Here are some things that may be able to fix your problem. Note that they may not work, because I do not exactly know what is causing your problem down to the core. Also I do not have enough information as to what is currently happening and what you have done before this happened. Or what you have already tried.
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You can try to do a password change, at least try this one twice because doing it once may not work at first.
If osu! is open, and you are actually already logged in but it says that authentication failed or that you cannot connect to multiplayer, try logging out and relogging.
Do an uninstall of osu! and install it again. I suggest you cut and paste some things such as beatmaps and skins to a seperate folder onto your desktop. So that after the re-installation you can reapply them again.
If you are doing the reinstall option, try starting the game in administrator mode after it is installed.
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Those solutions above may not work. If you cannot fix this problem, or nobody on the forums has a solution. Try contacting osu! support and they can probably narrow down some causes as to why this is happening. Also, since they are staff they should be able to try logging into your account themselves and see if the problem occurs for them. If it does not occur for them, they can possibly find the fix for you somehow.
As always, make sure to keep your osu! updated at the current recent patch/version. Using anything earlier has a slight chance to cause problems to your game in some situations.