Stephen's Birthday

Each concentric circle will have a power of 2 number of candles, and each circle will have a different power of 2 number in it. This turns the problem into which integers can be expressed as the sum of different powers of 2, to which the answer is $\boxed{\text{Yes}}$ as all numbers can be written in binary.

Taking liberties with the physical of the ability of the knife to cut infinitely many times, we permit the use of infinite series to solve this problem.

We have $\frac {1}{n-1} \equiv \ \frac{1}{n}+\frac{1}{n^2}+\frac{1}{n^3}+...$ :

This can be derived by a simple proof of convergence and then some simple algebra.

So start by splitting the cake into 128 pieces(possible because 128 is a power of 2) Then split one of those 128 pieces into 128, producing a piece of size 1/(128)^2. Take one of these 128 pieces and do the same. Continue this process forever. Adding all the pieces together, we get 1/127th of the cake.

But then we can take 1/127 of that piece, and so on, till we get to 1/126th of a cake.

This allows us to get pieces of size any n for n<128 by induction.

Actually its not getting pieces it's subdividing the cake, but the same applies.