What is the oldest open problem in mathematics? 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?
 A: This is not older than the rest, but old enough I believe: In 1775 Fagnano constructed periodic orbits for acutangular triangular billiards. The question about the existence of periodic orbits in general triangular (or polygonal) billiards (in the case of irrational angles) remains open. (
Troubetzkoy, Serge, Dual billiards, Fagnano orbits, and regular polygons, Am. Math. Mon. 116, No. 3, 251-260 (2009). arXiv:0704.0390
, jstor. ZBL1229.37033, MR2491981. ).
A: 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.)
A: The Congruent Number problem (Which integers are the areas of right triangles with rational sides?) dates back to an Arab manuscript written before 972 AD, according to https://www.jstor.org/stable/2320381.
A: Another unsolved problem from ancient Greek times is: which regular $n$-gons are constructible by ruler and compass? We know, since Gauss, that this problem reduces to finding all the Fermat primes, but we don't know that we have found them all yet.
A: The Perfect Cuboid problem was being considered in the early 18th century (according to https://mathworld.wolfram.com/EulerBrick.html )
I don't know if any ancient Greeks are on record as having considered the problem; but that doesn't seem beyond the bounds of possibility, although I would guess that in those times they were preoccupied mostly with 2-D problems.
A: Albrecht Dürer's conjecture states that every convex polytope has a non-overlapping edge unfolding (see here for the intro).  This problem was raised in 1525, revived by Shephard in 1975, and remains wide open.  
A: Not exactly what you are asking for, but a candidate for the longest time elapsing between the proposal and the solution of a problem: the Archimedes cattle problem, proposed by Archimedes and solved by A. Amthor in 1880. See https://en.wikipedia.org/wiki/Archimedes%27s_cattle_problem
A: According to Encyclopaedia Britannica,
"Greek mathematician Euclid (flourished c. 300 bce) gave the oldest known proof that there exist an infinite number of primes, and he conjectured that there are an infinite number of twin primes,"
which makes the twin prime conjecture remarkably old.
A: Zeno's paradoxes are among the oldest puzzles at the intersection of mathematics, philosophy, and physics (in alphabetical order). The traditional resolution of Zeno's paradoxes of motion involves modeling them in terms of the real line and interpreting the iterated procedure as an infinite series.
As pointed out in one of the comments, Heisenberg's uncertainty principle provides another way of accounting for the puzzle, by arguing that it has no physical meaning.
H. Jerome Keisler in his article "The hyperreal line" (207–237) in the collection
Real numbers, generalizations of the reals, and theories of continua. Edited by Philip Ehrlich. Synthese Library, 242. Kluwer Academic Publishers Group, Dordrecht, 1994
provides a different mathematical resolution of the puzzle in terms of the hyperreal continuum.
More recently (2013), Terry Tao notes the mathematical significance of these paradoxes by noting that they "make the important point that real analysis cannot be reduced to a branch of discrete mathematics, but requires additional tools in order to deal with the continuum" (see https://mathscinet.ams.org/mathscinet-getitem?mr=3026767).
In a review of Graham Oppy's book, John H. Mason makes the following intriguing comment, indicative of the richness of the issues involved: Have you ever briefly called upon Zeno's paradoxes when introducing the notion of limit to students? For example, the fact that Achilles really does catch the tortoise is only because he crosses distances halving in length in intervals of time also halving in length; the arrow does actually get to its target, even though it has to surmount an infinite number of decreasingly small intervals. This book addresses these and many other paradoxes involving infinitely large and infinitely small quantities with philosophical precision and reasoning. It reveals that there are much larger issues at stake than are perhaps commonly recognised, and certainly than are `dismissed' with the Cauchy-Weierstrass formalism of limits. See https://mathscinet.ams.org/mathscinet-getitem?mr=2238333
A: What about the time which elapsed between the question of squaring the circle ( which I was always taught was posed by the Greeks), and the proof that $\pi$ is transcendental in 1882(?) by Lindemann- admittedly not now an open problem, but an impressive time lapse.  
