"studying," not "learning." Although it's true that you sometimes study without learning anything.
While this is a grammatically correct English sentence, we simple do not say it in this way. It is better to say: why don't you study for the exam?
It would also be natural to ask 'Why aren't you studying for the exam?' if the exam was, say, the next day. But I agree definitely 'study' rather than 'learn'.
I'm happy to know that i'm not the only person, who thinks exactly the same. Though i'm not a native speaker, so it's not me who can judge.
You should be able to use a contraction as you already implied: "Why don't you ..." rather than "Why do not you ...", which doesn't quite work. ;-)
You could possibly say, "Why do you not study for the exam," but not "Why do not you." And "examen" is Spanish or French, not English.
I obviously can say " Why dontchu study for the exam " ... that's why I think that should be accepted :)
Why not " why are you not studying for an exam " ?
She said "le-mivkhan", not "la-.."
That's at least possible with the text, not the audio. For example, maybe the speaker assumes that students generally have one or more exams to study for simply because they're students and that's what students do! In that context, it would then make sense to ask, "Why aren't you studying for an exam?"
In short, by using "the exam", the speaker would assume that there is a specific, identifiable exam—one that can be (mentally) accessed and identified by both the speaker and the listener from the (communicative) context/situation. By using "an exam", the speaker would not be specifying a specific, identifiable exam.
As for the audio, I hear "la-mivkhan".
Edit: The second syllable in לומד might be interfering with how you hear the first syllable in למבחן. I've encountered such interference many, many times—especially for words and constructions that I haven't internalized sufficiently. That's one possibility, anyway!