Well I lived in SU for 15 years. There was joke that the Party declared that “in next 5 years we will catch US”. Smarter ones told “lets catch, but no bypass”. “Why not bypass also?” “Then they will see our naked ass”. Totally true. Most of economy was invested to military and this was only area with some world-class tech, and even that was often fake and/or with human sacrifices never published
The SU had a program of requiring world-class mathematicians (Kolmogorov, Arnold, Gelfand), physicists, etc. to teach one year in high school, and to think about the curriculum, and possibly write textbooks. Some of these were available in English via MIR publishers. Some of these books are unique moments in the history of mankind. The science education in SU trickled down throughout eastern europe. Indirectly this school of thought has been responsible for some of the best science and engineering in the world, sadly, mostly outside of the place that originated it.
I lived in Israel when all of the soviet Jews emigrated in the late 80's/early 90's. One of them got a job alongside my dad, who was an engineer, and they went on a business trip to the US together. When he came back, he said that if they "turned off the US and locked the door", it should still take Russia over 100 years to catch up.
However, during the early days of the space race, there was literally nothing the US could do to catch up with the Soviet Union space program. Their best hope was a former Nazi engineer.