5. For which value of the real number $a$ is $1+\frac{1}{2+\frac{1}{3+\frac{1}{a}}}=0$?

## You have 120 minutes to solve the problems.

Solve each problem independently on the provided paper, with each problem on a separate page. Do not sign the sheets; write only your code. Write your answers in pen, clearly and neatly. Draw graphs with a pencil. If you solve a problem in multiple ways, clearly indicate which solution the evaluator should grade. If you make a mistake, cross it out.

Each problem's solution will be graded on a scale of 0 to 6 points.

The National Competition Committee wishes you great success.

## 8th Mathematics Knowledge Competition for Students of Secondary Technical and Vocational Schools  National Competition, April 19, 2008

## Problems for 2nd Year Students
5. We simplify the continued fraction to the form $\frac{10 a+3}{7 a+2}$. This is equal to zero if the numerator is equal to 0. Thus, $10 a+3=0$ or $a=-\frac{3}{10}$.

![](https://cdn.mathpix.com/cropped/2024_06_07_2b0bf3255e176946d93cg-07.jpg?height=80&width=1639&top_left_y=1496&top_left_x=206)

![](https://cdn.mathpix.com/cropped/2024_06_07_2b0bf3255e176946d93cg-07.jpg?height=66&width=1642&top_left_y=1566&top_left_x=207)

![](https://cdn.mathpix.com/cropped/2024_06_07_2b0bf3255e176946d93cg-07.jpg?height=63&width=1639&top_left_y=1622&top_left_x=206)

![](https://cdn.mathpix.com/cropped/2024_06_07_2b0bf3255e176946d93cg-07.jpg?height=69&width=1639&top_left_y=1670&top_left_x=206)

![](https://cdn.mathpix.com/cropped/2024_06_07_2b0bf3255e176946d93cg-07.jpg?height=52&width=1639&top_left_y=1730&top_left_x=206)

![](https://cdn.mathpix.com/cropped/2024_06_07_2b0bf3255e176946d93cg-07.jpg?height=65&width=1636&top_left_y=1778&top_left_x=210)

## Second Year