VN-TST 2014

Find all $f:\mathbb{Z}\rightarrow\mathbb{Z}$ such that $$f(2m+f(m)+f(m)f(n))=nf(m)+m$$ $\forall m,n\in\mathbb{Z}$

In the Cartesian plane is given a set of points with integer coordinate $$T=\{ (x;y)\mid x,y\in\mathbb{Z} ; \ |x|,|y|\leq 20 ; \ (x;y)\ne (0;0)\}$$ We colour some points of $T$ such that for each point $(x;y)\in T$ then either $(x;y)$ or $(-x;-y)$ is coloured. Denote $N$ to be the number of couples ${(x_1;y_1),(x_2;y_2)}$ such that both $(x_1;y_1)$ and $(x_2;y_2)$ are coloured and $x_1\equiv 2x_2 \pmod {41}, y_1\equiv 2y_2 \pmod {41}$ . Find the all possible values of $N$ .

Find all polynomials $P(x),Q(x)$ which have integer coefficients and satify the following condtion: For the sequence $(x_n )$ defined by $$x_0=2014,x_{2n+1}=P(x_{2n}),x_{2n}=Q(x_{2n-1}) n\geq 1$$ for every positive integer $m$ is a divisor of some non-zero element of $(x_n )$

$m,n,p$ are positive integers which do not simultaneously equal to zero. $3$ D Cartesian space is divided into unit cubes by planes each perpendicular to one of $3$ axes and cutting corresponding axis at integer coordinates. Each unit cube is filled with an integer from $1$ to $60$ . A filling of integers is called Dien Bien if, for each rectangular box of size $\{2m+1,2n+1,2p+1\}$ , the number in the unit cube which has common centre with the rectangular box is the average of the $8$ numbers of the $8$ unit cubes at the $8$ corners of that rectangular box. How many Dien Bien fillings are there? Two fillings are the same if one filling can be transformed to the other filling via a translation.