Page and Post Category with same name – possible?

I have a URL string problem that I can’t find any info on:

I have a Page named “Projects.” So, the url string for this page is like so:
…/projects/
In this page I have thumbnails and brief descriptions to a bunch of projects.

I post a new project as a Post and I assign it the category “project.” So an individual project’s url is like so:
…/project/project-name

I would like to have the url string for individual projects be like so:
…/projects/project-name
but if I change the category name through Posts > Categories to “projects” it then disables the page with the same name.

Is there a workaround for this, so that I can have the category name be the same as the page name?