Let $ x$, $ y$, $ z$ be positive numbers. Find the minimum value of: $ (a)\quad \frac{x^2 + y^2 + z^2}{xy + yz}$ $ (b)\quad \frac{x^2 + y^2 + 2z^2}{xy + yz}$

For which $ n\in \mathbb{N}$ do there exist rational numbers $ a,b$ which are not integers such that both $ a + b$ and $ a^n + b^n$ are integers?

Point $ M$ is taken on side $ BC$ of a triangle $ ABC$ such that the centroid $ T_c$ of triangle $ ABM$ lies on the circumcircle of $ \triangle ACM$ and the centroid $ T_b$ of $ \triangle ACM$ lies on the circumcircle of $ \triangle ABM$. Prove that the medians of the triangles $ ABM$ and $ ACM$ from $ M$ are of the same length.

Let $ S$ be the set of all odd positive integers less than $ 30m$ which are not multiples of $ 5$, where $ m$ is a given positive integer. Find the smallest positive integer $ k$ such that each $ k$-element subset of $ S$ contains two distinct numbers, one of which divides the other.