Is it possible to arrange the numbers \(1, 2, 3, \ldots, 2024\) in a sequence such that the difference between any two adjacent numbers is greater than \(1\) but less than \(4\)?

Let \(u_n(t)\), \(n=1,2…\) be a sequence of concave functions on \(\mathbb{R}\). Let \(g(t)\) be a differentiable function on \(\mathbb{R}\). Assume \(\liminf_{n\to\infty} u_n(t) \geq g(t)\) for every \(t\) and \(\lim_{n\to \infty} u_n(0) = g(0)\). Suppose \(u_n'(0)\) exist for \(n=1,2,…\). Compare \(\lim_{n\to \infty} u_n'(0)\) and \(g'(0)\).

Count the number of distinct matrices \( A \), where two matrices are considered identical if one can be obtained from the other by rearranging rows and columns, that have the following properties:

1. \( A \) is a \( 7 \times 7 \) matrix and every entry of \( A \) is \( 0 \) or \( 1 \).
2. Each row of \( A\) contains exactly 3 non-zero entries.
3. For any two distinct rows \( i\) and \( j\) of \( A\), there exists exactly one column \( k \) such that \( A_{ik} \neq 0 \) and \( A_{jk} \neq 0 \).

Find all positive numbers \(a_1,…,a_{5}\) such that \(a_1^\frac{1}{n} + \cdots + a_{5}^\frac{1}{n}\) is integer for every integer \(n\geq 1.\)

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 \(\lim_{n\to \infty} f_n'(0)\).