Not a HW problem, but a "me re-thinking things" problem. Please tell me where my thinking is flawed:
You have an ice skater with no net external torques acting on him/her. (We are analyzing the time after they have to get an external torque on them by pushing off of the ground, and we are ignoring air resistance/friction slowing them back down.) Thus, their angular momentum is conserved. Thus, when they decrease their moment of inertia by moving their body in such a manner that more of their mass is closer to the axis of rotation (i.e., they pull their arms in towards their body), their angular velocity must increase, and vice versa.
But a change in angular velocity means that there must be an angular acceleration. If the ice skater then both a) has a moment of inertia about that axis of rotation, and b) is undergoing an angular acceleration of some magnitude, then, mathematically, there must be an external torque acting on them, right? But we're analyzing the problem under the assumption that the external torque is zero...I'm confused.
Conservation of angular momentum of the ice skater about the vertical axis of rotation that runs through their center of mass. (L = I*w.) The time-derivative of angular momentum equaling zero as the net external torque on the ice-skater system equals zero.