On June 6, 1999, at 1100 mountain daylight time, a Cessna 182B, N2276G, sustained substantial damage during landing roll at Salt Lake City International Airport, Salt Lake City, Utah, when the nose wheel collapsed. The private pilot and his two passengers were not injured. Visual meteorological conditions prevailed for this local area pleasure flight operating under Title 14 CFR Part 91 and no flight plan was filed. The flight departed at 0945. According to the pilot, as he lowered the nose landing gear to the runway during landing roll, the nose wheel started to shimmy followed by radical departure to the right. Full left brake and rudder had no effect and the nose wheel collapsed. The pilot said the aircraft propeller and left wing struck the ground and the aircraft came to rest nose down with the nose landing gear folded rearward and cocked at a 90 degree angle. Examination of the aircraft by a FAA airworthiness inspector provided information that the scissors bolt on the nose wheel landing gear torque link was missing. On examination of the landing area, the inspector, assisted by airport personnel, could find no evidence of the bolt, and the bolthole in the scissors assembly was not damaged or elongated.