Found problems: 4
Find all functions $f : Z \to Z$ for which $f(g(n)) - g(f(n))$ is independent on $n$ for any $g : Z \to Z$.
A linear binomial $l(z) = Az + B$ with complex coefficients $A$ and $B$ is given. It is known that the maximal value of $|l(z)|$ on the segment $-1 \leq x \leq 1$ $(y = 0)$ of the real line in the complex plane $z = x + iy$ is equal to $M.$ Prove that for every $z$
\[|l(z)| \leq M \rho,\]
where $\rho$ is the sum of distances from the point $P=z$ to the points $Q_1: z = 1$ and $Q_3: z = -1.$
The function $f(x, y)$ is defined for all real numbers $x, y$. It satisfies $f(x,0) = ax$ (where $a$ is a non-zero constant) and if $(c, d)$ and $(h, k)$ are distinct points such that $f(c, d) = f(h, k)$, then $f(x, y)$ is constant on the line through $(c, d)$ and $(h, k)$. Show that for any real $b$, the set of points such that $f(x, y) = b$ is a straight line and that all such lines are parallel. Show that $f(x, y) = ax + by$, for some constant $b$.
A linear binomial $l(z) = Az + B$ with complex coefficients $A$ and $B$ is given. It is known that the maximal value of $|l(z)|$ on the segment $-1 \leq x \leq 1$ $(y = 0)$ of the real line in the complex plane $z = x + iy$ is equal to $M.$ Prove that for every $z$
\[|l(z)| \leq M \rho,\]
where $\rho$ is the sum of distances from the point $P=z$ to the points $Q_1: z = 1$ and $Q_3: z = -1.$