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Let $n \in \mathbb{Z}$ such that $n \geq 2$. Let $\mathcal{S}_n$ be the permutation group on $n$ letters and $\mathcal{A}_n$ be the alternating group. We also denote $\mathbb{C}^*$ the group of non zero complex numbers under multiplication. Which of the following are correct statements? For every in...

We are aware of the Fresnel integral \begin{equation} \int_0^\infty \sin x^2 \, {\rm d}x = \frac{1}{2} \sqrt{\frac{\pi}{2}} \end{equation} The most common proof goes with complex analysis. Try to provide a proof with Real Analysis. There are at least $2$ proofs. The one is more elegant than the othe...

Denote by $\bar{\alpha}=0.\alpha \alpha \alpha \dots $. Find all $\alpha>0$ such that

$$\frac{1}{\alpha} = 0.\bar{a}$$

Let $p_n$ denote the $n$ -th prime number. Evaluate the sum

$$\mathcal{S} = \sum_{n=1}^{\infty} \frac{ \log p_n}{p_n^2 -1}$$

Since $\lim \limits_{x \rightarrow +\infty} f(x) =0$ the result follows immediately by making the change of variables $u=nx$ .

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Let

$$f(x) = \sin x \sin (2x) \sin (4x) \cdots \sin (2^n x)$$

Prove that

$$\left| {f(x)} \right| \le \frac{2}{{\sqrt 3 }}\left| {f(\frac{\pi }{3})} \right|$$

Thank you Papapetros Vaggelis. My solution is as follows. Since $\frac{1}{x} \; , \; \frac{1}{y} \; , \; \frac{1}{z} >0$ then the numbers \[\sqrt{\frac{1}{x} + \frac{1}{y}} \; , \; \sqrt{\frac{1}{x} +\frac{1}{z}} \; , \; \sqrt{\frac{1}{y} + \frac{1}{z}}\] could be sides of a triangle. The area of th...

Let $x, y,z >0$ satisfying $x+y+z=1$. Prove that

\[\frac{1}{x} + \frac{1}{y} + \frac{1}{z} \geq \sqrt{\frac{3}{xyz}}\]

Let $n \in \mathbb{N}$ and $\mathcal{B}_n$ denote the $n$ - th Bell number. Prove that

$$\sum_{k=0}^{\infty} \frac{k^n}{k!}=\mathcal{B}_n \cdot e$$