Bump. This problem feels really familiar; I think similar theme of cutting triangles into isosceles triangles have been in past problems.

We can prove, that if we cut non-isosceles triangle and get two isosceles triangles, than primary triangle has right angle. And now we can prove, that if cut non-isosceles triangle we can not have two right triangles or right triangle and isosceles. Also we can use idea from official solution.

Idea from official solution