# 2022-19 Inequality for twice differentiable functions

Let $$f : \mathbb{R} \to \mathbb{R}$$ be a twice differentiable function satisfying $$f(0) = 0$$ and $$0 \leq f'(x) \leq 1$$. Prove that
$\left( \int_0^1 f(x) dx \right)^2 \geq \int_0^1 [f(x)]^3 dx.$

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# 2022-13 Inequality involving sums with different powers

Prove for any $$x \geq 1$$ that

$\left( \sum_{n=0}^{\infty} (n+x)^{-2} \right)^2 \geq 2 \sum_{n=0}^{\infty} (n+x)^{-3}.$

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# 2021-10 Integral inequality

Let $$f: [0, 1] \to \mathbb{R}$$ be a continuous function satisfying
$\int_x^1 f(t) dt \geq \int_x^1 t\, dt$
for all $$x \in [0, 1]$$. Prove that
$\int_0^1 [f(t)]^2 dt \geq \int_0^1 t f(t) dt.$

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# 2020-10 An inequality with sin and log

Prove that
$\frac{x+\sin x}{2} \geq \log (1+x)$
for $$x > -1$$.

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# 2019-07 An inquality

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

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# 2018-04 An inequality

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 $$x_1x_2+x_2x_3+x_3x_4+\cdots + x_{n-1}x_n+x_nx_1$$?

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# 2016-23 Inequality on complex numbers

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 $$z_{n+1} = z_1$$.

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# 2015-13 Minimum

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

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# 2015-10 Product of sine functions

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 $\sin \left(\prod_{i=1}^n x_i^{w_i} \right) \ge \prod_{i=1}^n (\sin x_i)^{w_i}.$

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Prove that, for any sequences of real numbers $$\{ a_n \}$$ and $$\{ b_n \}$$, we have
$\sum_{m=1}^{\infty} \sum_{n=1}^{\infty} \frac{a_m b_n}{m+n} \leq \pi \left( \sum_{m=1}^{\infty} a_m^2 \right)^{1/2} \left( \sum_{n=1}^{\infty} b_n^2 \right)^{1/2}$