Determine all positive integers $n$ with at least $4$ factors such that $n$ is the sum the squares of its $4$ smallest factors.

Prove that if natural numbers $a,b,c,d$ satisfy the equality $ab = cd$, then $\frac{gcd(a,c)gcd(a,d)}{gcd(a,b,c,d)}= a$

Denote by $\phi(n)$ for all $n\in\mathbb{N}$ the number of positive integer smaller than $n$ and relatively prime to $n$. Also, denote by $\omega(n)$ for all $n\in\mathbb{N}$ the number of prime divisors of $n$. Given that $\phi(n)|n-1$ and $\omega(n)\leq 3$. Prove that $n$ is a prime number.

A construction rope is tied to two trees. It is straight and taut. It is then vibrated at a constant velocity $v_1$. The tension in the rope is then halved. Again, the rope is vibrated at a constant velocity $v_2$. The tension in the rope is then halved again. And, for the third time, the rope is vibrated at a constant velocity, this time $v_3$. The value of $\frac{v_1}{v_3}+\frac{v_3}{v_1}$ can be expressed as a positive number $\frac{m\sqrt{r}}{n}$, where $m$ and $n$ are relatively prime, and $r$ is not divisible by the square of any prime. Find $m+n+r$. If the number is rational, let $r=1$. [i](Ahaan Rungta, 2 points)[/i]

Find all the odd naturals whose indicator is the same as $1990$. We clarify that, if a natural decomposes into prime factors in the form $\Pi_{j=1}^r p_j^{a_j}$, define the [i]indicator [/i] as : $\phi (n) = r\Pi_{j=1}^r p_j^{a_j-1} (p_j + 1)$. [hide=official wording for first sentence]Encuentre todos los naturales impares cuyo indicador es el mismo que el de 1990.[/hide]

Find all positive integers $n$ such that the sum of the squares of the five smallest divisors of $n$ is a square.

Let $p\geq 3$ be a prime number and $0\leq r\leq p-3.$ Let $x_1,x_2,\ldots,x_{p-1+r}$ be integers satisfying \[\sum_{i=1}^{p-1+r}x_i^k\equiv r \bmod{p}\]for all $1\leq k\leq p-2.$ What are the possible remainders of numbers $x_2,x_2,\ldots,x_{p-1+r}$ modulo $p?$ [i]Proposed by Dávid Matolcsi, Budapest[/i]

a. Does there exist 4 distinct positive integers such that the sum of any 3 of them is prime? b. Does there exist 5 distinct positive integers such that the sum of any 3 of them is prime?

Denote by $l(n)$ the largest prime divisor of $n$. Let $a_{n+1} = a_n + l(a_n)$ be a recursively defined sequence of integers with $a_1 = 2$. Determine all natural numbers $m$ such that there exists some $i \in \mathbb{N}$ with $a_i = m^2$. [i]Proposed by Nikola Velov, North Macedonia[/i]

Let $m$ and $n$ be positive integers such that $m>n$. Define $x_k=\frac{m+k}{n+k}$ for $k=1,2,\ldots,n+1$. Prove that if all the numbers $x_1,x_2,\ldots,x_{n+1}$ are integers, then $x_1x_2\ldots x_{n+1}-1$ is divisible by an odd prime.

Find all real numbers $a > 1$ such that there exists an integer $k \ge 1$ such that the sequence $\{x_n\}_{n\ge 1}$ formed with the first $k$ digits of the number $\lfloor a^n\rfloor$ is periodical.

For a real number $x$, let $\lfloor x\rfloor$ stand for the largest integer that is less than or equal to $x$. Prove that \[ \left\lfloor{(n-1)!\over n(n+1)}\right\rfloor \] is even for every positive integer $n$.

Let $A=20132013...2013$ be formed by joining $2013$, $165$ times. Prove that $2013^2 \mid A$.

Is there a nine-digit number without zero digits, the remainder of dividing which on each of its digits is different?

Let $\pi$ be a permutation of the numbers from $1$ through $2012$. What is the maximum possible number of integers $n$ with $1 \le n \le 2011$ such that $\pi (n)$ divides $\pi (n + 1)$?

[b]p1.[/b] Find any integer solution of the puzzle: $WE+ST+RO+NG=128$ (different letters mean different digits between $1$ and $9$). [b]p2.[/b] A $5\times 6$ rectangle is drawn on the piece of graph paper (see the figure below). The side of each square on the graph paper is $1$ cm long. Cut the rectangle along the sides of the graph squares in two parts whose areas are equal but perimeters are different by $2$ cm. $\begin{tabular}{|l|l|l|l|l|l|} \hline & & & & & \\ \hline & & & & & \\ \hline & & & & & \\ \hline & & & & & \\ \hline \end{tabular}$ [b]p3.[/b] Three runners started simultaneously on a $1$ km long track. Each of them runs the whole distance at a constant speed. Runner $A$ is the fastest. When he runs $400$ meters then the total distance run by runners $B$ and $C$ together is $680$ meters. What is the total combined distance remaining for runners $B$ and $C$ when runner $A$ has $100$ meters left? [b]p4.[/b] There are three people in a room. Each person is either a knight who always tells the truth or a liar who always tells lies. The first person said «We are all liars». The second replied «Only you are a liar». Is the third person a liar or a knight? [b]p5.[/b] A $5\times 8$ rectangle is divided into forty $1\times 1$ square boxes (see the figure below). Choose 24 such boxes and one diagonal in each chosen box so that these diagonals don't have common points. $\begin{tabular}{|l|l|l|l|l|l|l|l|} \hline & & & & & & & \\ \hline & & & & & & & \\ \hline & & & & & & & \\ \hline & & & & & & & \\ \hline \end{tabular}$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For $a_1 = 3$, define the sequence $a_1, a_2, a_3, \ldots$ for $n \geq 1$ as $$na_{n+1}=2(n+1)a_n-n-2.$$ Prove that for any odd prime $p$, there exist positive integer $m,$ such that $p|a_m$ and $p|a_{m+1}.$

Find, as a function of $\, n, \,$ the sum of the digits of \[ 9 \times 99 \times 9999 \times \cdots \times \left( 10^{2^n} - 1 \right), \] where each factor has twice as many digits as the previous one.

Let $n$ be a positive integer. Let $a$ be an integer such that $\gcd (a,n)=1$. Prove that \[\frac{a^{\phi (n)}-1}{n}=\sum_{i\in R}\frac{1}{ai}\left[\frac{ai}{n}\right]\pmod{n}\] where $R$ is the reduced residue system of $n$ with each element a positive integer at most $n$.

Prove that if the sum of $x^5,y^5$ and $z^5$, where $x,y$ and $z$ are integer numbers, is divisible by $25$ then the sum of some two of them is divisible by $25$.

Determine whether there exists an infinite sequence of nonzero digits $a_1 , a_2 , a_3 , \cdots $ and a positive integer $N$ such that for every integer $k > N$, the number $\overline{a_k a_{k-1}\cdots a_1 }$ is a perfect square.

Let $A_n$ be the number of arranged n-tuples of natural numbers $(a_1,a_2…a_n)$, such that $\frac{1}{a_1} +\frac{1}{a_2} +...+\frac{1}{a_n} =1$. Find the parity of $A_{68}$.

Let $a\geq 2$ be an integer. Prove that there exists a positive integer $b$ with the following property: For each positive integer $n$, there is a prime number $p$ (possibly depending on $a,b,n$) such that $a^n + b$ is divisible by $p$, but not divisible by $p^2$.

Find all primes $p,q$ such that $p$ divides $30q-1$ and $q$ divides $30p-1$.

Six free cells are given in a row. Players $A$ and $B$ alternately write digits from $0$ to $9$ in empty cells, with $A$ starting. When all the cells are filled, one considers the obtained six-digit number $z$. Player $B$ wins if $z$ is divisible by a given natural number $n$, and loses otherwise. For which values of $n$ not exceeding $20$ can $B$ win independently of his opponent’s moves?