Let \( f: [0, \infty) \to \mathbb{R} \) be a function satisfying the following conditions:
(1) For any \( x, y \geq 0 \), \( f(x+y) \geq f(x)+f(y) \).
(2) For any \( x \in [0, 2] \), \( f(x) \geq x^2 – x \).
Prove that, for any positive integer \( M \) and positive reals \( n_1, n_2, \cdots, n_M \) with \( n_1 + n_2 + \cdots + n_M = M \), we have
\[ f(n_1) + f(n_2) + \cdots + f(n_M) \geq 0. \]
Determine all positive integers \(\ell\) such that \[ \sum_{n=1}^\infty \frac{n^3}{(n+1)(n+2)(n+3)\cdots (n+\ell)}\] converges and if it converges, then compute its value.
Let \( f \) be a real-valued continuous function on \( [ 0, 1] \). For a positive integer \( n \), define
\[
B_n(f; x) = \sum_{j=0}^n f( \frac{j}{n}) {n \choose j} x^j (1-x)^{n-j}.
\]
Prove that \( B_n (f; x) \) converges to \( f \) uniformly on \( [0, 1 ] \) as \( n \to \infty \).
Find all polynomials \( P(x) = a_n x^n + \cdots + a_1 x + a_0 \) satisfying (i) \( a_n \neq 0 \), (ii) \( (a_0, a_1, \cdots, a_n) \) is a permutation of \( (0, 1, \cdots, n) \), and (iii) all zeros of \( P(x) \) are rational.
Find all field automorphisms of the field of real numbers \( \mathbb{R} \). (A field automorphism of a field \( F \) is a bijective map \( \sigma : F \to F \) that preserves all of \( F \)’s algebraic properties.)
Let \( f(z) = z + e^{-z} \). Prove that, for any real number \( \lambda > 1 \), there exists a unique \( w \in H = \{ z \in \mathbb{C} : \text{Re } z > 0 \} \) such that \( f(w) = \lambda \).
Let \( A, B, C = A+B \) be \( N \times N \) Hermitian matrices. Let \( \alpha_1 \geq \cdots \geq \alpha_N \), \( \beta_1 \geq \cdots \geq \beta_N \), \( \gamma_1 \geq \cdots \geq \gamma_N \) be the eigenvalues of \( A, B, C \), respectively. For any \( 1 \leq i, j \leq N \) with \( i+j -1 \leq N \), prove that
\[ \gamma_{i+j-1} \leq \alpha_i + \beta_j \]
Suppose that a function \( f:[0, 1] \to (0, \infty) \) satisfies that
\[ \int_0^1 f(x) dx = 1. \]
Prove the following inequality.
\[ \left( \int_0^1 |f(x)-1| dx \right)^2 \leq 2 \int_0^1 f(x) \log f(x) dx. \]
Let \( R \) be a ring of characteristic zero. Assume further that \( na \neq 0 \) for a positive integer \( n \) and \( a \in R \) unless \( a = 0 \). Suppose that \( e, f, g \in R \) are idempotent (with respect to the multiplication) and satisfy \( e + f + g = 0 \). Show that \( e = f = g = 0 \). (An element \( a \) is idempotent if \( a^2 = a \). )
A real sequence \( x_1, x_2, x_3, \cdots \) satisfies the relation \( x_{n+2} = x_{n+1} + x_n \) for \( n = 1, 2, 3, \cdots \). If a number \( r \) satisfies \( x_i = x_j = r \) for some \( i \) and \( j \) \( (i \neq j) \), we say that \( r \) is a repeated number in this sequence. Prove that there can be more than \( 2013 \) repeated numbers in such a sequence, but it is impossible to have infinitely many repeated numbers.