Prove that there exist infinitely many integers $n$ such that $n^2+1$ is squarefree.

Prove that for every square-free integer $n>1$, there exists a prime number $p$ and an integer $m$ satisfying \[ p \mid n \quad \text{and} \quad n \mid p^2+p\cdot m^p. \]

Determine all squarefree positive integers $n\geq 2$ such that \[\frac{1}{d_1}+\frac{1}{d_2}+\cdots+\frac{1}{d_k}\]is a positive integer, where $d_1,d_2,\ldots,d_k$ are all the positive divisors of $n$.

Prove that there exist infinitely many integers $n$ such that $n^2+1$ is squarefree.

a) Prove that $\lfloor x\rfloor$ is odd iff $\Big\lfloor 2\{\frac{x}{2}\}\Big\rfloor=1$ ($\lfloor x\rfloor$ denotes the largest integer less than or equal to $x$ and $\{x\}=x-\lfloor x\rfloor$). b) Let $n$ be a natural number. Find the number of [i]square free[/i] numbers $a$, such that $\Big\lfloor\frac{n}{\sqrt{a}}\Big\rfloor$ is odd. (A natural number is [i]square free[/i] if it's not divisible by any square of a prime number).

Let $P (n)$ and $Q (n)$ be two polynomials (not constant) whose coefficients are integers not negative. For each positive integer $n$, define $x_n = 2016^{P (n)} + Q (n)$. Prove that there exist infinite primes $p$ for which there is a positive integer $m$, squarefree, such that $p | x_m$. Clarification: A positive integer is squarefree if it is not divisible by the square of any prime number.

Prove that for every square-free integer $n>1$, there exists a prime number $p$ and an integer $m$ satisfying \[ p \mid n \quad \text{and} \quad n \mid p^2+p\cdot m^p. \]

$9.3$ A natural number is called square-free, if it is not divisible by the square of any prime number. For a natural number $a$, we consider the number $f(a) = a^{a+1} + 1$. Prove that: a) if $a$ is even, then $f(a)$ is not square-free b) there exist infinitely many odd $a$ for which $f(a)$ is not square-free $9.4$ We will call a generalized $2n$-parallelogram a convex polygon with $2n$ sides, so that, traversed consecutively, the $k$th side is parallel and equal to the $(n+k)$th side for $k=1, 2, ... , n$. In a rectangular coordinate system, a generalized parallelogram is given with $50$ vertices, each with integer coordinates. Prove that its area is at least $300$.

Consider the equation $3y^4 + 4cy^3 + 2xy + 48 = 0$, where $c$ is an integer parameter. Determine all values of $c$ for which the number of integral solutions $(x,y)$ satisfying the conditions (i) and (ii) is maximal: (i) $|x|$ is a square of an integer; (ii) $y$ is a squarefree number.

Vukasin, Dimitrije, Dusan, Stefan and Filip asked their teacher to guess three consecutive positive integers, after these true statements: Vukasin: " The sum of the digits of one number is prime number. The sum of the digits of another of the other two is, an even perfect number.($n$ is perfect if $\sigma\left(n\right)=2n$). The sum of the digits of the third number equals to the number of it's positive divisors". Dimitrije:"Everyone of those three numbers has at most two digits equal to $1$ in their decimal representation". Dusan:"If we add $11$ to exactly one of them, then we have a perfect square of an integer" Stefan:"Everyone of them has exactly one prime divisor less than $10$". Filip:"The three numbers are square free". Professor found the right answer. Which numbers did he mention?