B3. Solve the equation $4 \cdot 25^{x} + 5 \cdot 16^{x} = 9 \cdot 20^{x}$.

## 22nd Mathematics Knowledge Competition for Students of Secondary Technical and Vocational Schools National Competition, April 23, 2022

## Problems for 4th 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. Write your answers for section A in the left table, leave the right table blank.
![](https://cdn.mathpix.com/cropped/2024_06_07_37ea02f5a530d200baf4g-14.jpg?height=168&width=804&top_left_y=590&top_left_x=638)
B3. First, we write $25^{x}=5^{2 x}, 16^{x}=4^{2 x}$ and $20^{x}=(4 \cdot 5)^{x}$. We then transform the equation into the form $4 \cdot 5^{2 x}-4 \cdot(4 \cdot 5)^{x}+5 \cdot 4^{2 x}-5 \cdot(4 \cdot 5)^{x}=0$ and factor out the common factor $4 \cdot 5^{x}\left(5^{x}-4^{x}\right)+$ $5 \cdot 4^{x}\left(4^{x}-5^{x}\right)=0$. We then transform the equation to the form $\left(5^{x}-4^{x}\right)\left(4 \cdot 5^{x}-5 \cdot 4^{x}\right)=0$. From the first parenthesis, we get the solution $x_{1}=0$, and from the second, the solution $x_{2}=1$.

![](https://cdn.mathpix.com/cropped/2024_06_07_37ea02f5a530d200baf4g-23.jpg?height=63&width=1739&top_left_y=1716&top_left_x=176)

![](https://cdn.mathpix.com/cropped/2024_06_07_37ea02f5a530d200baf4g-23.jpg?height=60&width=1742&top_left_y=1769&top_left_x=174)

Identifying that the equation can be rewritten from a trinomial to a polynomial ................................ 1 point

![](https://cdn.mathpix.com/cropped/2024_06_07_37ea02f5a530d200baf4g-23.jpg?height=57&width=1744&top_left_y=1870&top_left_x=173)

![](https://cdn.mathpix.com/cropped/2024_06_07_37ea02f5a530d200baf4g-23.jpg?height=60&width=1742&top_left_y=1917&top_left_x=174)

![](https://cdn.mathpix.com/cropped/2024_06_07_37ea02f5a530d200baf4g-23.jpg?height=58&width=1739&top_left_y=1967&top_left_x=176)

![](https://cdn.mathpix.com/cropped/2024_06_07_37ea02f5a530d200baf4g-23.jpg?height=60&width=1739&top_left_y=2020&top_left_x=176)

If the solutions are guessed, the student gets only two points.
}

# dMFA 

22nd competition in mathematics for students of secondary technical and vocational schools National competition, April 23, 2022

Solutions to problems for 4th year students

![](https://cdn.mathpix.com/cropped/2024_06_07_37ea02f5a530d200baf4g-24.jpg?height=148&width=251&top_left_y=474&top_left_x=914)