If you've not sold it yet...
Temporarily connect to one of the other TVs with an HDMI cable and boot up
Turn off HDMI mode auto-selection
Change down to 720 at 50Hz and confirm you want to keep the change
Shutdown the device, then plug it into your normal TV
Assuming you now see video output, head back to the HDMI mode option, and try 1080 at 50 and 60Hz. If you don't confirm the change each time (if you can't see anything any more for example), then the system will revert to the resolution you booted up with
To change the relevant settings, try the following (exact labels differ between firmware releases)
Start from the device's homescreen (so boot up and don't run any apps)
If you can see a Settings shortcut, click it
If not, open the list of all installed apps, and look for Settings or Settings Mbox and click it
If you now see a colourful tabbed interface, head to the second tab and turn off the auto-select of HDMI modes. The entry below that normally handles manually specifying 720/1080 etc
If you see a monochrome-like interface, look for Display or Screen for the relevant settings