I wish for Athens and Beijing they just left the signs in only Greek and only Chinese respectively. It'd be more reflective of the host country's culture, plus having 3 languages on on sign is just too much, especially for the countries that have to have the really long names like Former Yugoslav Republic of Macedonia or United States of America.
I really like your LCD idea! Have the name in English *splash effect* name in French. Maybe do another spalsh effect and then show the outline of the country, although that probably wouldn't work for countries with disputed borders like India and Pakistan. It wouldn't work really well for Oceanian countries that have a bigillion little islands, but none of them are going to be in the winter Olympics anyway.