## SUBJECT IV

Determine the functions $\quad f, g:(0, \infty) \rightarrow R$ 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 $x \in(0, \infty)$.

NOTE:

All problems are mandatory.

Each problem is worth a maximum of 7 points.

Time allowed: 3 hours

## SOCIETY OF MATHEMATICAL SCIENCES - SĂLAJ BRANCH

## NATIONAL OLYMPIAD $\breve{\text { OF MATHEMATICS }}$

local stage, February 9, 2013

## 10th grade
## SUBJECT IV

Determine the functions $\quad f, g:(0, \infty) \rightarrow R$ 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 $x \in(0, \infty)$.

E-mail: isjsalaj@isj.sj.edu.ro

Grading Rubric

By substituting $x$ with $3x$ in the inequalities $\qquad$

$$
f\left(\frac{x}{3}\right)+2 \leq \log _{3} x \quad \text { and } \quad g\left(\frac{x}{3}\right) \leq \log _{3} x
$$

we obtain:

$f(x) \leq \log _{3} x-1 \quad$ and $\quad g(x) \leq \log _{3} x+1, \forall x \in(0, \infty)$.

2 points

Since $\quad f(x) \geq \log _{3} x-1$

and $f(x) \leq \log _{3} x-1 \Rightarrow f(x)=\log _{3} x-1$

2 points

respectively:

from the hypothesis $1+\log _{3} x \leq g(x)$

$$
\text { and } g(x) \leq \log _{3} x+1, \forall x \in(0, \infty) \Rightarrow g(x)=\log _{3} x+1
$$

2 points