Given sets $A = \{1, 4, 5, 6, 7, 9, 11, 16, 17\}$, $B = \{2, 3, 8, 10, 12, 13, 14, 15, 18\}$, if a positive integer leaves a remainder (the smallest non-negative remainder) that belongs to $A$ when divided by 19, then that positive integer is called an $\alpha$ number. If a positive integer leaves a remainder that belongs to $B$ when divided by 19, then that positive integer is called a $\beta$ number. (1) For what positive integer $n$, among all its positive divisors, are the numbers of $\alpha$ divisors and $\beta$ divisors equal? (2) For which positive integers $k$, are the numbers of $\alpha$ divisors less than the numbers of $\beta$ divisors? For which positive integers $l$, are the numbers of $\alpha$ divisors greater than the numbers of $\beta$ divisors?

Let $p$ be a prime number and $a$ be an integer. Prove that if $2^p +3^p = a^n$ for some integer $n$, then $n = 1$.

Let $ b, m, n$ be positive integers such that $ b > 1$ and $ m \neq n.$ Prove that if $ b^m \minus{} 1$ and $ b^n \minus{} 1$ have the same prime divisors, then $ b \plus{} 1$ is a power of 2.

Find the number of positive $6$ digit integers, such that the sum of their digits is $9$, and four of its digits are $2,0,0,4.$ [hide= original wording] before finding a typo .. Find the number of positive $6$ digit integers, such that the sum of their digits is $9$, and four of its digits are $1,0,0,4.$ Posts 2 and 3 reply to this wording [/hide]

$2010$ ships deliver bananas, lemons and pineapples from South America to Russia. The total number of bananas on each ship equals the number of lemons on all other ships combined, while the total number of lemons on each ship equals the total number of pineapples on all other ships combined. Prove that the total number of fruits is a multiple of $31$.

Let $d(n)$ denote the number of positive divisors of the positive integer $n$.Determine those numbers $n$ for which $d(n^3)=5d(n)$.

Let $n \ge 2$ be a positive integer. Prove that the following assertions are equivalent: a) for all integer $x$ coprime with n the congruence $x^6 \equiv 1$ (mod $n$) hold, b) $n$ divides $504$.

Let $a$ and $k$ be positive integers. Prove that for every positive integer $d$ there exists a positive integer $n$ such that $d$ divides $ka^n + n.$

Determine whether exists positive integer $n$ such that the number $A=8^n+47$ is prime.

Let $p > 3$ be a prime such that $p\equiv 3 \pmod 4.$ Given a positive integer $a_0$ define the sequence $a_0, a_1, \ldots $ of integers by $a_n = a^{2^n}_{n-1}$ for all $n = 1, 2,\ldots.$ Prove that it is possible to choose $a_0$ such that the subsequence $a_N , a_{N+1}, a_{N+2}, \ldots $ is not constant modulo $p$ for any positive integer $N.$

In the middle of the number 1996 (i.e. between $19$ and $96$) you need to insert several digits so that the resulting number is divisible by $1997$. In this case, you need to get by with the least number of inserted digits.

A positive integer $d$ is [i]special[/i] if every integer can be represented as $a^2 + b^2 - dc^2$ for some integers $a, b, c$. [list] [*]Find the smallest positive integer that is not special. [*]Prove 2017 is special. [/list]

Let $x, y$ be integers and $p$ be a prime for which \[ x^2-3xy+p^2y^2=12p \] Find all triples $(x,y,p)$.

Prove that there are infinitely many natural numbers, the arithmetic mean and geometric mean of the divisors which are both integers.

$(POL 3)$ Given a polynomial $f(x)$ with integer coefficients whose value is divisible by $3$ for three integers $k, k + 1,$ and $k + 2$. Prove that $f(m)$ is divisible by $3$ for all integers $m.$

For which $ n \geq 2, n \in \mathbb{N}$ are there positive integers $ A_1, A_2, \ldots, A_n$ which are not the same pairwise and have the property that the product $ \prod^n_{i \equal{} 1} (A_i \plus{} k)$ is a power for each natural number $ k.$

Suppose that $k \ge 2$ and $n_{1}, n_{2}, \cdots, n_{k}\ge 1$ be natural numbers having the property \[n_{2}\; \vert \; 2^{n_{1}}-1, n_{3}\; \vert \; 2^{n_{2}}-1, \cdots, n_{k}\; \vert \; 2^{n_{k-1}}-1, n_{1}\; \vert \; 2^{n_{k}}-1.\] Show that $n_{1}=n_{2}=\cdots=n_{k}=1$.

Find all functions $f:\mathbb{N}\setminus\{1\} \rightarrow\mathbb{N}$ such that for all distinct $x,y\in \mathbb{N}$ with $y\ge 2018$, $$\gcd(f(x),y)\cdot \mathrm{lcm}(x,f(y))=f(x)f(y).$$

For $\pi\leq\theta<2\pi$, let \[ P=\dfrac12\cos\theta-\dfrac14\sin2\theta-\dfrac18\cos3\theta+\dfrac1{16}\sin4\theta+\dfrac1{32}\cos5\theta-\dfrac1{64}\sin6\theta-\dfrac1{128}\cos7\theta+\ldots \] and \[ Q=1-\dfrac12\sin\theta-\dfrac14\cos2\theta+\dfrac1{8}\sin3\theta+\dfrac1{16}\cos4\theta-\dfrac1{32}\sin5\theta-\dfrac1{64}\cos6\theta+\dfrac1{128}\sin7\theta +\ldots \] so that $\tfrac PQ = \tfrac{2\sqrt2}7$. Then $\sin\theta = -\tfrac mn$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Let $k$ be a fixed integer greater than 1, and let ${m=4k^2-5}$. Show that there exist positive integers $a$ and $b$ such that the sequence $(x_n)$ defined by \[x_0=a,\quad x_1=b,\quad x_{n+2}=x_{n+1}+x_n\quad\text{for}\quad n=0,1,2,\dots,\] has all of its terms relatively prime to $m$. [i]Proposed by Jaroslaw Wroblewski, Poland[/i]

The Fibonacci sequence is given by $x_1 = x_2 = 1$ and $x_{k+2} = x_{k+1} + x_k$ for each $k \in N$. (a) Prove that there are Fibonacci numbes that end in a $9$ in the decimal system. (b) Determine for which $n$ can a Fibonacci number end in $n$ $9$-s in the decimal system.

There are no integers $ a,b,c $ that satisfy $ \left( a+b\sqrt{-3}\right)^{17}=c+\sqrt{-3} . $ [i]Dorin Andrica, Mihai Piticari[/i]

Prove that ${d((n^2 +1)}^2)$ does not become monotonic from any given point onwards.

Determine the least integer $n > 1$ such that the quadratic mean of the first $n$ positive integers is an integer. [i]Note: the quadratic mean of $a_1, a_2, \dots , a_n$ is defined to be $\sqrt{\frac{a_1^2+a_2^2+\cdots+a_n^2}{n}}$.[/i]

A frog jumps around on the integers on the number line. If it lands on an even number $n$, it jumps to the number $\frac{n}{2}$ . If it lands on an odd number $n$, it jumps to the number $n + 5$. At some point it lands on the number $25$. At which numbers may it have been three jumps ago?