A sequence starts at some rational number $x_1>1$, and is subsequently defined using the recurrence relation \[x_{n+1}=\frac{x_n\cdot n}{\lfloor x_n\cdot n\rfloor }\]Show that $k>0$ exists with $x_k=1$.

Triangle $\triangle ABC$ is inscribed in circle $\Omega$. Let $I$ denote its incenter and $I_A$ its $A$-excenter. Let $N$ denote the midpoint of arc $BAC$. Line $NI_A$ meets $\Omega$ a second time at $T$. The perpendicular to $AI$ at $I$ meets sides $AC$ and $AB$ at $E$ and $F$ respectively. The circumcircle of $\triangle BFT$ meets $BI_A$ a second time at $P$, and the circumcircle of $\triangle CET$ meets $CI_A$ a second time at $Q$. Prove that $PQ$ passes through the antipodal to $A$ on $\Omega$.

For a positive integer \( n \geq 2 \), does there exist positive integer solutions to the following system of equations? \[ \begin{cases} a^n - 2b^n = 1, \\ b^n - 2c^n = 1. \end{cases} \]

A graph with $10^{100}$ vertices satisfies the following condition: Any simple odd cycle has length > 100. Prove there is an independent set in the graph of size at least $\frac{10^{100}}{102}$

Let $ABCD$ be a cyclic quadrilateral with circumcenter $O$. The internal angle bisectors of \(\angle DAB\), \(\angle ABC\), \(\angle BCD\), \(\angle CDA\) create a convex quadrilateral $Q_1$. The external bisectors of the same angles create another convex quadrilateral $Q_2$. Prove $Q_1$, $Q_2$ are cyclic, and that $O$ is the midpoint of their circumcenters.