Let \(z_1\) and \(z_2\) be fixed points of a complex plane. Give a geometric description of the sets of all points \(z\) that satisfy the conditions:
a) \(\operatorname{arg} \frac{z - z_1}{z - z_2} = 0\);
b) \(\operatorname{arg} \frac{z_1 - z}{z - z_2} = 0\).
The numerical function \(f\) is such that for any \(x\) and \(y\) the equality \(f (x + y) = f (x) + f (y) + 80xy\) holds. Find \(f(1)\) if \(f(0.25) = 2\).
We consider a function \(y = f (x)\) defined on the whole set of real numbers and satisfying \(f (x + k) \times (1 - f (x)) = 1 + f (x)\) for some number \(k \ne 0\). Prove that \(f (x)\) is a periodic function.
The function \(f (x)\) for each real value of \(x\in (-\infty, + \infty)\) satisfies the equality \(f (x) + (x + 1/2) \times f (1 - x) = 1\).
a) Find \(f (0)\) and \(f (1)\). b) Find all such functions \(f (x)\).
The function \(F\) is given on the whole real axis, and for each \(x\) the equality holds: \(F (x + 1) F (x) + F (x + 1) + 1 = 0\).
Prove that the function \(F\) can not be continuous.