What is the oldest open problem in mathematics? By old, I am referring to the date the problem was stated.

Browsing Wikipedia list of open problems, it seems that the Goldbach conjecture (1742, every even integer greater than 2 is the sum of two primes) is a good candidate. 

The Kepler conjecture about sphere packing is from 1611 but I think this is finally solved (anybody confirms?). There may still be some open problem stated at that time on the same subject, that is not solved. Also there are problems about cuboids that Euler may have stated and are not yet solved, but I am not sure about that.

A related question: can we say that we have solved all problems 
handed down by the mathematicians from Antiquity?

Existence or nonexistence of odd perfect numbers.
Update: Goes back at least to Nicomachus of Gerasa around 100 AD, according to  J J O'Connor and E F Robertson.  Nichomachus also asked about infinitude of perfect numbers.
(Goes back at least to Descartes 1638 https://mathworld.wolfram.com/OddPerfectNumber.html and arguably all the way back to Euclid.)
