B4. Solve the equation: $\log \left(4+2^{x+2}\right)=\log 4+\log \left(5 \cdot 2^{4-x}-1\right)$.

## PROBLEMS FOR THE FOURTH YEAR

Before you are two sets of problems. Problems 1 to 6 of the first set are solved by selecting the correct answer from the five proposed answers on this sheet and writing it in the table under the corresponding number. Only one answer is correct. A correct answer will be scored with two points, while a wrong answer will result in the deduction of one point.

Problems 1 to 4 of the second set are solved on the provided paper. The solution to each of these problems will be evaluated on a scale of 0 to 6 points. Do not sign the sheets where you will solve the problems, write only your code. Write your solution clearly and legibly with a pen.

The time for solving is 90 minutes.

THE NATIONAL COMPETITION COMMITTEE WISHES YOU GREAT SUCCESS.

![](https://cdn.mathpix.com/cropped/2024_06_07_4401d59a37898c9fd2b4g-08.jpg?height=174&width=488&top_left_y=1004&top_left_x=801)

## PART I
B4. By antilogarithmizing we get $4+2^{x+2}=4 \cdot\left(5 \cdot 2^{4-x}-1\right)$. When we remove the parentheses, we get $4+2^{x+2}=20 \cdot 2^{4-x}-4$ or $4+2^{x} \cdot 2^{2}=\frac{20 \cdot 2^{4}}{2^{x}}-4$, which we rearrange to $2^{x}+\left(2^{x}\right)^{2}=5 \cdot 2^{4}-2^{x}$. We introduce a new variable $2^{x}=t$. We get the equation $t^{2}+2 t-80=0$, which we factorize to $(t+10)(t-8)=0$. The solution $t=-10$ is not valid, as a power with a positive base cannot have a negative value. We substitute the solution $t=8$ into $2^{x}=t$ and get $x=3$. We verify the result.

![](https://cdn.mathpix.com/cropped/2024_06_07_4401d59a37898c9fd2b4g-14.jpg?height=62&width=1639&top_left_y=1985&top_left_x=274)

![](https://cdn.mathpix.com/cropped/2024_06_07_4401d59a37898c9fd2b4g-14.jpg?height=63&width=1639&top_left_y=2033&top_left_x=274)

![](https://cdn.mathpix.com/cropped/2024_06_07_4401d59a37898c9fd2b4g-14.jpg?height=60&width=1639&top_left_y=2083&top_left_x=274)

Solutions: $t_{1}=8$ and $t_{2}=-10$ (the latter is not valid).....................................................................................

![](https://cdn.mathpix.com/cropped/2024_06_07_4401d59a37898c9fd2b4g-14.jpg?height=48&width=1642&top_left_y=2189&top_left_x=273)

Verification completed.................................................................................................

## Fourth Year

## I. PART

| Task | A1 | A2 | A3 | A4 | A5 | A6 |
| :--- | ---: | ---: | ---: | ---: | ---: | ---: |
| Answer | C | E | D | B | A | E |