Let $\mathbb{N}$ be the set of positive integers. Determine all positive integers $k$ for which there exist functions $f:\mathbb{N} \to \mathbb{N}$ and $g: \mathbb{N}\to \mathbb{N}$ such that $g$ assumes infinitely many values and such that $$ f^{g(n)}(n)=f(n)+k$$holds for every positive integer $n$. (Remark. Here, $f^{i}$ denotes the function $f$ applied $i$ times i.e $f^{i}(j)=f(f(\dots f(j)\dots ))$.)

We call a positive integer $N$ contagious if there are $1000$ consecutive non-negative integers such that the sum of all their digits is $N$. Find all contagious positive integers.

Let $ABC$ be an acute scalene triangle with circumcircle $\omega$ and incenter $I$. Suppose the orthocenter $H$ of $BIC$ lies inside $\omega$. Let $M$ be the midpoint of the longer arc $BC$ of $\omega$. Let $N$ be the midpoint of the shorter arc $AM$ of $\omega$. Prove that there exists a circle tangent to $\omega$ at $N$ and tangent to the circumcircles of $BHI$ and $CHI$.

Find all positive integers $n$ for which there exist positive integers $x_1, x_2, \dots, x_n$ such that $$ \frac{1}{x_1^2}+\frac{2}{x_2^2}+\frac{2^2}{x_3^2}+\cdots +\frac{2^{n-1}}{x_n^2}=1.$$