Library / Ukrainian National Mathematical Olympiad /40 of 201

Ukrainian National Mathematical Olympiad

Algebra Difficulty 5.2 AIME, harder Prove it Ukraine

Determine if there exist a function $f : (0; 1) \rightarrow (2018; +\infty)$ , so that the following conditions hold:
* $f(x \cdot y) = f(x) \cdot f(y)$ for any $x, y \in (0; 1)$ ;
* for any $y \in (2018; +\infty)$ there exists $x \in (0; 1)$ such that $f(x) = y$ ?

Solution

Suppose such a function exists. Since for any $x \in (0; 1)$ equation $x = \sqrt{x} \cdot \sqrt{x}$ holds, then $f(x) = f(\sqrt{x} \cdot \sqrt{x}) = f(\sqrt{x}) \cdot f(\sqrt{x}) > 2018^2$ , that contradicts the fact that $E_f = (2018, +\infty)$ .

Want a route through all this instead of an archive? The track puts 2,604 problems in a working order, from Junior Challenge level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.