Is it permissible to create a website for past exams at lower prices to compete with other websites, and to purchase content from one of these websites (which is merely a compilation of past exam questions) to use as a source for our website?

There is no objection to creating a website for this permissible purpose, nor to lowering the price to compete with other websites. Similarly, there is no impediment to purchasing content from a website to use as a source for questions, especially since the questions are available to everyone.