I saw this phrase and I was kinda boggled by it. I know that it means "let the games begin", but it doesn't seem like imperativ to me, and the use of "mögen" is just mysterious here. Is there an omitted wir or sie? Is it just one old impression or some general usage rule that I'm missing?
All modal verbs have some tricky uses, sometimes differing on the basis of grammatical mood.
Mögen die Spiele beginnen!
Möge das Fest beginnen!
By comparing these examples, one can see that i) the verb is in the present subjunctive (Konjunktiv I) and that ii) the noun phrase that follows is the subject.
The meaning is that of a wish ("optative", Wunschsatz). This use of mögen is pretty old-fashioned. That same meaning can often be expressed by the subjunctive alone, without mögen.
Möge Gott dir beistehen!
Stehe Gott dir bei!