In a research statement for PhD admissions, should I start with the context, or directly with the idea for my research? I'm applying for a PhD in engineering and I was asked by the advisor I have chosen to write down a short program (~10 pages) about the "research" I will conduct if I get into the PhD program.

The advisor provided ten choices where the future PhD program will lay. The topics are very large and well studied in the literature. Our "task" is to choose one of them, find out something "new" inside of this big sea and write it down. I did a lot of research on Scholar and downloaded more or less 20 free papers I found interesting. I collected data and results, thought a bit about the problem and found a ( maybe ) new application that fits my interests and past studies and I think it could bring some novelty in the field I'm applying. 

Now my question is, which is the best way to format the proposal?


Should I start with the context of the research, point where previous studies went and what kind of results yielded and now focus on my brand new idea?
Should I skip 1) and start in medias res with the heart of the matter?
Now my question is, which is the best way to format the proposal?


Different writers could balance these issues in different ways.  The key is moderation: you have to give enough context to make your proposal understandable, but you also shouldn't go overboard (it's a terrible idea to start a research proposal by saying "Since the dawn of history, transportation has been one of humanity's greatest challenges. The wheel and axle were a great advance...").  If you never mention until page 5 that you even have new ideas, then readers may give up in disgust before reaching that point.  On the other hand, too little context can be just as bad, since the proposal will do you no good if it's not understandable.  Unfortunately, there's no universal rule to decide the right amount.

One strategy is to look at the introductions to the research papers you've found.  If your introduction is briefer or more abrupt, then it's not reasonable to expect anyone to follow it.  If anything, a research statement should generally be accessible to a broader audience, and that may mean somewhat more background and explanation.

It can also be useful to cycle between background and new material.  For example: brief introduction, brief description of your new idea, longer account of context, more details on new idea.

Given that you are writing this for a specific professor, it's worth asking him/her for more guidance.  If you phrase it too generally ("How do I do this?") it might give a poor impression, but I think it would be safe to ask a question that shows you are seriously working on it.  For example, "I'm thinking of organizing my proposal like this: [Insert one or two paragraphs about your topic and how you plan to arrange the proposal.] Does that sound like what you had in mind?  If not, I'd be happy to rethink the organization."