What is the smallest value that the sum of the digits of the number $3n^2+n+1,$ $n\in\mathbb{N}$ can take?

Let $n\geq 3$ be a positive integer. Consider an $n\times n$ square. In each cell of the square, one of the numbers from the set $M=\{1,2,\ldots,2n-1\}$ is to be written. One such filling is called good if, for every index $1\leq i\leq n,$ row no. $i$ and column no. $i,$ together, contain all the elements of $M$. Prove that there exists $n\geq 3$ for which a good filling exists. Prove that for $n=2017$ there is no good filling of the $n\times n$ square.

Consider an acute triangle $ABC$ in which $A_1, B_1,$ and $C_1$ are the feet of the altitudes from $A, B,$ and $C,$ respectively, and $H$ is the orthocenter. The perpendiculars from $H$ onto $A_1C_1$ and $A_1B_1$ intersect lines $AB$ and $AC$ at $P$ and $Q,$ respectively. Prove that the line perpendicular to $B_1C_1$ that passes through $A$ also contains the midpoint of the line segment $PQ$.

Determine all triples of positive integers $(x,y,z)$ such that $x^4+y^4 =2z^2$ and $x$ and $y$ are relatively prime.

Find all polynomials $P(x)$ with integer coefficients such that $a^2+b^2-c^2$ divides $P(a)+P(b)-P(c)$, for all integers $a,b,c$.

Let n be a positive interger. Let n real numbers be wrote on a paper. We call a "transformation" :choosing 2 numbers $a,b$ and replace both of them with $a*b$. Find all n for which after a finite number of transformations and any n real numbers, we can have the same number written n times on the paper.

Let $O,H$ be the circumcenter and the orthocenter of triangle $ABC$. Let $F$ be the foot of the perpendicular from C onto AB, and $M$ the midpoint of $CH$. Let N be the foot of the perpendicular from C onto the parallel through H at $OM$. Let $D$ be on $AB$ such that $CA=CD$. Let $BN$ intersect $CD$ at $P$. Let $PH$ intersect $CA$ at $Q$. Prove that $QF\perp OF$.

Let us have an infinite grid of unit squares. We write in every unit square a real number, such that the absolute value of the sum of the numbers from any $n*n$ square is less or equal than $1$. Prove that the absolute value of the sum of the numbers from any $m*n$ rectangular is less or equal than $4$.