Let \(A\) be a \(16 \times 16\) matrix whose entries are either \(1\) or \(-1\). What is the maximum value of the determinant of \(A\)?

Let \(f_n(t)\), \(n=1,2…\) be a sequence of concave functions on \(\mathbb{R}\). Assume \(\liminf_{n\to\infty} f_n(t) \geq 2024\,t^{5}+3\) for \(t\in [-1, 1]\) and \(\lim_{n\to \infty} f_n(0) = 3\). Suppose \(f_n'(0)\) exist for \(n=1,2,…\). Compute \(\lim_{n\to \infty} f_n'(0)\).

A complex number \(z \in S^1 \smallsetminus \{1\} \) is called a Knotennullstelle if there exists a Laurent polynomial \(p(t) \in \mathbb{Z} [t,t^{-1}]\) such that \(p(1) =\pm 1\) and \(p(z)=0\). Prove that the collection of all Knotennullstelle numbers is a discrete subset of \(\mathbb{C}\).

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 \(\{ 1,2, \ldots, 15 \}\)?

Consider a function \(f: \{1,2,\dots, n\}\rightarrow \mathbb{R}\) satisfying the following for all \(1\leq a,b,c \leq n-2\) with \(a+b+c\leq n\).

\[ f(a+b)+f(a+c)+f(b+c) – f(a)-f(b)-f(c)-f(a+b+c) \geq 0 \text{ and } f(1)=f(n)=0.\]

Prove or disprove this: all such functions \(f\) always have only nonnegative values on its domain.

Acknowledgement: This problem arises during a research discussion between June Huh, Jaehoon Kim and Matt Larson.

Does there exist a nontrivial subgroup \(G\) of \( GL(10, \mathbb{C}) \) such that each element in \(G\) is diagonalizable but the set of all the elements of \(G\) is not simultaneously diagonalizable?