Denote \(P = \{(x, y, z) \in \mathbb{R^3}: 10< x,y,z <31\}\). Suppose a function \(f (v): \mathbb{R^3} \to \mathbb{R_{\geq 0}}\) satisfies:
(a) \(f(\lambda v) = \lambda^{25} f(v)\) for all \(v\in P\) and \(0<\lambda \in \mathbb{R}\),
(b) \(f(v+w) \geq f(v)\) for every \(v, w \in P\),
(c) \(f (v)\) is locally bounded.
Show that \(f (v)\) is locally Lipschitz in \(P\).

For given \(a, b \in \mathbb{R}\) and \(c \in \mathbb{Z}\), find all function \(f: \mathbb{R} \to \mathbb{R}\) which is continuous at 0 and satisfies
\[
f(ax) = f(bx) + x^c \quad \forall x\in \mathbb{R}\setminus \{0\}.
\]

We write \(tx = (tx_0,…,tx_5)\) for \(x=(x_0,…,x_5)\in \mathbb{R^{6}}\) and \(t\in \mathbb{R}\). Find all real multivariate polynomials \(P(x)\) in \(x\) satisfying the following properties:
(a) \(P(tx) = t^d P(x)\) for all \(t\in \mathbb{R}\) and \(x\in \mathbb{R}^{6}\), where \(0\leq d \leq 15\) is an integer;
(b) \(P(x) =0\) if \(x_i = x_j\) with \(i\neq j\).

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

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 \(g'(0)\).

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 \(f_n(t)\), \(n=1,2…\) 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 \(\lim_{n\to \infty} f_n'(0)\).

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  \({\rm Re} (P’(w)) = 0\).

Can we find a sequence \(a_i, i=0,1,2,…\) with the following property: for each given integer \(n\geq 0\), we have \[\lim_{L\to +\infty}\sum_{i=0}^L 2^{ni} |a_i|\leq 23^{(n+11)^{10}} \quad \text{ and }\quad \lim_{L\to +\infty}\sum_{i=0}^L 2^{ni} a_i = (-1)^n ?\]