It is given the sequence defined by $$\{a_{n+2}=6a_{n+1}-a_n\}_{n \in \mathbb{Z}_{>0}},a_1=1, a_2=7 \text{.}$$Find all $n$ such that there exists an integer $m$ for which $a_n=2m^2-1$.
2016 Olympic Revenge
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.
Let $T$ the set of the infinite sequences of integers. For two given elements in $T$: $(a_{1},a_{2},a_{3},...)$ and $(b_{1},b_{2},b_{3},...)$, define the sum $(a_{1},a_{2},a_{3},...)+(b_{1},b_{2},b_{3},...)=(a_{1}+b_{1},a_{2}+b_{2},a_{3}+b_{3},...)$. Let $f: T\rightarrow$ $\mathbb{Z}$ a function such that: i) If $x\in T$ has exactly one of your terms equal $1$ and all the others equal $0$, then $f(x)=0$. ii)$f(x+y)=f(x)+f(y)$, for all $x,y\in T$. Prove that $f(x)=0$ for all $x\in T$