OLYMPIC-REVENGE 2016

Let $S$ a finite subset of $\mathbb{N}$ . For every positive integer $i$ , let $A_{i}$ the number of partitions of $i$ with all parts in $\mathbb{N}-S$ . Prove that there exists $M\in \mathbb{N}$ such that $A_{i+1}>A_{i}$ for all $i>M$ . ( $\mathbb{N}$ is the set of positive integers)

Let $\Gamma$ a fixed circunference. Find all finite sets $S$ of points in $\Gamma$ such that: For each point $P\in \Gamma$ , there exists a partition of $S$ in sets $A$ and $B$ ( $A\cup B=S$ , $A\cap B=\phi$ ) such that $\sum_{X\in A}PX = \sum_{Y\in B}PY$ .

Let $\Omega$ and $\Gamma$ two circumferences such that $\Omega$ is in interior of $\Gamma$ . Let $P$ a point on $\Gamma$ . Define points $A$ and $B$ distinct of $P$ on $\Gamma$ such that $PA$ and $PB$ are tangentes to $\Omega$ . Prove that when $P$ varies on $\Gamma$ , the line $AB$ is tangent to a fixed circunference.