"Those trees that stand on the white mountain are green."
Translation:Azok a fák zöldek, amelyek a fehér hegyen állnak.
Is there any reason that the relative pronoun here must be amelyek, and not just amik?
Yes, the noun that it refers to ("fák") is present in the sentence, therefore the correct usage is "amelyek". When the noun is missing, that's when you use "amik".
"Azok a fák, amelyek..."
This is just the main rule, there was a deeper discussion here, for example:
Yes, of course, the spoken language is always more forgiving. And the use of "ami" is spreading like wild fire. It has its place, and the rules for its use are not as black and white as I might have described it here and there. But I think a language course should teach the rules most of all, and let the students learn to break them on their own. :) Teaching should always aim higher, because no student will learn 100% of what is being taught. Knowing the rules will give you a strong foundation, and the rest is up to you. :)