B3. Four times the quotient of the difference and the sum of a real number $a$ and its reciprocal, decreased by twice the difference between the number $a$ and its reciprocal. Write the desired expression, simplify it, and calculate for which values of the number $a$ the value of the given expression is 0.

## 21st Mathematics Knowledge Competition for Students of Secondary Technical and Vocational Schools National Competition, May 15, 2021

## Problems for 2nd Year

Time for solving: 120 minutes. In section A, we will award three points for each correct answer, and deduct one point for each incorrect answer. Enter your answers for section A in the left table, leave the right table blank.
![](https://cdn.mathpix.com/cropped/2024_06_07_5bf9d67aee417dc7ba6ag-06.jpg?height=168&width=802&top_left_y=590&top_left_x=640)
B3. Write the expression 4. $\frac{a-a^{-1}}{a+a^{-1}}-2 \cdot\left(a-a^{-1}\right)$. Simplify the expression $4 \cdot \frac{a-\frac{1}{a}}{a+\frac{1}{a}}-2 \cdot\left(a-\frac{1}{a}\right)=4 \cdot \frac{a^{2}-1}{a^{2}+1}-2 \cdot \frac{a^{2}-1}{a}=\frac{4 a\left(a^{2}-1\right)-2\left(a^{2}+1\right)\left(a^{2}-1\right)}{\left(a^{2}+1\right) a}=\frac{2\left(a^{2}-1\right)\left(2 a-a^{2}-1\right)}{\left(a^{2}+1\right) a}=\frac{-2\left(a^{2}-1\right)\left(a^{2}-2 a+1\right)}{\left(a^{2}+1\right) a}=\frac{-2(a-1)^{3}(a+1)}{\left(a^{2}+1\right) a}, a \neq 0$. The value of the given expression is 0 for $a=-1$ and $a=1$.

![](https://cdn.mathpix.com/cropped/2024_06_07_5bf9d67aee417dc7ba6ag-19.jpg?height=86&width=1757&top_left_y=448&top_left_x=158)

![](https://cdn.mathpix.com/cropped/2024_06_07_5bf9d67aee417dc7ba6ag-19.jpg?height=66&width=1742&top_left_y=504&top_left_x=174)

![](https://cdn.mathpix.com/cropped/2024_06_07_5bf9d67aee417dc7ba6ag-19.jpg?height=91&width=1742&top_left_y=563&top_left_x=174)

![](https://cdn.mathpix.com/cropped/2024_06_07_5bf9d67aee417dc7ba6ag-19.jpg?height=89&width=1742&top_left_y=641&top_left_x=174)

![](https://cdn.mathpix.com/cropped/2024_06_07_5bf9d67aee417dc7ba6ag-19.jpg?height=80&width=1742&top_left_y=714&top_left_x=174)

![](https://cdn.mathpix.com/cropped/2024_06_07_5bf9d67aee417dc7ba6ag-19.jpg?height=83&width=1742&top_left_y=775&top_left_x=174)

Calculation that the value of the expression is 0 for $a=-1$ and $a=1 \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots 1+1$ point

## 21st Mathematics Knowledge Competition for Students of Secondary Technical and Vocational Schools  National Competition, May 15, 2021  Solutions to Problems for 2nd Year

| A1 | A2 | A3 |
| :---: | :---: | :---: |
| E | B | B |