Školjka

%V0 Find all ordered pairs $(m,n)$ where $m$ and $n$ are positive integers such that $\frac {n^3 + 1}{mn - 1}$ is an integer.

%V0 The positive integers $a$ and $b$ are such that the numbers $15a + 16b$ and $16a - 15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares?

%V0 Determine all pairs $(x,y)$ of positive integers such that $x^{2}y+x+y$ is divisible by $xy^{2}+y+7$.

%V0 Find all the pairs of positive integers $(x,p)$ such that p is a prime, $x \leq 2p$ and $x^{p-1}$ is a divisor of $(p-1)^{x}+1$.

%V0 Let $n\geq2$ be a positive integer, with divisors $1=d_1<d_2<\,\ldots<d_k=n$. Prove that $d_1d_2+d_2d_3+\,\ldots\,+d_{k-1}d_k$ is always less than $n^2$, and determine when it is a divisor of $n^2$.

%V0 Let $n$ be a positive integer and let $a_1$, $a_2$, $a_3$, ..., $a_k$ ($k \geqslant 2$) be distinct integers in the set $\left\{1,\,2,\,\ldots,\,n\right\}$ such that $n$ divides $a_i \left(a_{i + 1} - 1\right)$ for $i = 1,\,2,\,\ldots,\,k - 1$. Prove that $n$ does not divide $a_k \left(a_1 - 1\right)$. Proposed by Ross Atkins, Australia