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Out of curiosity (I know nothing about space exploration and do not have strong background in physics either), just to try to build a tiny space-shuttle that would have a chance, what would be the total cost?

ps. These competitions are probably for universities or high-tech laboratories (maybe Google would put a team together...).

Cubesat mission costs depend pretty heavily on launch options. For this, it isn't entirely clear if it's included in the challenge. But you can generally expect to pay somewhere in the neighborhood of $100-150k for launch to LEO. Design and construction costs probably aren't very large for materials, but in labor (FTE). If the work is being done by skilled students as part of a university project, they could probably build and test a working cubesat for under $20k

So a whole cubesat mission can actually fit well within the limits of a sub 200k grant. But again, like most things, it's the labor costs that can make or break that.

What's your source on the LEO cost? (anecdotally) Pre-SpaceX I remember a Pegasus piggyback bill for a 3U running upwards of $600k and through the grapevine (from microsatellite array startups) that SpaceX was charging $400-500k as of a few years ago.

Have the 1U launch costs fallen that dramatically?

Did they switch the labels on the Lunar & Deep Space Derbies?