Solve the equation: \(|x-2005| + |2005-x|=2006\).
Does there exist a function \(f (x)\) defined for all \(x \in \mathbb{R}\) and for all \(x, y \in \mathbb{R}\) satisfying the inequality \(| f (x + y) + \sin x + \sin y | < 2\)?
The quadratic trinomials \(f (x)\) and \(g (x)\) are such that \(f' (x) g' (x) \geq | f (x) | + | g (x) |\) for all real \(x\). Prove that the product \(f (x) g (x)\) is equal to the square of some trinomial.
Two people play a game with the following rules: one of them guesses a set of integers \((x_1, x_2, \dots , x_n)\) which are single-valued digits and can be either positive or negative. The second person is allowed to ask what is the sum \(a_1x_1 + \dots + a_nx_n\), where \((a_1, \dots ,a_n)\) is any set. What is the smallest number of questions for which the guesser recognizes the intended set?
Solve the inequality: \(|x + 2000| <|x - 2001|\).