I have recently seen this post about the theorem von Neumann proved in five minutes.
Searching for more information about that, it is certain that I can't find any references for what the theorem was.
But as I looked through the comments, I found it very convincing that it was an open problem which Pólya frequently asked in classes.
Since Pólya was a famous problem solver, he could access many open problems in that era.
What I'm curious about is what kind of open problems were in the 1920s?
Were there math magazines or problem books or math competitions?