Find all pairs of natural numbers $(n,k)$ for which $(n+1)^{k}-1 = n!$.

Find all triples of natural numbers $(a, b, c)$ for which the number $$2^a + 2^b + 2^c + 3$$ is the square of an integer.

Prove that any natural number smaller or equal than the factorial of a natural number $ n $ is the sum of at most $ n $ distinct divisors of the factorial of $ n. $

Find all functions $f:\mathbb{Z}^+\rightarrow \mathbb{Z}^+$ satisfying $$m!!+n!!\mid f(m)!!+f(n)!!$$for each $m,n\in \mathbb{Z}^+$, where $n!!=(n!)!$ for all $n\in \mathbb{Z}^+$. [i]Proposed by DVDthe1st[/i]

A natural number $N$ is written in its decimal representation . It is known that for each digit in this representation , this digit divides exactly into the number $N$ (the digit $0$ is not encountered). What is the maximum number of different digits which there can be in such a representation of $N$? (S . Fomin, Leningrad)

We will say that a natural number $n$ is [i]adequate[/i] if there exist $n$ integers $a_1,a_2,\ldots ,a_n$ (which are not necessarily positive and can be repeated) such that$$a_1+a_2+\cdots +a_n=a_1a_2 \cdots a_n=n.$$Determine all [i]adequate[/i] numbers.

Let $a$ be a fixed integer. Find all integer solutions $x, y, z$ of the system \[5x + (a + 2)y + (a + 2)z = a,\]\[(2a + 4)x + (a^2 + 3)y + (2a + 2)z = 3a - 1,\]\[(2a + 4)x + (2a + 2)y + (a^2 + 3)z = a + 1.\]

[b]p1.[/b] A tennis net is made of strings tied up together which make a grid consisting of small squares as shown below. [img]https://cdn.artofproblemsolving.com/attachments/9/4/72077777d57408d9fff0ea5e79be5ecb6fe8c3.png[/img] The size of the net is $100\times 10$ small squares. What is the maximal number of sides of small squares which can be cut without breaking the net into two separate pieces? (The side is cut only in the middle, not at the ends). [b]p2.[/b] What number is bigger $2^{300}$ or $3^{200}$ ? [b]p3.[/b] All noble knights participating in a medieval tournament in Camelot used nicknames. In the tournament each knight had combats with all other knights. In each combat one knight won and the second one lost. At the end of tournament the losers reported their real names to the winners and to the winners of their winners. Was there a person who knew the real names of all knights? [b]p4.[/b] Two players Tom and Sid play the following game. There are two piles of rocks, $10$ rocks in the first pile and $12$ rocks in the second pile. Each of the players in his turn can take either any amount of rocks from one pile or the same amount of rocks from both piles. The winner is the player who takes the last rock. Who does win in this game if Tom starts the game? [b]p5.[/b] There is an interesting $5$-digit integer. With a $1$ after it, it is three times as large as with a $1$ before it. What is the number? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A sequence $(a_n)$ positive integers is determined by equalities $a_1=20,a_2=22$ and $a_{n+1}=4a_n^2+5a_{n-1}^3$ for all $n \geq 2$. Find the maximum power of two which divides $a_{2023}$.

Determine all natural numbers $n \geq 2$ with at most four natural divisors, which have the property that for any two distinct proper divisors $d_1$ and $d_2$ of $n$, the positive integer $d_1-d_2$ divides $n$.

Let $n$ be a positive integer having at least two different prime factors. Show that there exists a permutation $a_1, a_2, \dots , a_n$ of the integers $1, 2, \dots , n$ such that \[\sum_{k=1}^{n} k \cdot \cos \frac{2 \pi a_k}{n}=0.\]

Find all pairs $(a, b)$ of coprime positive integers, such that $a<b$ and $$b \mid (n+2)a^{n+1002}-(n+1)a^{n+1001}-na^{n+1000}$$ for all positive integers $n$.

What is the hundreds digit of $2011^{2011}$? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 9 $

The numbers $a_1,a_2$ and $a_3$ are distinct positive integers, such that (i) $a_1$ is a divisor of $a_2+a_3+a_2a_3$; (ii) $a_2$ is a divisor of $a_3+a_1+a_3a_1$; (iii) $a_3$ is a divisor of $a_1+a_2+a_1a_2$. Prove that $a_1,a_2$ and $a_3$ cannot all be prime.

Given an even positive integer $m$, find all positive integers $n$ for which there exists a bijection $f:[n]\to [n]$ so that, for all $x,y\in [n]$ for which $n\mid mx-y$, $$(n+1)\mid f(x)^m-f(y).$$ Note: For a positive integer $n$, we let $[n] = \{1,2,\dots, n\}$. [i]Proposed by Milan Haiman and Carl Schildkraut[/i]

Every positive integer is either [i]nice [/i] or [i]naughty[/i], and the Oracle of Numbers knows which are which. However, the Oracle will not directly tell you whether a number is [i]nice [/i] or [i]naughty[/i]. The only questions the Oracle will answer are questions of the form “What is the sum of all nice divisors of $n$?,” where $n$ is a number of the questioner’s choice. For instance, suppose ([i]just [/i] for this example) that $2$ and $3$ are nice, while $1$ and $6$ are [i]naughty[/i]. In that case, if you asked the Oracle, “What is the sum of all nice divisors of $6$?,” the Oracle’s answer would be $5$. Show that for any given positive integer $n$ less than $1$ million, you can determine whether $n$ is [i]nice [/i] or [i]naughty [/i] by asking the Oracle at most four questions.

Call a natural number $n$ a [i]magic [/i] number if the number obtained by putting $n$ on the right of any natural number is divisible by $n$. Find the number of magic numbers less than $500$. Justify your answer

Find three distinct natural numbers such that the sum of their reciprocals is an integer.

Find all positive integers $n$ such that there exist two permutations $a_0,a_1,\ldots,a_{n-1}$ and $b_0,b_1,\ldots,b_{n-1}$ of the set $\lbrace0,1,\ldots,n-1\rbrace$, satisfying the condition $$ia_i\equiv b_i\pmod{n}$$ for all $0\le i\le n-1$. [i]Proposed by Fysty[/i]

Let $m$ positive integers $a_1, \dots , a_m$ be given. Prove that there exist fewer than $2^m$ positive integers $b_1, \dots , b_n$ such that all sums of distinct $b_k$’s are distinct and all $a_i \ (i \leq m)$ occur among them.

Anton wrote $4$ positive integers on the board. Oleksii calculated their product, while Fedir calculated the sum of their fourth powers. Is it possible that Oleksii's number and Fedir's number have the same number of digits and that these numbers are written as digit-reversals of each other? [i]Proposed by Fedir Yudin and Mykhailo Shtandenko[/i]

Is it true that for any permulation $a_1,a_2.....,a_{2002}$ of $1,2....,2002$ there are positive integers $m,n$ of the same parity such that $0<m<n<2003$ and $a_m+a_n=2a_{\frac {m+n}{2}}$

Prove that the sum $1^3 + 2^3 +...+ n^3$ is a perfect square for all $n$.

Let a pair of positive integers $(n, m)$ that are relatively prime be called [i]intertwined[/i] if among any two divisors of $n$ greater than $1$, there exists a divisor of $m$ and among any two divisors of $m$ greater than $1$, there exists a divisor of $n$. For example, pair $(63, 64)$ is intertwined. [b]a)[/b] Find the largest integer $k$ for which there exists an intertwined pair $(n, m)$ such that the product $nm$ is equal to the product of the first $k$ prime numbers. [b]b)[/b] Prove that there does [b]not[/b] exist an intertwined pair $(n, m)$ such that the product $nm$ is the product of $2025$ distinct prime numbers. [b]c)[/b] Prove that there exists an intertwined pair $(n, m)$ such that the number of divisors of $n$ is greater than $2025$. [i]Proposed by Stijn Cambie, Belgium[/i]

Does there exist an infinite set $S$ consisted of positive integers,so that for any $x,y,z,w\in S,x<y,z<w$,if $(x,y)\ne (z,w)$,then $\gcd(xy+2022,zw+2022)=1$?