Prove or disprove that every homomorphism \( \pi_1(X) \to \pi_1(X)\) can be realized as the induced homomorphism of a continuous map \(X \to X\).

Let \(g(t): [0,+\infty) \to [0,+\infty)\) be a decreasing continuous function. Assume \(g(0)=1\), and for every \(s, t \geq 0 \) \[t^{11}g(s+t) \leq 2024 \; [g(s)]^2.\] Show that \(g(11) = g(12)\).

Let \( f(n) \) denote the number of possible sequences of length \( n \), where each term is either \(0, 1,\) or \(-1\), such that the product of every three consecutive numbers is nonnegative. Compute \( f(33)\).

Let \(A= [a_{ij}]_{1\leq i,j\leq 5}\) be a \(5\times 5\) positive definite (real) matrix. Show that the matrix \([a_{ij}/(i+j)]\) is also positive definite.

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 \(g'(0)\).