Can I paraphrase this sentence (Numerous examples exist of translation problems in advertising.) like this:
There exist many examples which have problems in advertising translation.
Exist is vi; if it is used with an object. It has to be followed by a prep. Is that right?
Perhaps you could say:
Problems with translating advertisements exist in many forms.
Many examples of problems in advertising translation exist.
But why use the word "exist"?
Just say "there are problems . . . ."