Školjka

%V0 Find all $x,y$ and $z$ in positive integer: $z + y^{2} + x^{3} = xyz$ and $x = \gcd(y,z)$.

%V0 Does there exist an integer $n > 1$ which satisfies the following condition? The set of positive integers can be partitioned into $n$ nonempty subsets, such that an arbitrary sum of $n - 1$ integers, one taken from each of any $n - 1$ of the subsets, lies in the remaining subset.

%V0 Let $p$ be an odd prime. Determine positive integers $x$ and $y$ for which $x \leq y$ and $\sqrt{2p} - \sqrt{x} - \sqrt{y}$ is non-negative and as small as possible.

%V0 Prove that the equation $\frac {x^{7} - 1}{x - 1} = y^{5} - 1$ doesn't have integer solutions!

%V0 Let $a > b > 1$ be relatively prime positive integers. Define the weight of an integer $c$, denoted by $w(c)$ to be the minimal possible value of $|x| + |y|$ taken over all pairs of integers $x$ and $y$ such that $ax + by = c$. An integer $c$ is called a local champion if $w(c) \geq w(c \pm a)$ and $w(c) \geq w(c \pm b)$. Find all local champions and determine their number.

%V0 Let $k$ be a positive integer. Show that if there exists a sequence $a_0$, $a_1$, ... of integers satisfying the condition $$a_n=\frac{a_{n-1}+n^k}{n} \text{,} \qquad \forall n \geqslant 1 \text{,}$$ then $k-2$ is divisible by $3$. Proposed by Turkey