## Problem 4:

Determine the functions $f, g:(0, \infty) \rightarrow \mathbb{R}$ with the property that:

$f\left(\frac{x}{3}\right)+2 \leq \log _{3} x \leq g(x)-1$ and $g\left(\frac{x}{3}\right) \leq \log _{3} x \leq f(x)+1$, for any $x \in(0, \infty)$.

Gazeta Matematică 4 / 2012

Note: All problems are mandatory.

Time allowed: 3 hours

Each problem is worth 7 points.

## Mathematics Olympiad Local Stage, Caraş-Severin, 16.02.2013

## 10th Grade  SCORING GUIDE
## Problem 4:

![](https://cdn.mathpix.com/cropped/2024_06_07_d2b47b3b67070d597e58g-2.jpg?height=66&width=1739&top_left_y=2097&top_left_x=181)

Observe that $f(x) \geq \log _{3} x-1$ and $g(x) \geq \log _{3} x+1, \forall x \in(0, \infty)$..............................................................3pts

Obtain that $f(x)=\log _{3} x-1$ and $g(x)=\log _{3} x+1$.............................................................................2pts