Vincent2
Our solved tag has the following (relevant) permissions allotted to these groups:
View Discussions: Everyone
Start Discussions: Admins / Mods
Start Discussions without Approval: Admins / Mods
If you set your site's tag up like that, it should function the same. This will mean that only Admins and Mods can use or add the Solved tag on a discussion.
You can adjust which groups you allow on your site as needed.