B2. Let $a$ and $b$ be real numbers such that $\frac{3 a}{a+b}+\frac{2 b}{a+2 b}=1$. Determine all possible values of the expression $\frac{2 a-3 b}{2 a+b}$.

## 63rd Mathematical Competition for High School Students in Slovenia

Selection Competition, March 21, 2019

## Problems for 2nd Year

Time for solving: 45 minutes.
B2. In the given equation, we eliminate the fractions to get $3 a(a+2 b)+2 b(a+b)=(a+b)(a+2 b)$, or $3 a^{2}+6 a b+2 a b+2 b^{2}=a^{2}+2 a b+a b+2 b^{2}$. We then rearrange the equation to $2 a^{2}+5 a b=0$ and factor the left side to $a(2 a+5 b=0)$. If $a=0$, then $b \neq 0$, otherwise the fractions $\frac{3 a}{a+b}$ and $\frac{2 b}{a+2 b}$ would not be defined, and thus $\frac{2 a-3 b}{2 a+b}=\frac{-3 b}{b}=-3$. If $a \neq 0$, then $2 a+5 b=0$ or $2 a=-5 b$. In this case, $b \neq 0$ and thus $\frac{2 a-3 b}{2 a+b}=\frac{-5 b-3 b}{-5 b+b}=\frac{-8 b}{-4 b}=2$. The possible values of the expression $\frac{2 a-3 b}{2 a+b}$ are -3 and 2.

Reformulating the expression to $3 a(a+2 b)+2 b(a+b)=(a+b)(a+2 b)$.............. 2 points

![](https://cdn.mathpix.com/cropped/2024_06_07_81d52baa66c72127156fg-11.jpg?height=63&width=1576&top_left_y=637&top_left_x=274)

![](https://cdn.mathpix.com/cropped/2024_06_07_81d52baa66c72127156fg-11.jpg?height=60&width=1579&top_left_y=690&top_left_x=273)

![](https://cdn.mathpix.com/cropped/2024_06_07_81d52baa66c72127156fg-11.jpg?height=57&width=1579&top_left_y=740&top_left_x=273)

![](https://cdn.mathpix.com/cropped/2024_06_07_81d52baa66c72127156fg-11.jpg?height=60&width=1579&top_left_y=787&top_left_x=273)

![](https://cdn.mathpix.com/cropped/2024_06_07_81d52baa66c72127156fg-11.jpg?height=66&width=1579&top_left_y=841&top_left_x=273)

Writing the conditions for the second solution:

$a \neq 0$

1 point

$2 a=-5 b$

2 points

![](https://cdn.mathpix.com/cropped/2024_06_07_81d52baa66c72127156fg-11.jpg?height=60&width=1573&top_left_y=1038&top_left_x=276)

![](https://cdn.mathpix.com/cropped/2024_06_07_81d52baa66c72127156fg-11.jpg?height=68&width=1576&top_left_y=1091&top_left_x=274)

Writing the final solution . ..............................................................................

## 63rd Mathematical Competition for High School Students in Slovenia

Selection Competition, March 21, 2019

## Each mathematically correct and complete solution is worth $\mathbf{2 0}$ points, even if the solving process is different from the official solution.

## Solutions for 2nd Year