B4. Simplify the expression

$$
\left(1+\frac{1+\frac{1+a}{1-3 a}}{1-3 \cdot \frac{1+a}{1-3 a}}\right):\left(1-3 \cdot \frac{1+\frac{1+a}{1-3 a}}{1-3 \cdot \frac{1+a}{1-3 a}}\right)
$$

## PROBLEMS FOR THE SECOND 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 and writing it in the table below 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 attached paper. The solution to each of these problems will be evaluated on a scale from 0 to 6 points. Write only your code on the sheets where you will solve the problems. 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_8d4f73ada7862f989e67g-04.jpg?height=168&width=488&top_left_y=1001&top_left_x=801)

## PART I
B4. First, we simplify the double fraction, which is equal to $\frac{a-1}{3 a+1}$. Then we simplify the expression in the first parenthesis, which is equal to $\frac{4 a}{3 a+1}$. The expression in the second parenthesis is equal to $\frac{4}{3 a+1}$. We perform the division of fractions, simplify, and get the result $a$.

Simplification of the double fraction: $\frac{1+\frac{1+a}{1-3 a}}{1-3 \cdot \frac{1+a}{1-3 a}}=\frac{a-1}{3 a+1}$. .1 point

Determination that the double fraction in the second parenthesis is equal to: $\frac{a-1}{3 a+1} \ldots$ .1 point

Simplification of the expression in the first parenthesis: $1+\frac{a-1}{3 a+1}=\frac{4 a}{3 a+1} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . \ldots \ldots \ldots$ point

Simplification of the expression in the second parenthesis: $1-3 \cdot \frac{a-1}{3 a+1}=\frac{4}{3 a+1} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots$ point

![](https://cdn.mathpix.com/cropped/2024_06_07_8d4f73ada7862f989e67g-11.jpg?height=65&width=1641&top_left_y=2446&top_left_x=276)

![](https://cdn.mathpix.com/cropped/2024_06_07_8d4f73ada7862f989e67g-11.jpg?height=46&width=1639&top_left_y=2510&top_left_x=274)

## Second Year

## I. PART

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