## SUBJECT I

a) Solve the equation $x^{\log _{2} 9}=x^{2} \cdot 3^{\log _{2} x}-x^{\log _{2} 3}$

b) If $x \in[2, \infty)$, calculate $\left[\log _{[x]} x\right]$, where $[a]$ represents the integer part of $a$.
Solution:

![](https://cdn.mathpix.com/cropped/2024_06_07_65693c76e450f9300622g-1.jpg?height=66&width=1656&top_left_y=1235&top_left_x=223)

![](https://cdn.mathpix.com/cropped/2024_06_07_65693c76e450f9300622g-1.jpg?height=63&width=1590&top_left_y=1325&top_left_x=290)

![](https://cdn.mathpix.com/cropped/2024_06_07_65693c76e450f9300622g-1.jpg?height=52&width=1582&top_left_y=1393&top_left_x=297)

Unique solution \( x=2 \) .............................................................................1p

b) Let \([x]=k, k \in \mathbb{N}^{*} \backslash\{1\}\)

![](https://cdn.mathpix.com/cropped/2024_06_07_65693c76e450f9300622g-1.jpg?height=65&width=1730&top_left_y=1598&top_left_x=163)

![](https://cdn.mathpix.com/cropped/2024_06_07_65693c76e450f9300622g-1.jpg?height=66&width=1722&top_left_y=1663&top_left_x=173)

![](https://cdn.mathpix.com/cropped/2024_06_07_65693c76e450f9300622g-1.jpg?height=77&width=1730&top_left_y=1732&top_left_x=163)

![](https://cdn.mathpix.com/cropped/2024_06_07_65693c76e450f9300622g-2.jpg?height=300&width=797&top_left_y=113&top_left_x=127)

## MATHEMATICS OLYMPIAD  LOCAL STAGE - 21.02.2016  GRADING SCALE  GRADE 10