Let \(P(z) = z^3 + c_1 z^2 + c_2 z+ c_3\) be a complex polynomial in \(\mathbb{C}\). Its complex derivative is given by \(P’(z) = 3z^{2} +2c_1z+c_{2}.\) Assume that there exist two points a, b in the open unit disc of complex plane such that P(a) = P(b) =0. Show that there is a point w belonging to the line segment joining a and b such that \({\rm Re} (P’(w)) = 0\).
A permutation \(\phi \colon \{ 1,2, \ldots, n \} \to \{ 1,2, \ldots, n \}\) is called a well-mixed if \(\phi (\{1,2, \ldots, k \}) \neq \{1,2, \ldots, k \}\) for each \(k<n\). What is the number of well-mixed permutations of \(\{ 1,2, \ldots, 15 \}\)?
Suppose that we roll \(n\) (6-sided, fair) dice. Let \(S_n\) be the sum of their faces. Find all positive integers \(k\) such that the probability that \(k\) divides \(S_n\) is \(1/k\) for all \(n \geq 1\).