B1. Express $\frac{x+3}{x^{2}-4}, \frac{x-1}{x-2}$ and $\frac{x+3}{x+2}$. Determine $x$ so that the second expression is the arithmetic mean of the first and the third.
B1. The condition of the problem is satisfied if $\frac{x-1}{x-2}=\frac{\frac{x+3}{x^{2}-4}+\frac{x+3}{x+2}}{2}$, and the equation can be rearranged to $x^{2}=1$, which has solutions $x_{1}=1$ and $x_{2}=-1$.

![](https://cdn.mathpix.com/cropped/2024_06_07_ce887fced8b2a2bf0894g-15.jpg?height=123&width=1642&top_left_y=1435&top_left_x=273)

Condition $x \neq \pm 2$.........................................................................................................................................

Elimination of fractions ........................................................................................................................................

Correct solving of the equation ......................................................................................................................................

Conclusion $x^{2}-1=0$................................................................................................................................

![](https://cdn.mathpix.com/cropped/2024_06_07_ce887fced8b2a2bf0894g-15.jpg?height=57&width=1636&top_left_y=1742&top_left_x=276)