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Do you "logout"(click the x) or Quit the application (CMD + Q) and does it work with the other way????
I am on a mac as well and I do NOT have this issue, I just tried to quit out and it asked me if I wanted to log out. Maybe try to repair the client.
MacBook Pro (Retina, 15-inch, Mid 2015)
mac OS 10.12.2 Beta
Ok think I may have replicated your issue, I Force Quit (not CMD + Q) out of League and now I can not get it to launch via the shortcut.
I think my issue is by force quitting the application, The "lockfile" never got deleted. Try to open one of these files
/Applications/League of Legends.app/Contents/LoL/LeagueClient
/Applications/League of Legends.app/Contents/LoL/Play League of Legends
The first is the link to the Alpha client, the second is the link to the legacy client.
If that does not work, try opening up one of the links inside one of the deploy folders:
/Applications/League of Legends.app/Contents/LoL/RADS/projects/league_client/releases/
open up on of the "0.0." files, then go into
/deploy/
click on "LeagueClient"
Opening up from there worked after I was getting locked out
Let me know if that helps.