Different blogs, same posts

I have a master blog where I post everything from all my other sites. I would like to add a news section to one of my other websites that automatically updates when I post to a specific category on my main blog (and only that category).

For example:
MyBlog includes categories for website1, website2, website3, etc. I would like to create a news section (or blog) for website1 that would feature only what has been added to the website1 category.

Is this possible? How do I go about doing it?