No actual duels were fought, alas. Rather, contestants shot at a dummy dressed in a frock coat.
The photo you posted is not the Olympics, but in 1909 at Panzer's Gymnasium, Carnegie Hall, 57th St. & 7th Av. in NYC. Established 1899.
The dueling guns and their wax shot.
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Here's an article explaining the short lived sport:
Wax Bullet Duels Were the Paintball Fights of 1909