Does the Shift+F1 while the game is open, open OiG? If it does bring up OiG perhaps try inviting directly through the Friends list.
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If it doesn't come up.
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Disable any other overlay programs (Fraps, RiviaTuner, Raptr, etcetc)
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Try uninstalling and reinstalling Origin client (this does not remove data in \Origin Games\)
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CleanBoot
http://support.microsoft.com/kb/929135
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Graphics drivers up to date?
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Run Origin client as Admin
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Try changing from Fullscreen to Borderless.... or try flipping the v-sync setting (found a google result saying this worked for someone)
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If you minimize the game while it's running, can you directly invite from the main Origin window's friends list? Probably not the neatest solution but better than not inviting I guess.
