Is there a simple characterization for an integer partition $(s_1,\dots,s_k)$, such that a polygon with these sides is inscribed in a circle with integer radius?
This is what I got so far: All pythagorean triplets $(\sqrt{a^2+b^2},a,b)$ of course have this property, and if $r=\frac{1}{2}\sqrt{a^2+b^2}$ is integer, then the 5-tuple $(a,b,r,r,r)$ has this property, since the diameter is $2r$.
Note that this question is equivalent to asking when the equation $$ \cos\left(\sum_{j=1}^k \arccos\left( 1 - \frac{a_j²}{2r^2}\right) \right) = 1 $$ has an integer solution $r\geq 1$.