As shown in figure, given right $\vartriangle ABC$ with $\angle C=90^o$. $I$ is the incenter. The line $BI$ intersects segment $AC$ at the point $D$ . The line passing through $D$ parallel to $AI$ intersects $BC$ at point $E$. The line $EI$ intersects segment $AB$ at point $F$. Prove that $DF \perp AI$.

Positive integers $x_1,x_2,...,x_n$ ($n \in N_+$) satisfy $x_1^2 +x_2^2+...+x_n^2=111$, find the maximum possible value of $S =\frac{x_1 +x_2+...+x_n}{n}$.

Suppose $S= \{x|x=a^2+ab+b^2,a,b \in Z\}$. Prove that: (1) If $m \in S$, $3|m$ , then $\frac{m}{3} \in S$ (2) If $m,n \in S$ , then $mn\in S$.

There are $n$ ($n \ge 4$) straight lines on the plane. For two straight lines $a$ and $b$, if there are at least two straight lines among the remaining $n-2$ lines that intersect both straight lines $a$ and $b$, then $a$ and $b$ are called a congruent pair of staight lines, otherwise it is called a separated pair of straight lines. If the number of congruent pairs of straight line among $n$ straight lines is $2012$ more than the number of separated pairs of straight line , find the smallest possible value of $n$ (the order of the two straight lines in a pair is not counted).

Let $\{a_n\}$ be the sequance with $a_0=0$, $a_n=\frac{1}{a_{n-1}-2}$ ($n\in N_+$). Select an arbitrary term $a_k$ in the sequence $\{a_n\}$ and construct the sequence $\{b_n\}$: $b_0=a_k$, $b_n=\frac{2b_{n-1}+1} {b_{n-1}}$ ($n\in N_+$) . Determine whether the sequence $\{b_n\}$ is a finite sequence or an infinite sequence and give proof.

Prove that\[(1+\frac{1}{3})(1+\frac{1}{3^2})\cdots(1+\frac{1}{3^n})< 2.\]

As shown in figure , in the pentagon $ABCDE$, $BC = DE$, $CD \parallel BE$, $AB>AE$. If $\angle BAC = \angle DAE$ and $\frac{AB}{BD}=\frac{AE}{ED}$. Prove that $AC$ bisects the line segment $BE$.

Assume $p$ is a prime number. If there is a positive integer $a$ such that $p!|(a^p + 1)$, prove that : (1) $(a+1, \frac{a^p+1}{a+1}) = p$ (2) $\frac{a^p+1}{a+1}$ has no prime factors less than $p$. (3) $p!|(a +1) $.