Why, do you think there's something dodgy about there being no application fee? According to this:
Carnegie Mellon is ranked joint 33rd of pure math departments, with a score of 3.6/5.
And yes, according to their web page there is no application fee:
I did a bit of searching around. For universitities of a similar ranking to Carnegie Mellon here are the application fees:
Ohio State: $5
Univ. of Utah: $55
University of California--Davis: $70
Rice University: $70
Pennsylvania State University--University Park: $45
Indiana University--Bloomington: $50
OK, so in my limited search I could not find another university with no application fee. However the other universities have fees ranging from $5 to $125, quite a wide distribution for only 7 universities don't you think?