Prove that

\[

\frac{x+\sin x}{2} \geq \log (1+x)

\]

for \( x > -1 \).

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Prove that

\[

\frac{x+\sin x}{2} \geq \log (1+x)

\]

for \( x > -1 \).

Suppose that \( f: \mathbb{R} \to \mathbb{R} \) is differentiable and \( \max_{ x \in \mathbb{R}} |f(x)| = M < \infty \). Prove that \[ \int_{-\infty}^{\infty} (|f'|^2 + |f|^2) \geq 2M^2. \]

Let \(x_1,x_2,\ldots,x_n\) be reals such that \(x_1+x_2+\cdots+x_n=n\) and \(x_1^2+x_2^2+\cdots +x_n^2=n+1\). What is the maximum of \(x_1x_2+x_2x_3+x_3x_4+\cdots + x_{n-1}x_n+x_nx_1\)?

Suppose that \( z_1, z_2, \dots, z_n \) are complex numbers satisfying \( \sum_{k=1}^n z_k = 0 \). Prove that

\[

\sum_{k=1}^n |z_{k+1} – z_k|^2 \geq 4 \sin^2 \left( \frac{\pi}{n} \right) \sum_{k=1}^n |z_k|^2,

\]

where we let \( z_{n+1} = z_1 \).

Find the minimum value of

\[

\int_{\mathbb{R}} f(x) \log f(x) dx

\]

among functions \(f: \mathbb{R} \rightarrow \mathbb{R}^+ \cup \{ 0 \}\) that satisfy the condition

\[

\int_{\mathbb{R}} f(x) dx = \int_{\mathbb{R}} x^2 f(x) dx = 1.

\]

Let \(w_1,w_2,\ldots,w_n\) be positive real numbers such that \( \sum_{i=1}^n w_i=1\). Prove that if \(x_1,x_2,\ldots,x_n\in [0,\pi]\), then \[ \sin \left(\prod_{i=1}^n x_i^{w_i} \right) \ge \prod_{i=1}^n (\sin x_i)^{w_i}.\]

Prove that, for any sequences of real numbers \( \{ a_n \} \) and \( \{ b_n \} \), we have

\[

\sum_{m=1}^{\infty} \sum_{n=1}^{\infty} \frac{a_m b_n}{m+n} \leq \pi \left( \sum_{m=1}^{\infty} a_m^2 \right)^{1/2} \left( \sum_{n=1}^{\infty} b_n^2 \right)^{1/2}

\]

Suppose that \( a_1, a_2, \cdots \) are positive real numbers. Prove that

\[

\sum_{n=1}^{\infty} (a_1 a_2 \cdots a_n)^{1/n} \leq e \sum_{n=1}^{\infty} a_n \,.

\]

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 \( N > 1000 \) be an integer. Define a sequence \( A_n \) by

\[

A_0 = 1, \, A_1 = 0, \, A_{2k+1} = \frac{2k}{2k+1} A_{2k} + \frac{1}{2k+1} A_{2k-1}, \, A_{2k} = \frac{2k-1}{2k} \frac{A_{2k-1}}{N} + \frac{1}{2k} A_{2k-2}.

\]

Show that the following inequality holds for any integer \( k \) with \( 1 \leq k \leq (1/2) N^{1/3} \).

\[

A_{2k-2} \leq \frac{1}{\sqrt{(2k-2)!}}.

\]