(a) Prove that there exist infinitely many natural numbers $n$ such that the sum of the digits of $11n$ is twice the sum of the digits of $n$. (b) Prove that there exist infinitely many natural numbers $n$ such that the sum of the digits of $4n + 3$ is equal to the sum of the digits of $n$. (c) Prove that for any natural number $n$, it is possible to find $n$ consecutive numbers such that none of them is prime.

Let $ p$ be a prime number. Solve in $ \mathbb{N}_0\times\mathbb{N}_0$ the equation $ x^3\plus{}y^3\minus{}3xy\equal{}p\minus{}1$.

Let $n$ be a positive integer and $M = \{1, 2, 3, 4, 5, 6\}$. Two persons $A$ and $B$ play in the following Way: $A$ writes down a digit from $M$, $B$ appends a digit from $M$, and so it becomes alternately one digit from $M$ is appended until the $2n$-digit decimal representation of a number has been created. If this number is divisible by $9$, $B$ wins, otherwise $A$ wins. For which $n$ can $A$ and for which $n$ can $B$ force the win?

Of the integers $a$, $b$, and $c$ that satisfy $0 < c < b < a$ and $$a^3 - b^3 - c^3 - abc + 1 = 2022,$$ let $c'$ be the least value of $ c$ appearing in any solution, let $a'$ be the least value of $a$ appearing in any solution with $c = c'$, and let $b'$ be the value of $b$ in the solution where $c = c'$ and $a = a'$. Find $a' + b' + c'$.

Let $p\geq5$ be a prime and $S_k=1^k+2^k+...+(p-1)^k,\forall k\in\mathbb{N}.$ Prove that there is an infinity of numbers $n\in\mathbb{N}$ such that $p^3$ divides $S_n$ and $ p $ divides $S_{n-1}$ and $S_{n-2}.$

Let $a, b$ and $c$ be natural numbers. Determine the smallest value that the following expression can take: $$\frac{a}{gcd\,\,(a + b, a - c)} + \frac{b}{gcd\,\,(b + c, b - a)} + \frac{c}{gcd\,\,(c + a, c - b)}.$$ . Remark: $gcd \,\, (6, 0) = 6$ and $gcd\,\,(3, -6) = 3$.

Let $m,n,k$ be positive integers and $1+m+n \sqrt 3=(2+ \sqrt 3)^{2k+1}$. Prove that $m$ is a perfect square.

A list of numbers $a_1,a_2,\ldots,a_m$ contains an arithmetic trio $a_i, a_j, a_k$ if $i < j < k$ and $2a_j = a_i + a_k$. Let $n$ be a positive integer. Show that the numbers $1, 2, 3, \ldots, n$ can be reordered in a list that does not contain arithmetic trios.

a)Prove that $\frac{1}{2}+\frac{1}{3}+...+\frac{1}{{{2}^{m}}}<m$, for any $m\in {{\mathbb{N}}^{*}}$. b)Let ${{p}_{1}},{{p}_{2}},...,{{p}_{n}}$ be the prime numbers less than ${{2}^{100}}$. Prove that $\frac{1}{{{p}_{1}}}+\frac{1}{{{p}_{2}}}+...+\frac{1}{{{p}_{n}}}<10$

Mr. Jones has eight children of different ages. On a family trip his oldest child, who is 9, spots a license plate with a 4-digit number in which each of two digits appears two times. "Look, daddy!" she exclaims. "That number is evenly divisible by the age of each of us kids!" "That's right," replies Mr. Jones, "and the last two digits just happen to be my age." Which of the following is not the age of one of Mr. Jones's children? $ \textbf{(A) } 4 \qquad \textbf{(B) } 5 \qquad \textbf{(C) } 6 \qquad \textbf{(D) } 7 \qquad \textbf{(E) } 8$

Find the smallest positive integer $n$ such that [list][*] $n$ has exactly $144$ distinct positive divisors, [*] there are ten consecutive integers among the positive divisors of $n$. [/list]

Prove that every positive rational number can be represented in the form $\dfrac{a^{3}+b^{3}}{c^{3}+d^{3}}$ where a,b,c,d are positive integers.

A positive integer $a$ is called a [i]double number[/i] if it has an even number of digits (in base 10) and its base 10 representation has the form $a = a_1a_2 \cdots a_k a_1 a_2 \cdots a_k$ with $0 \le a_i \le 9$ for $1 \le i \le k$, and $a_1 \ne 0$. For example, $283283$ is a double number. Determine whether or not there are infinitely many double numbers $a$ such that $a + 1$ is a square and $a + 1$ is not a power of $10$.

Let $d(n)$ be the number of positive divisors of a positive integer $n$ (including $1$ and $n$). Find all values of $n$ such that $n + d(n) = d(n)^2$.

Let $A=\{1,3,3^2,\ldots, 3^{2014}\}$. We obtain a partition of $A$ if $A$ is written as a disjoint union of nonempty subsets. [list=a] [*]Prove that there is no partition of $A$ such that the product of elements in each subset is a square. [*]Prove that there exists a partition of $A$ such that the sum of elements in each subset is a square.[/list]

The sequence of integers $a_1, a_2, ,,$ is defined as follows: $$a_n=\begin{cases} 0\,\,\,\, if\,\,\,\, n\,\,\,\, has\,\,\,\, an\,\,\,\, even\,\,\,\, number\,\,\,\, of\,\,\,\, divisors\,\,\,\, greater\,\,\,\, than\,\,\,\, 2014 \\ 1 \,\,\,\, if \,\,\,\, n \,\,\,\, has \,\,\,\, an \,\,\,\, odd \,\,\,\, number \,\,\,\, of \,\,\,\, divisors \,\,\,\, greater \,\,\,\, than \,\,\,\, 2014\end{cases}$$ Show that the sequence $a_n$ never becomes periodic.

Let $k$ and $n$ be two given distinct positive integers greater than $1$. There are finitely many (not necessarily distinct) integers written on the blackboard. Kázmér is allowed to erase $k$ consecutive elements of an arithmetic sequence with a difference not divisible by $k$. Similarly, Nándor is allowed to erase $n$ consecutive elements of an arithmetic sequence with a difference that is not divisible by $n$. The initial numbers on the blackboard have the property that both Kázmér and Nándor can erase all of them (independently from each other) in a finite number of steps. Prove that the difference of biggest and the smallest number on the blackboard is at least $\varphi(n)+\varphi(k)$. [i]Proposed by Boldizsár Varga, Budapest[/i]

Find all prime numbers $p$ such that the number $$3^p+4^p+5^p+9^p-98$$ has at most $6$ positive divisors.

Let $A$ be a set of $8$ elements, and $B := (B_1,...,B_7)$ be an ordered $7$-tuple of subsets of $A$. Let $N$ be the number of such $7$-tuples $B$ such that there exists a unique $4$-element subset $I \subseteq \{1,2,...,7\}$ for which the intersection $\cap _{ i\in I} B_i$ is nonempty. Find the remainder when $N$ is divided by $67$.

Let $p$ be a prime number. Find all pairs $(x, y)$ of positive integers such that $x^3 + y^3 - 3xy = p -1$.

Find all ordered triples $(a,b,c)$ of positive integers such that the value of the expression \[\left (b-\frac{1}{a}\right )\left (c-\frac{1}{b}\right )\left (a-\frac{1}{c}\right )\] is an integer.

Find all pairs of positive integers $ (a, b)$ such that \[ ab \equal{} gcd(a, b) \plus{} lcm(a, b). \]

We call a number greater than $25$, [i] semi-prime[/i] if it is the sum of some two different prime numbers. What is the greatest number of consecutive natural numbers that can be [i]semi-prime[/i]?

Find all positive integers $m$ with the next property: If $d$ is a positive integer less or equal to $m$ and it isn't coprime to $m$ , then there exist positive integers $a_{1}, a_{2}$,. . ., $a_{2019}$ (where all of them are coprimes to $m$) such that $m+a_{1}d+a_{2}d^{2}+\cdot \cdot \cdot+a_{2019}d^{2019}$ is a perfect power.

$3.$ Positive integers $a$ and $b$ are such that $a+b=\frac{a}{b}+\frac{b}{a}.$ What is the value of $a^2+b^2 ?$