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The question in the title is clear enough to not to write additional details here.

Only to say, as far as I know, Goldbach´s conjecture and the problem of existence of odd perfect numbers are among the oldest unsolved problems that I know of.

What problems do you know of?

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The questions of existence of odd perfect numbers and are there infinitely many even perfect numbers go back to Euclid (although I'm sure Euclid did not explicitly state these problems). There are not many advanced mathematical texts before Euclid, so they have to be the oldest unsolved problem.

On the other hand if you count the questions like "who wins with the optimal strategy in a particular game (e.g. Go)?" as an open problem, then it could be even older.

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    $\begingroup$ Euclid does not mention these problems either explicitly or implicitly. The earliest source that can be interpreted as suggesting that there are infinitely many perfect numbers and they are all even is Nichomachus c. 100 AD, four centuries later. As for the perfect numbers themselves, Pythagoreans were already interested in them two centuries before Euclid. $\endgroup$ – Conifold Dec 15 '17 at 3:36

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