B4. Describe the set of all points $(x, y)$ that satisfy the condition

$$
7^{x+y}=7^{x+2} \cdot 7^{2 x+1}
$$

Represent the set in a coordinate system.

## 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 awarded 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 attached 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 allowed for solving is 90 minutes.

THE NATIONAL COMPETITION COMMITTEE WISHES YOU GREAT SUCCESS.

![](https://cdn.mathpix.com/cropped/2024_06_07_8d4f73ada7862f989e67g-08.jpg?height=171&width=488&top_left_y=997&top_left_x=801)

## PART I
B4. We rearrange the exponential equation $7^{x+y}=7^{3 x+3}$. We equate the exponents and rearrange. We get $y=2 x+3$, which represents the equation of a line.

Rearrangement of the equation: $7^{x+y}=7^{3 x+3} \ldots \ldots \ldots \ldots \ldots \ldots \ldots . \ldots$ point

Writing the equality of exponents $x+y=3 x+3 \ldots \ldots \ldots .1$ point

Rearrangement of the expression $y=2 x+3 \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots$ point

Conclusion that this is the equation of a line ..................... 1 point

Drawing the line in the coordinate system............. 2 points

![](https://cdn.mathpix.com/cropped/2024_06_07_8d4f73ada7862f989e67g-15.jpg?height=514&width=528&top_left_y=220&top_left_x=1392)

## Fourth Year

I. PART

| Task | 1 | 2 | 3 | 4 | 5 | 6 |
| :--- | :---: | :---: | :---: | :---: | :---: | :---: |
| Answer | $\mathrm{D}$ | $\mathrm{B}$ | $\mathrm{C}$ | $\mathrm{D}$ | $\mathrm{B}$ | $\mathrm{A}$ |