how to report a problem with a purchase

What type of problem ? If you've been over-charged then you can contact iTunes support via this page : http://www.apple.com/support/itunes/contact/ - click on Contact iTunes Store Support on the right-hand side of the page, then Purchases, Billing & Redemption

If the item doesn't work, then depending upon the item type (app, ibook, music etc) then you might be able to re-download it via the Purchased tab in the App Store or iTunes store app : re-downloading . Music, films and tv shows can't be re-downloaded in all countries. If it's an app then you should also try contacting the app's developer before using the 'report a problem'

If you aren't in a country where you can re-download the item or if it re-downloads in the same state then try the 'report a problem' link from your purchase history : log into your account on your computer's iTunes via Store > View My Account (Store > View My Apple ID on iTunes 11) and you should then see a Purchase History section with a 'see all' link to the right of it ; click on that and you should see a list of your purchases ; find that item and use the 'Report a Problem' link and fill in details about the problem (iTunes support should reply within, I think, about 24 hours).

Some people have had a problem with the 'report a problem' link (it's been taking people to this site on a browser instead of showing a form in iTunes) - if it does that to you then try contacting iTunes support via this page : http://www.apple.com/support/itunes/contact/- click on Contact iTunes Store Support on the right-hand side of the page.