“Of the cleanly-formulated Hilbert problems, problems 3, 7, 10, 11, 13, 14, 17, 19, 20, and 21 have a resolution that is accepted by consensus. On the other hand, problems 1, 2, 5, 9, 15, 18+, and 22 have solutions that have partial acceptance, but there exists some controversy as to whether it resolves the problem.”

I think some Wikipedia editor should properly fix this. Arnold is claimed to have solved this problem.

Marco

“A variant of this problem, searching for a solution within the universe of continuous functions, was solved by Andrei Kolmogorov and Vladimir Arnold. It is not difficult to show that the problem has a positive solution within the space of single-valued analytic functions (Raudenbush). Some authors argue that Hilbert intended for a solution within the space of (multi-valued) algebraic functions, thus continuing his own work on algebraic functions and being a question about a possible extension of the Galois theory (see, for example, Abhyankar,[14] Vitushkin,[15] Chebotarev [16] and others). It appears from one of Hilbert’s papers [17] that this was his original intention for the problem.
The language of Hilbert there is “…Existenz von algebraischen Funktionen…”, i.e., “…existence of algebraic functions…”. As such, the problem is still unresolved.”
(http://en.wikipedia.org/wiki/Hilbert's_problems)