Actually, Hagar, there was an official competition and rules were established between the US and the USSR.
None of the American suborbital flights counted because they were below the 100 mile or 100 km threshold established by the rules (I'm going to research this and get you some links, so be patient).
None of the Vostok flights counted because the cosmonauts did not return to Earth with their vehicles. The design of the Vostoks did allow them survive impact. The solution was that the cosmonaut opened the hatch at about 20,000 feet, jumped out, then parachuted to Earth.
I'm not kidding and I don't expect you to believe me. Therefore, links will follow.
Edit: Here's a link to get you started
http://www.vibrationdata.com/space/Yuri.htm"As part of the flight plan, Gagarin exited the spacecraft at an altitude of about 20,000 feet and then parachuted to the ground. He landed near Saratov in the Volgograd region.
The Soviet government apparently kept this parachute detail secret for many years. The Soviets sought to give an impression that the Vostok spacecraft made a soft landing with Gagarin still inside. The Federation Aeronautique International required that a pilot land with his vehicle in order to claim a complete flight for the record books."