As shown in the figure, given $\vartriangle ABC$ with $AB \perp AC$, $AB=BC$, $D$ is the midpoint of the side $AB$, $DF\perp DE$, $DE=DF$ and $BE \perp EC$. Prove that $\angle AFD= \angle CEF$.

(1) Find the smallest positive integer $a$ such that $221|3^a -2^a$, (2) Let $A=\{n\in N^*: 211|1+2^n+3^n+4^n\}$. Are there infinitely many numbers $n$ such that both $n$ and $n+1$ belong to set $A$?

Let $\{a_n\}$ be a sequence of positive terms such that $a_{n+1}=a_n+ \frac{n^2}{a_n}$ . Let $b_n =a_n-n$ . (1) Are there infinitely many $n$ such that $b_n \ge 0$ ? (2) Prove that there is a positive number $M$ such that $\sum^{\infty}_{n=3} \frac{b_n}{n+1}<M$.

$22$ mathematicians are meeting together. Each mathematician has at least $3$ friends (friends are mutual). And each mathematician can pass his or her information to any mathematician through the transfer between friends. Is it possible to divide these $22$ mathematicians into $2$-person groups (that is, two people in each group, a total of $11$ groups), so that the mathematicians in each group are friends? original wording in Chinese仃22位数学家一起开会.每位数学家都至少有3个朋友(朋友是相互的).而且每 位数学家都可以通过朋友之间的传递.将门已的资料传给任意一位数学家.问：是否一定可 以将这22位数学家两两分组(即每组两人,共11组),使得每组的数学家都是朋友？