Which of the problems from the Scottish Book (pdf of English version) by Stefan Banach are still open? I know that one of the problems was solved by Per Enflo for which he got a live goose from Stanislaw Mazur.

The book version edited by Daniel Mauldin (from 1982) has commentaries on the problems as of that date. 


Luis Montejano solved problem 68 in 1990 (he also solved the limiting 0 density case of problem 19, but I think this is already mentioned in Mauldin's book). The paper is called About a problem of Ulam concerning flat sections of manifolds and appeared in Commentarii Mathematici Helvetici. 


Second problem in The Scottish Book (Edit: is open? GRP). Let $X$ be a compact metric space. If there exists a finitely additive Borel measure $\mu$ such that $\mu(X)=1$ and if $A,B\subset X$ are congruent then $\mu(A)=\mu(B)$. Remark. We say that that the sets $A,B \subset X$ are $congruent$ if there exists a distance preserving bijection from $A$ to $B$, not necessarily defined on the whole space $X$. 2nd edit. I think the best reference frame, related to the above mentioned problem, is a book by Stan Wagon "The BanachTarski Paradox". On page 31 of this book there is described recent progress (up to 1985) toward a solution of the problem. 

