CWMO 2011

Given that $0 < x,y < 1$ , find the maximum value of $\frac{xy(1-x-y)}{(x+y)(1-x)(1-y)}$

Let $M$ be a subset of $\{1,2,3... 2011\}$ satisfying the following condition: For any three elements in $M$ , there exist two of them $a$ and $b$ such that $a|b$ or $b|a$ . Determine the maximum value of $|M|$ where $|M|$ denotes the number of elements in $M$

In a circle $\Gamma_{1}$ , centered at $O$ , $AB$ and $CD$ are two unequal in length chords intersecting at $E$ inside $\Gamma_{1}$ . A circle $\Gamma_{2}$ , centered at $I$ is tangent to $\Gamma_{1}$ internally at $F$ , and also tangent to $AB$ at $G$ and $CD$ at $H$ . A line $l$ through $O$ intersects $AB$ and $CD$ at $P$ and $Q$ respectively such that $EP = EQ$ . The line $EF$ intersects $l$ at $M$ . Prove that the line through $M$ parallel to $AB$ is tangent to $\Gamma_{1}$

Does there exist any odd integer $n \geq 3$ and $n$ distinct prime numbers $p_1 , p_2, \cdots p_n$ such that all $p_i + p_{i+1} (i = 1,2,\cdots , n$ and $p_{n+1} = p_{1})$ are perfect squares?

Let $a,b,c > 0$ , prove that $$\frac{(a-b)^2}{(c+a)(c+b)} + \frac{(b-c)^2}{(a+b)(a+c)} + \frac{(c-a)^2}{(b+c)(b+a)} \geq \frac{(a-b)^2}{a^2+b^2+c^2}$$

Find all pairs of integers $(a,b)$ such that $n|( a^n + b^{n+1})$ for all positive integer $n$