It is known that the general term $\{a_n\}$ of the sequence is $a_n =(\sqrt3 +\sqrt2)^{2n}$ ($n \in N*$), let $b_n= a_n +\frac{1}{a_n}$ . (1) Find the recurrence relation between $b_{n+2}$, $b_{n+1}$, $b_n$. (2) Find the unit digit of the integer part of $a_{2011}$.

As shown in figure , the inscribed circle of $ABC$ is intersects $BC$, $CA$, $AB$ at points $D$, $E$, $F$, repectively, and $P$ is a point inside the inscribed circle. The line segments $PA$, $PB$ and $PC$ intersect respectively the inscribed circle at points $X$, $Y$ and $Z$. Prove that the three lines $XD$, $YE$ and $ZF$ have a common point.

Find all positive integer solutions $(x, y, z)$ of the equation $1 + 2^x \cdot 7^y=z^2$.

Assume the $n$ sets $A_1, A_2..., A_n$ are a partition of the set $A=\{1,2,...,29\}$, and the sum of any elements in $A_i$ , $(i=1,2,...,n)$ is not equal to $30$. Find the smallest possible value of $n$.

If the positive integers $a, b, c$ satisfy $a^2+b^2=c^2$, then $(a, b, c)$ is called a Pythagorean triple. Find all Pythagorean triples containing $30$.

As shown in figure, from a point $P$ exterior of circle $\odot O$, we draw tangent $PA$ and the secant $PBC$. Let $AD \perp PO$ Prove that $AC$ is tangent to the circumcircle of $\vartriangle ABD$.

In $\triangle ABC$ , then \[\frac{1}{1+\cos^2 A+\cos^2 B}+\frac{1}{1+\cos^2 B+\cos^2 C}+\frac{1}{1+\cos^2 C+\cos^2 A}\le 2\]

It is known that $n$ is a positive integer, and the real number $x$ satisfies $$|1-|2-...|(n-1)-|n-x||...||=x.$$Find the value of $x$.