When Rutland leaves Haversack's office the Japanese flag is on his left (and Union Jack on his right) in the next shot it is on his right (and Union Jack on his left).
When Christine (Samantha Eggar) backs out of the taxi after her ride with Steve (Jim Hutton), her hat falls off to the side. In the very next shot, her hat is back on her head again, with her hand holding it in place.
After Julius gets Steve to sign the waiver to keep everyone's names out of the press and heads back to the taxi carrying Sir William and Miss Easton, a front view shows the taxi is empty except for the driver. After ending his conversation with the reporter dressed as a competitor, Miss Easton and Sir William are back in the taxi.
At the race, Julius says the fastest time for the 50 km race-walk is "four hours and fifty-three minutes." At the time of the 1964 Olympics, the men's 50 km race-walk world record was actually 4:00:50. (At the time of the film's release, the record was 3:55:36.)
At the race, the announcer says an Australian race-walker took 9th place with a time of 4 hours 53 minutes. 9th place was actually taken by Charles Sowa of Luxembourg with 4:20:37.2. 4:53:00 would have taken 28th place.
Towards the end of the race scene, in Tokyo, Cary Grant walks past a yellow and red storefront sign the appears to say "TUBAME PACU" upside down in English letters. This was clearly a last minute cover-up by the American film crew using materials they had at hand.