I prefer yours. I'm assuming that you would go on to end it to produce something something like:
I believe that the question of why people choose to enter this type of competition is an interesting one.
And I'd be equally happy with an "as to" form, such as:
I believe that the question as to why people choose to enter this type of competition is an interesting one.
I suspect the reason I prefer those over your friend's is that I'm vaguely uneasy with the fact that the text implies that it contains a portion that is a question when in fact that is not strictly true.
Consider: the overall form is as follows:
I believe that [SOMETHING] is an interesting question
where [SOMETHING] represents the text fragment, 'why people choose to enter this type of competition'. The problem is, that text fragment is not a question. Contrast your friend's offering with the following:
I believe that, "Why do people enter this type of competition?" is an interesting question.
In that case, the text fragment actually is a question, and so my unease is resolved. However, the resolution is done inelegantly, by brute force of typography. Yours is more subtle, which is why I'd go that route myself.
All that said, I don't think your friend is strictly wrong. I'm not a language examiner, but I'd be surprised if they were to lose marks for their form.