Let $ABC$ be an acute-angled triangle inscribed in circle $c(O,R)$ with $AB<AC<BC$, and $c_1$ be the inscribed circle of $ABC$ which intersects $AB, AC, BC$ at $F, E, D$ respectivelly. Let $A', B', C'$ be points which lie on $c$ such that the quadrilaterals $AEFA', BDFB', CDEC'$ are inscribable. (1) Prove that $DEA'B'$ is inscribable. (2) Prove that $DA', EB', FC'$ are concurrent.

Prove that the number $A=\frac{(4n)!}{(2n)!n!}$ is an integer and divisible by $2^{n+1}$, where $n$ is a positive integer.

Find all fuctions $f,g:\mathbb{R}\rightarrow \mathbb{R}$ such that: $f(x-3f(y))=xf(y)-yf(x)+g(x) \forall x,y\in\mathbb{R}$ and $g(1)=-8$

Some positive integers are initially written on a board, where each $2$ of them are different. Each time we can do the following moves: (1) If there are 2 numbers (written in the board) in the form $n, n+1$ we can erase them and write down $n-2$ (2) If there are 2 numbers (written in the board) in the form $n, n+4$ we can erase them and write down $n-1$ After some moves, there might appear negative numbers. Find the maximum value of the integer $c$ such that: Independetly of the starting numbers, each number which appears in any move is greater or equal to $c$