Rules: Send mail to author when value in date field has passed
Assuming you are using Drupal 7. 

You need to enable the Rules Scheduler module, it comes with Rules.

Then for Rule event use After saving new content. 

Add condition, content is of type: Contest

Exit out of your Rule, and now create a Rule Component of Data Type: Node 

Add condition, content is of type: Contest

Action Send E-mail.

Now exit the component and go back to your rule, for action, schedule rule component, and use your end-date field for schedule evaluation date.

I highly suggest you watch Introducing Rules Scheduler and I am sure my answer will make more sense to you.