5. Can a regular triangle be divided into 2025 equal regular triangles? Justify your answer.

## Answers and solutions

Problem 1 Answer: $54^{0}$

Problem 2 Answer: 1) 35 numbers, 2) 10003

Problem 3 Answer: the exponent is 2003.

Problem 4 Answer:

1) $n=4 t+1, t=0,1,2, \ldots$
2) 250

Problem 5 Answer: yes, the side of the sought triangle is $\frac{1}{45}$ of the side of the given triangle.
Solution: Let the side length of the triangles in the partition be $\frac{1}{n}$ of the side length of the given triangle. Then, in the first strip, there is 1 triangle, in the second strip, there are 2 more, i.e., 3 triangles, in the third strip, there are 2 more, i.e., 5, and so on. In the $k$-th strip, there are $2 k-1$ triangles. The total number of triangles is

$1+3+5+\ldots+(2 k-1)+\ldots+(2 n-1)=n^{2}$

The number 2025 is the square of the number 45.

![](https://cdn.mathpix.com/cropped/2024_05_06_a2a9f04ae562a0d0b324g-2.jpg?height=560&width=645&top_left_y=1639&top_left_x=1274)