Is theodicy considered an open problem or a solved one, according to catholicism? Do catholics consider the problem of theodicy (Why God allows evil on this world) an open question, or is it "solved", seeing that so many people have tried to answer it. If it is not solved, what is the reason?
Yes, it is an open question.

Yes, it is a "solved" question.

No, it is not an OR question.

The question you are asking "Is Theodicy an "open" question or a solved one?" is based on a faulty premise.  Namely, "open" does not mean "unanswered" or "unsolved" - it means one that is continually being investigated.  Thus, the opposite of an "open" problem isn't a "solved" one, but rather an unasked one.

At best, one can ask, "Is there a consensus position?" and the answer is "within a fairly broad spectrum, yes."  Yes, there are several "solutions" that theologians have posited that each have a significant consensus behind them.  Go find them - that's a book.

In order to highlight the faulty premise, let's use another question as an example: "Did God create the world ex nihilo or did it evolve?" As contentious as the question is, no one can say there are no "solutions" to it. Non-theists have posed several answers, Theists have posed several answers, and there are even significant groups that seek a "theistic evolution" answer in the middle.  Is it an open question? Sure! People wrestle with the origins question everyday. And is it a solved one? For many people, again, the answer is "Yes!"  They have weighed the evidence and personally picked the "solution" they feel is most compelling.

In theology, there are very, very few "closed" questions. John Dominic Crosson believes the literalness of the Resurrection to be an "open question," despite Paul's very obvious answer to "If there is no resurrection, then we are to be pitied above all men!"  The closest thing I can think of as a "closed" question is the literal existence of a King David. For many years, theologians thought it an "open question" as to whether or not David existed.  When archeologists found a steele mentioning the House of David, the evidence pretty thoroughly closed any further debate, because it is really hard to argue with tangible evidence to the contrary - and yet there are still some who claim it is "open"

Back to the "problem" of theodicy - or "Why does a Good God allow evil?"  There are several solutions to the problem - but that does not mean it is unasked.  It is thus open and solved at the same time.  And frankly, until there is no more evil, or until people see God as not good, it will always be asked. It will and must be open to new queries as long as man asks questions.