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Most contests are run in Forum or Survey items. Your participation in these types of items is logged in your My Account/(My Activity) section: Forum Posts and Survey Responses. Both logs have date and item info in the list and are searchable (for example, searching for entry or contest).
Note that anything older than 2 months is not likely to be for a contest that is still in process.
If this is information you will want going forward, it's probably easier to keep a log of such information by hand, such as in an item or in one of your Notepad pages.
Hope that helps.