5. For the function $f(x)=\frac{3-2a}{a+5}x+\frac{2a-1}{3-a}$, determine the parameter $a$ such that the graph of the function intersects the y-axis at the origin and the function is decreasing.

Solve each problem independently on the provided paper, with each problem on its own page.

Do not sign the sheets, write only your code.

Write your answers clearly and neatly with a pen. Each problem's solution will be graded on a scale from 0 to 6 points.

You have 120 minutes to complete the problems.

3rd Mathematical Competition for Students of Secondary Technical and Vocational Schools, Ljubljana, April 12, 2003

## PROBLEMS FOR 3RD YEAR STUDENTS
5. The given function is decreasing if $\frac{3-2 a}{a+5}<0$. The first inequality is satisfied if the numerator and denominator of the fraction $\frac{3-2 a}{a+5}$ have different signs, which is for $a>\frac{3}{2}$. The second inequality is satisfied if the numerator and denominator of the fraction $\frac{2 a-1}{3-a}$ have the same sign, which is for $\frac{1}{2}<a<3$. Both inequalities are satisfied if $\frac{3}{2}<a<3$.

![](https://cdn.mathpix.com/cropped/2024_06_07_06886e410b39c0216094g-10.jpg?height=83&width=1639&top_left_y=1495&top_left_x=206)

![](https://cdn.mathpix.com/cropped/2024_06_07_06886e410b39c0216094g-10.jpg?height=60&width=1642&top_left_y=1555&top_left_x=204)

![](https://cdn.mathpix.com/cropped/2024_06_07_06886e410b39c0216094g-10.jpg?height=66&width=1642&top_left_y=1600&top_left_x=207)

![](https://cdn.mathpix.com/cropped/2024_06_07_06886e410b39c0216094g-10.jpg?height=65&width=1636&top_left_y=1655&top_left_x=207)

## Third Year