2025 Cayley Problem 24

A circle has area A, a regular octagon has area \frac{1}{2}A, and a regular dodecagon has area \frac{1}{2}A. The circle has radius 3000. The distance from the centre of the octagon to each of its vertices is x, as shown. Also, the distance from the centre of the dodecagon to each of its vertices is y.

What is the integer closest to x - y?

(A regular polygon is a polygon all of whose side lengths are equal and all of whose interior angles are equal. An octagon has 8 sides and a dodecagon has 12 sides.)

If there are no answer choices shown, enter a numerical answer.

Full credit to this problem is given to the CEMC, you may view all Cayley contests here.