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 $M$

Let $n \geq 2$ be a given integer $a)$ Prove that one can arrange all the subsets of the set $\{1,2... ,n\}$ as a sequence of subsets $A_{1}, A_{2},\cdots , A_{2^{n}}$, such that $|A_{i+1}| = |A_{i}| + 1$ or $|A_{i}| - 1$ where $i = 1,2,3,\cdots , 2^{n}$ and $A_{2^{n} + 1} = A_{1}$ $b)$ Determine all possible values of the sum $\sum \limits_{i = 1}^{2^n} (-1)^{i}S(A_{i})$ where $S(A_{i})$ denotes the sum of all elements in $A_{i}$ and $S(\emptyset) = 0$, for any subset sequence $A_{1},A_{2},\cdots ,A_{2^n}$ satisfying the condition in $a)$

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 \[\frac{(a-b)^2}{(c+a)(c+b)} + \frac{(b-c)^2}{(a+b)(a+c)} + \frac{(c-a)^2}{(b+c)(b+a)} \geq \frac{(a-b)^2}{a^2+b^2+c^2}\]

In triangle $ABC$ with $AB>AC$ and incenter $I$, the incircle touches $BC,CA,AB$ at $D,E,F$ respectively. $M$ is the midpoint of $BC$, and the altitude at $A$ meets $BC$ at $H$. Ray $AI$ meets lines $DE$ and $DF$ at $K$ and $L$, respectively. Prove that the points $M,L,H,K$ are concyclic.

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