Task 2.

Maximum 15 points

Solve the inequality:

$$
\sqrt{2 x+8}+\sqrt{10-2 x} \geq \log _{2}\left(4 x^{2}-4 x+65\right)
$$
# Solution:

It is not difficult to understand that

$$
\log _{2}\left(4 x^{2}-4 x+65\right)=\log _{2}\left((2 x-1)^{2}+64\right) \geq \log _{2} 64=6
$$

Let's estimate the expression $\sqrt{2 x+8}+\sqrt{10-2 x}$. Consider the vectors $\overrightarrow{\boldsymbol{a}}(1,1)$ and $\vec{b}(\sqrt{2 x+8}, \sqrt{10-2 x})$. We have

$$
\begin{gathered}
|\vec{a} \cdot \vec{b}| \leq|\vec{a}| \cdot|\vec{b}| \\
\Leftrightarrow \sqrt{2 x+8}+\sqrt{10-2 x} \leq \sqrt{1+1} \cdot \sqrt{2 x+8+10-2 x}=6
\end{gathered}
$$

Therefore, the inequality is equivalent to the system

$$
\left\{\begin{array} { l } 
{ \operatorname { l o g } _ { 2 } ( 4 x ^ { 2 } - 4 x + 6 5 ) = 6 } \\
{ \sqrt { 2 x + 8 } + \sqrt { 1 0 - 2 x } = 6 }
\end{array} \Leftrightarrow \left\{\begin{array}{c}
2 x-1=0 \\
\sqrt{2 x+8}+\sqrt{10-2 x}=6
\end{array}\right.\right.
$$

Answer: $\frac{1}{2}$.

## Evaluation Criteria:

- 15 points - the correct answer is obtained with proper justification.
- 10 points - the solution contains a correct sequence of steps leading to the correct answer, but one arithmetic error is made during the solution.
- 5 points - the solution contains generally correct steps, the correct answer is obtained, but the solution contains inaccuracies and/or is not fully justified.
- 0 points - the solution does not meet any of the criteria provided.