CN-LEAGUE 2000

In acute-angled triangle $ABC,$ $E,F$ are on the side $BC,$ such that $\angle BAE=\angle CAF,$ and let $M,N$ be the projections of $F$ onto $AB,AC,$ respectively. The line $AE$ intersects $\odot (ABC)$ at $D$ (different from point $A$ ). Prove that $S_{AMDN}=S_{\triangle ABC}.$

Define the sequence $a_1, a_2, \ldots$ and $b_1, b_2, \ldots$ as $a_0=1,a_1=4,a_2=49$ and for $n \geq 0$ $$\begin{cases} a_{n+1}=7a_n+6b_n-3, b_{n+1}=8a_n+7b_n-4. \end{cases}$$ Prove that for any non-negative integer $n,$ $a_n$ is a perfect square.

There are $n$ people, and given that any $2$ of them have contacted with each other at most once. In any group of $n-2$ of them, any one person of the group has contacted with other people in this group for $3^k$ times, where $k$ is a non-negative integer. Determine all the possible value of $n.$