JoM Journal

Let $p$ be a prime and let $\mathbb{F}_p=\mathbb{Z}_p$. Find all $p \times p$ matrices $A$ and $B$ over $\mathbb{F}_p$ such that $AB - BA = \mathbb{I}$.
Let $A$ be a complex matrix with real determinant. Prove that $A$ is the product of $4$ Hermitian matrices.
Let $A \in \mathcal{M}_{n} \left( \mathbb{R} \right)$ be an invertible matrix. Prove that $$\det A = \frac{1}{n!}\begin{vmatrix} \mathrm{tr}(A) & 1 &0 & 0&\cdots &0\\ \mathrm{tr}(A^2) & \mathrm{tr}(A) &2 & 0&\cdots &0\\ \mathrm{tr}(A^3) & \mathrm{tr}(A^2) &\mathrm{tr}(A) & 3&\cdots &0\\ \vdots & \vdots & \vdots & \ddots & \ddots & \vdots\\ \mathrm{tr}(A^{n-1}) & \mathrm{tr}(A^{n-2}) & \cdots & \cdots & \mathrm{tr}(A) &n-1\\ \mathrm{tr}(A^{n}) & \mathrm{tr}(A^{n-1}) &\mathrm{tr}(A^{n-2}) & \cdots & \cdots & \mathrm{tr}(A) \\ \end{vmatrix}$$
Let $n \in \mathbb{N}$ and $1 \leq k \leq n$. Evaluate the determinant $$\mathcal{D} = \begin{vmatrix} \binom{n}{0} & \binom{n}{1} & \cdots & \binom{n}{k}\\ \binom{n+1}{0} & \binom{n+1}{1} & \cdots & \binom{n+1}{k} \\ \vdots &\vdots & \ddots & \vdots \\ \binom{n+k}{0} & \binom{n+k}{1} & \cdots & \binom{n+k}{k} \end{vmatrix}$$
Prove that $\displaystyle \int_0^{\infty}\frac{\arctan x -x e^{-x}}{x^2}\,\mathrm{d}x=1+\gamma$ where $\gamma$ denotes the Euler – Mascheroni constant.
Let $a, b>0$ and $d = \gcd(a, b)$. Prove that $$\int_{0}^{1} \left(\{ax\}-\frac{1}{2}\right)\left(\{bx\}-\frac{1}{2} \right) \, \mathrm{d}x=\frac{d^2}{12ab}$$
Let $n \in \mathbb{N}$. Prove that $$\int_{0}^{1} \frac{\ln^n \frac{1+x}{1-x}}{\sqrt{1-x^2}} \frac{\mathrm{d}x}{1+x} = 2^n \Gamma(n+1)$$ where $\Gamma$ denotes the Euler's Gamma function.
Let $\alpha \in \mathbb{R}$. Prove that $\displaystyle \sum_{n=1}^\infty 2^{2n}\sin^4\frac a{2^n}=a^2-\sin^2a$.
Let $r_n$ be a sequence of all rational numbers in $(0,1)$. Show that the series $\sum \limits_{n=2}^\infty|r_n-r_{n-1}|$ diverges.
Let $\gamma_n = \mathcal{H}_n - \ln n$. Evaluate the limit $\ell = \lim \limits_{n \rightarrow +\infty} n \left( \gamma_n - \gamma \right)$.
For $\alpha \in \left(0, \frac{\pi}{2} \right)$ prove that $\displaystyle \alpha > \frac{3\sin \alpha}{2 +\cos \alpha}$.
For $x \geq 0$ prove that $2 \sinh x + \tanh x \geq 3x$.
Prove that in any triangle the following inequality $\displaystyle \sum \tan^2 \frac{A}{2} \geq 2 - 8 \prod \sin \frac{A}{2}$ holds.
Prove the following double inequality, where the sum and product are cyclic over the angles $A, B, C$ of a triangle $$\sum \sin^2 A \leq 2 + 16 \prod \sin^2 \frac{A}{2} \leq \frac{9}{4}$$
Prove that in any triangle $ABC$ the following double inequality $$\frac{4}{9} \sum \sin B \sin C \leq \prod \cos \frac{B-C}{2} \leq \frac{2}{3} \sum \cos A$$ holds.

Solutions should be submitted at the e-mail tolaso at tolaso.com.gr and will be published in the next edition. Solutions should be submitted by February 28 , 2021. Solutions should be written in $\mathrm{\LaTeX}$ . Code should be clear and easy reading. Finally, all solutions should be concise.