Just to be clear, the reason you are getting this error isn't because of a problem with Java on your Mac. The problem is with a certificate that has been used to sign one or more of the JAR files that make up the TopCoder Java application.
The certificate expired on March 25th this year.
If you still want to run the application, even though the certificate has expired, then you need to add it to the Exception Site List within Java Control Panel.
You may want to update Java on your Mac first though. You can get the latest download from Oracle here, Apple don't supply their own Java runtime anymore.
- Launch Java Control Panel from System Preferences
- Select the Security tab
- Click on "Edit Site List..."
- Click on "Add"
- Enter "http://www.topcoder.com" in the Location field
- Click OK
- Click Apply
Now when you try to open the JNLP file you'll get a different message. Just check the "I accept the risk and want to run this application" and click Run.
At which point the application should successfully download and run.