ANSWER
The "Geometric Problems of Antiquity" were three problems the ancient Greeks attempted to solve using only a compass and a straightedge. Unfortunately for Greek mathematicians, these tasks were impossible. (Which is to say, impossible given the constraint of only using a compass and a straightedge.) Nevertheless, mathematicians attempted for thousands of years to solve them, only because, while they seemed impossible, nobody knew for sure whether they really were impossible or not. Hope springs eternal. It wasn't until the 19th century that mathematicians were able to prove that these three problems were unsolvable, at least unsolvable using only a compass and a straightedge, thus freeing mathematicians to concentrate on bigger problems (like contemplating the number of angels that can dance on the head of a pin). The three problems are:
1) Circle Squaring - Construct a square with the same area as a given circle.
2) Cube Duplication - Construct a cube with double the volume of a given cube.
3) Angle Trisection - Divide any given angle into three equal angles.
If you are wondering why they are unsolvable, it is because the solution to each involves an irrational number, and irrational numbers do not lend themselves to rational solutions. There is a more exact explanation involving pi, the cube root of two, polynomial equations, an alter at Delos, and other such things. However, unless you hold a graduate degree in mathematics, chances are these explanations will be somewhat difficult to understand. (At least they were for me.)
For more information: http://mathworld.wolfram.com/GeometricProblemsofAntiquity.html
WHO GOT IT RIGHT: Kristy Morgan, Kathi Reynolds, JP Weigt, Robert Walker, Ryan Scannell, Marc Quinlivan, Bob Milligan, Robin Campbell, Marika Thiessen, Andie J, Brian Bement, Arne Haaning, Christine Severen, Mark Collins, Luz Velasco, and Brian Ashby.
