Why do universities share test questions? I have had quite a few take home exams, mostly because they required you to be online and were a lot longer than 2 hour exams. 

When taking these exams and answering questions (all essay questions) I've typed in the question into Google and bam. I find the same questions used from some PDF associated with, often, a different university and the PDF has either the answer or guidelines as to whether or not the student gave a close enough answer. 

This has happened in 3 courses so far and I brought it up to a professor a year ago and she had no idea that the PDF she got test questions for had the answers on the same PDF online. 

I assume it isn't students putting these PDF's up because the PDF's often have more questions than are on the exam and will have pages with a university name, class, and symbol which are not given on the exams handed out. 

So, why would a university use questions from another university when a simple copy and paste reveals the answers or at least guides as to how to answer?
I am not aware of any university having agreements with other universities about exchanging exam questions and i doubt that such agreements would exist in many places. I would wager a bet that the same is pretty much true on departmental level.

On the level of individual teachers there might be many reasons why exam questions pop up in different places. One has been mentioned by Brian Borchers in a comment (questions taken from textbooks). Secondly, many first- and second-year courses teach pretty standard material which often has not changed for quite a while (e.g. basic linear algebra in many disciplines). For such courses (and even some higher-level courses) the set of questions that are useful as exam questions tends to be limited and so i would not find it surprising to find that most if not all questions have been asked somewhere and have been discussed on the web. Thirdly, it is not unheard of that teachers re-use exam questions from previous years. Fourth: sometimes teachers talk to each other and exchange experiences and material.