A sequence of real numbers $ \{a_{n}\}$ is defineds by $ a_{0}\neq 0,1$, $ a_1=1-a_0$,$ a_{n+1}=1-a_n(1-a_n)$, $ n=1,2,...$. Prove that for any positive integer $ n$, we have $ a_{0}a_{1}...a_{n}(\frac{1}{a_0}+\frac{1}{a_1}+...+\frac{1}{a_n})=1$

In triangle $ ABC$, $ AB=AC$, the inscribed circle $ I$ touches $ BC, CA, AB$ at points $ D,E$ and $ F$ respectively. $ P$ is a point on arc $ EF$ opposite $ D$. Line $ BP$ intersects circle $ I$ at another point $ Q$, lines $ EP$, $ EQ$ meet line $ BC$ at $ M, N$ respectively. Prove that (1) $ P, F, B, M$ concyclic (2)$ \frac{EM}{EN} = \frac{BD}{BP}$ (P.S. Can anyone help me with using GeoGebra, the incircle function of the plugin doesn't work with my computer.)

Given an integer $ m\geq$ 2, m positive integers $ a_1,a_2,...a_m$. Prove that there exist infinitely many positive integers n, such that $ a_{1}1^{n} + a_{2}2^{n} + ... + a_{m}m^{n}$ is composite.

Given an integer $ m\geq 2$, and two real numbers $ a,b$ with $ a > 0$ and $ b\neq 0$. The sequence $ \{x_n\}$ is such that $ x_1 = b$ and $ x_{n + 1} = ax^{m}_{n} + b$, $ n = 1,2,...$. Prove that (1)when $ b < 0$ and m is even, the sequence is bounded if and only if $ ab^{m - 1}\geq - 2$; (2)when $ b < 0$ and m is odd, or when $ b > 0$ the sequence is bounded if and only if $ ab^{m - 1}\geq\frac {(m - 1)^{m - 1}}{m^m}$.

Four frogs are positioned at four points on a straight line such that the distance between any two neighbouring points is 1 unit length. Suppose the every frog can jump to its corresponding point of reflection, by taking any one of the other 3 frogs as the reference point. Prove that, there is no such case that the distance between any two neighbouring points, where the frogs stay, are all equal to 2008 unit length.

Given $ x,y,z\in (0,1)$ satisfying that $ \sqrt{\frac{1 - x}{yz}} + \sqrt{\frac{1 - y}{xz}} + \sqrt{\frac{1 - z}{xy}} = 2$. Find the maximum value of $ xyz$.

For a given positive integer $n$, find the greatest positive integer $k$, such that there exist three sets of $k$ non-negative distinct integers, $A=\{x_1,x_2,\cdots,x_k\}, B=\{y_1,y_2,\cdots,y_k\}$ and $C=\{z_1,z_2,\cdots,z_k\}$ with $ x_j+y_j+z_j=n$ for any $ 1\leq j\leq k$. [Moderator edit: LaTeXified]

Let P be an interior point of a regular n-gon $ A_1 A_2 ...A_n$, the lines $ A_i P$ meet the regular n-gon at another point $ B_i$, where $ i=1,2,...,n$. Prove that sums of all $ PA_i\geq$ sum of all $ PB_i$.