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.

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 $n$.

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.

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?

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).

$9.3$ 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 $300$.

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 \[ p \mid n \quad \text{and} \quad n \mid p^2+p\cdot m^p. \]