Solve the system of equations $\begin{cases}6x-y+z^2=3\\ x^2-y^2-2z=-1\quad\quad (x,y,z\in\mathbb{R}.)\\ 6x^2-3y^2-y-2z^2=0\end{cases}$.

a) Let $(a_n)$ be the sequence defined by $a_n=\ln (2n^2+1)-\ln (n^2+n+1)\,\,\forall n\geq 1.$ Prove that the set $\{n\in\mathbb{N}|\,\{a_n\}<\dfrac{1}{2}\}$ is a finite set; b) Let $(b_n)$ be the sequence defined by $a_n=\ln (2n^2+1)+\ln (n^2+n+1)\,\,\forall n\geq 1$. Prove that the set $\{n\in\mathbb{N}|\,\{b_n\}<\dfrac{1}{2016}\}$ is an infinite set.

Let $ABC$ be an acute triange with $B,C$ fixed. Let $D$ be the midpoint of $BC$ and $E,F$ be the foot of the perpendiculars from $D$ to $AB,AC$, respectively. a) Let $O$ be the circumcenter of triangle $ABC$ and $M=EF\cap AO, N=EF\cap BC$. Prove that the circumcircle of triangle $AMN$ passes through a fixed point; b) Assume that tangents of the circumcircle of triangle $AEF$ at $E,F$ are intersecting at $T$. Prove that $T$ is on a fixed line.

Let $m$ and $n$ be positive integers. A people planted two kind of different trees on a plot tabular grid size $ m \times n $ (each square plant one tree.) A plant called inpressive if two conditions following conditions are met simultaneously: i) The number of trees in each of kind is equal; ii) In each row the number of tree of each kind is diffrenent not less than a half of number of cells on that row and In each colum the number of tree of each kind is diffrenent not less than a half of number of cells on that colum. a) Find an inpressive plant when $m=n=2016$; b) Prove that if there at least a inpressive plant then $4|m$ and $4|n$.

Find all $a\in\mathbb{R}$ such that there is function $f:\mathbb{R}\to\mathbb{R}$ i) $f(1)=2016$ ii) $f(x+y+f(y))=f(x)+ay\quad\forall x,y\in\mathbb{R}$

Given a triangle $ABC$ inscribed by circumcircle $(O)$. The angles at $B,C$ are acute angle. Let $M$ on the arc $BC$ that doesn't contain $A$ such that $AM$ is not perpendicular to $BC$. $AM$ meets the perpendicular bisector of $BC$ at $T$. The circumcircle $(AOT)$ meets $(O)$ at $N$ ($N\ne A$). a) Prove that $\angle{BAM}=\angle{CAN}$. b) Let $I$ be the incenter and $G$ be the foor of the angle bisector of $\angle{BAC}$. $AI,MI,NI$ intersect $(O)$ at $D,E,F$ respectively. Let ${P}=DF\cap AM, {Q}=DE\cap AN$. The circle passes through $P$ and touches $AD$ at $I$ meets $DF$ at $H$ ($H\ne D$).The circle passes through $Q$ and touches $AD$ at $I$ meets $DE$ at $K$ ($K\ne D$). Prove that the circumcircle $(GHK)$ touches $BC$.

a) Prove that if $n$ is an odd perfect number then $n$ has the following form \[ n=p^sm^2 \]where $p$ is prime has form $4k+1$, $s$ is positive integers has form $4h+1$, and $m\in\mathbb{Z}^+$, $m$ is not divisible by $p$. b) Find all $n\in\mathbb{Z}^+$, $n>1$ such that $n-1$ and $\frac{n(n+1)}{2}$ is perfect number