Good problems (or at least problems I enjoy)

Let a \(k\)-cycle of an ordered list \( (a_{1}, a_{2}, \dots, a_{n}) \) be the ordered list \( (a_{n-k+1}, \dots, a_{n}, a_{1}, a_{2}, \dots, a_{n-k}) \). Given an ordered list of \(n\) numbers, determine in linear time how many of the \(k\)-cycles are such that all \(n\) partial sums are nonnegative.
Five points are placed on a sphere. What is the largest \(N\) such that no matter where the points are placed, there is a closed hemisphere containing \(N\) of the points?
There are \(n\) coins arranged in a line from left to right. If there are exactly \(k > 0\) coins showing heads, we flip the \(k\)th coin from the left; otherwise, we stop. Show that for any initial configuration of the coins, we will stop after a finite number of flips.