How much does it hurt one's application to try the GRE again? I'm applying for mathematics PhD programs and I got 162 on the quantitative GRE exam (I didn't prepare for it at all). As I thought this would probably be a good grade and I don't like spending 200 bucks for nothing, I reported the scores to some universities I would really like to go to (e.g. UIUC, Berkeley). However, it ends up being in the 82 percentile, which I'm not sure is great. I think I have a reasonably strong profile (strong letters of recommendation, lots of graduate classes, high GPA, scholarship to study for a couple months at a high-regarded european university etc.), but I'm worried that this not-so-great quant score would keep me away from top mathematics programs. If I retake the test and get a better grade, will my previous score hurt me somehow?
The first score will very likely not matter if you do better the second time.  Most programs look only at the best score a student has on their record.  Officially, that's what my program does.  Unofficially, if you had a really low score the first time you too the exam, it would raise a red flag, and we might go looking for some other reason not to admit you.  However, an 82 percentile on the first try, if you take the test again and do better, is not going alarm anyone.