The whole thing was about national pride on both sides. The soviets didn't admit they were striving for the same thing because they never wanted to be seen to lose. Their pattern was always the same:

• They didn't say they were working towards having the first satellite. They just announced it when they were successfully.
• They didn't say they were working towards having the first living animal in space. They just announced it when they were successfully.
• They didn't say they were working towards having the first man in space. They just announced it when they were successfully.
• They didn't say they were working towards having the first man on the moon. They just denied it when they were unsuccessful.

However, the Soviet lunar program was confirmed many years after the fact under Gorbachev's policy of Glastnost when the Soviet Union fell. The Soviet Lunar program is fact. Their lunar landers were built just months after the US. Some still exist. There's one on loan for display at Disneyland in Paris. I've seen another at the London Science Museum. Russia loans them out to show how advanced they were at the time. To take pride in what they accomplished, and rightly so.

This is all very public, yet you're trying to convince me that 50yo face saving propaganda is the truth?