy the relations \[ xy - zt = x + y = z + t. \] Is it possible that both $xy$ and $zt$ are perfect squares?

Let be a prime number $ p, $ the quotient ring $ R=\mathbb{Z}[X,Y]/(pX,pY), $ and a prime ideal $ I\supset pA $ that is not maximal. Show that the ring $ \left\{ r/i|r\in R, i\in I \right\} $ is factorial.

For any positive integer $n$, let $s(n)$ be the sum of its digits, written in decimal base. Prove that for each integer $n \ge 1$ there exists a positive integer $x$ such that the fraction $\frac{x + k}{s(x + k)}$ is not integral, for each integer $k$ with $0 \le k \le n$.

Find all prime numbers $p$ and $q$ such that $p^2|q^3+1$ and $q^2|p^6-1$

Find all positive integers $n$ such that $n^4 - n^3 + 3n^2 + 5$ is a perfect square.

p1. Let $AB$ be the diameter of the circle and $P$ is a point outside the circle. The lines $PQ$ and $PR$ are tangent to the circles at points $Q$ and $R$. The lines $PH$ is perpendicular on line $AB$ at $H$ . Line $PH$ intersects $AR$ at $S$. If $\angle QPH =40^o$ and $\angle QSA =30^o$, find $\angle RPS$. p2. There is a meeting consisting of $40$ seats attended by $16$ invited guests. If each invited guest must be limited to at least $ 1$ chair, then determine the number of arrangements. p3. In the crossword puzzle, in the following crossword puzzle, each box can only be filled with numbers from $ 1$ to $9$. [img]https://cdn.artofproblemsolving.com/attachments/2/e/224b79c25305e8ed9a8a4da51059f961b9fbf8.png[/img] Across: 1. Composite factor of $1001$ 3. Non-polyndromic numbers 5. $p\times q^3$, with $p\ne q$ and $p,q$ primes Down: 1. $a-1$ and $b+1$ , $a\ne b$ and $p,q$ primes 2. multiple of $9$ 4. $p^3 \times q$, with $p\ne q$ and $p,q$ primes p4. Given a function $f:R \to R$ and a function $g:R \to R$, so that it fulfills the following figure: [img]https://cdn.artofproblemsolving.com/attachments/b/9/fb8e4e08861a3572412ae958828dce1c1e137a.png[/img] Find the number of values ​​of $x$, such that $(f(x))^2-2g(x)-x \in\{-10,-9,-8,…,9,10\}$. p5. In a garden that is rectangular in shape, there is a watchtower in each corner and in the garden there is a monitoring tower. Small areas will be made in the shape of a triangle so that the corner points are towers (free of monitoring and/or supervisory towers). Let $k(m,n)$ be the number of small areas created if there are $m$ control towers and $n$ monitoring towers. a. Find the values ​​of $k(4,1)$, $k(4,2)$, $k(4,3)$, and $k(4,4)$ b. Find the general formula $k(m,n)$ with $m$ and $n$ natural numbers .

For integers $n>1$, define $f(n)$ to be the sum of all postive divisors of $n$ that are less than $n$. Prove that for any positive integer $k$, there exists a positive integer $n>1$ such that $n<f(n)<f^2(n)<\cdots<f^k(n)$, where $f^i(n)=f(f^{i-1}(n))$ for $i>1$ and $f^1(n)=f(n)$.

Show that for any $n$ we can find a set $X$ of $n$ distinct integers greater than 1, such that the average of the elements of any subset of $X$ is a square, cube or higher power.

$find$ all non negative integers $m$, $n$ such that $mn-1$ divides $n^3-1$

A divisor $d$ of a positive integer $n$ is said to be a [i]close[/i] divisor of $n$ if $\sqrt{n}<d<2\sqrt{n}$. Does there exist a positive integer with exactly $2020$ close divisors?

Let $S$ be a set of $n$ distinct real numbers, and $A_S$ set of arithemtic means of two distinct numbers from $S$. For given $n \geq 2$ find minimal number of elements in $A_S$

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 $m +n$

Solve in integers the equation : $x^2y+y^2=x^3$

Prove that there are positive integers $a_1, a_2,\dots, a_{2020}$ such that $$\dfrac{1}{a_1}+\dfrac{1}{2a_2}+\dfrac{1}{3a_3}+\dots+\dfrac{1}{2020a_{2020}}=1.$$

Show that if an odd prime number $ p$ can be expressed in the form $ x^5\minus{}y^5$ for some integers $ x,y,$ then: $ \sqrt{\frac{4p\plus{}1}{5}}\equal{}\frac{v^2\plus{}1}{2}$ for some odd integer $ v$.

Decide, whether exists positive rational number $w$, which isn't integer, such that $w^w$ is a rational number.

(a) Does there exist a polynomial $ P(x)$ with coefficients in integers, such that $ P(d) \equal{} \frac{2008}{d}$ holds for all positive divisors of $ 2008$? (b) For which positive integers $ n$ does a polynomial $ P(x)$ with coefficients in integers exists, such that $ P(d) \equal{} \frac{n}{d}$ holds for all positive divisors of $ n$?

Let $n$ be a positive integer with $n = p^{2010}q^{2010}$ for two odd primes $p$ and $q$. Show that there exist exactly $\sqrt[2010]{n}$ positive integers $x \le n$ such that $p^{2010}|x^p - 1$ and $q^{2010}|x^q - 1$.

Let $n > 1$ be a positive integer and $p$ a prime number such that $n | (p - 1) $and $p | (n^6 - 1)$. Prove that at least one of the numbers $p- n$ and $p + n$ is a perfect square.

The sequence $\left(a_{n}\right)_{n\in\mathbb{N}}$ is defined recursively as $a_{0}=a_{1}=1$, $a_{n+2}=5a_{n+1}-a_{n}-1$, $\forall n\in\mathbb{N}$ Prove that $$a_{n}\mid a_{n+1}^{2}+a_{n+1}+1$$ for any $n\in\mathbb{N}$

Let $n$ be a positive integer, $n\geq 2$. Let $M=\{0,1,2,\ldots n-1\}$. For an integer nonzero number $a$ we define the function $f_{a}: M\longrightarrow M$, such that $f_{a}(x)$ is the remainder when dividing $ax$ at $n$. Find a necessary and sufficient condition such that $f_{a}$ is bijective. And if $f_{a}$ is bijective and $n$ is a prime number, prove that $a^{n(n-1)}-1$ is divisible by $n^{2}$.

Let an even positive integer $2k$ be given. Find such relatively prime positive integers $x, y$ that maximize the product $xy$.

Numbers $d(n,m)$, with $m, n$ integers, $0 \leq m \leq n$, are defined by $d(n, 0) = d(n, n) = 0$ for all $n \geq 0$ and \[md(n,m) = md(n-1,m)+(2n-m)d(n-1,m-1) \text{ for all } 0 < m < n.\] Prove that all the $d(n,m)$ are integers.