Determine if there exists a triangle that can be cut into 101 congruent triangles.
Answer: Yes, there is.

Choose an arbitrary positive integer \( m \) and draw a height in the right triangle with the ratio of legs \( 1: m \). This height cuts the triangle into two similar triangles with a similarity coefficient of \( m \). The larger of these can further be divided into \( m^2 \) smaller equal triangles by splitting all sides into \( m \) equal parts and connecting corresponding points with parallel lines. Thus, a triangle can be split into \( m^2 + 1 \) equal triangles.

The figure shows this for \( m = 4 \), but in our problem, we must take \( m = 10 \).

![](https://cdn.mathpix.com/cropped/2024_04_17_4ead9738494cc20c28d3g-13.jpg?height=215&width=866&top_left_y=1089&top_left_x=595)