Školjka

%V0 Determine all pairs $(x,y)$ of positive integers satisfying the equation $$x!+y!=x^{y}\text{.}$$

%V0 A positive integer $N$ is called balanced, if $N=1$ or if $N$ can be written as a product of an even number of not necessarily distinct primes. Given positive integers $a$ and $b$, consider the polynomial $P$ defined by $P\!\left(x\right) = \left(x+a\right)\left(x+b\right)$. a) Prove that there exist distinct positive integers $a$ and $b$ such that all the number $P\!\left(1\right)$, $P\!\left(2\right)$, ..., $P\!\left(50\right)$ are balanced. b) Prove that if $P\!\left(n\right)$ is balanced for all positive integers $n$, then $a=b$. Proposed by Jorge Tipe, Peru

%V0 For every integer $k \geq 2,$ prove that $2^{3k}$ divides the number $$\binom{2^{k + 1}}{2^{k}} - \binom{2^{k}}{2^{k - 1}}$$ but $2^{3k + 1}$ does not. Author: unknown author, Poland

%V0 Find all pairs of natural numbers $(a, b)$ such that $7^a - 3^b$ divides $a^4 + b^2$. Author: Stephan Wagner, Austria

%V0 Let $a,b,n$ be positive integers, $b > 1$ and $b^n-1|a.$ Show that the representation of the number $a$ in the base $b$ contains at least $n$ digits different from zero.

%V0 A natural number $n$ is said to have the property $P,$ if, for all $a, n^2$ divides $a^n - 1$ whenever $n$ divides $a^n - 1.$ a.) Show that every prime number $n$ has property $P.$ b.) Show that there are infinitely many composite numbers $n$ that possess property $P.$