Evaluate the integral:

$$\mathcal{J} = \int_0^\pi \frac{{\rm d}x}{1-2a \cos x + a^2} \quad , \quad \left| a \right| <1$$

$$\mathcal{J} = \int_0^\pi \frac{{\rm d}x}{1-2a \cos x + a^2} \quad , \quad \left| a \right| <1$$

**Solution**This is an open invitation to the new brand site Joy of Mathematics . We have expanded it a bit and of course equipped it a bit with new features. The new team is waiting you over to this new site.

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The Website Team of JoM.

The Website Team of JoM.

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Evaluate the integral:

$$\mathcal{J} = \int_0^\pi \frac{{\rm d}x}{1-2a \cos x + a^2} \quad , \quad \left| a \right| <1$$

**Solution**

$$\mathcal{J} = \int_0^\pi \frac{{\rm d}x}{1-2a \cos x + a^2} \quad , \quad \left| a \right| <1$$

Let $\alpha, \beta $ be positive irrational numbers such that $\displaystyle \frac{1}{\alpha} + \frac{1}{\beta}=1$. Evaluate the (pseudo) sum:

$$\sum_{n=1}^{\infty} \left(\frac{1}{\lfloor n\alpha\rfloor^2}+\frac{1}{\lfloor n\beta\rfloor^2}\right)$$

**Solution**

$$\sum_{n=1}^{\infty} \left(\frac{1}{\lfloor n\alpha\rfloor^2}+\frac{1}{\lfloor n\beta\rfloor^2}\right)$$

Prove that there does not exist a sequence $\{ p_n(z)\}_{n \in \mathbb{N}}$ of complex polynomials such that $p_n(z) \rightarrow \frac{1}{z}$ uniformly on $\mathcal{C}_R=\{ z \in \mathbb{C} \mid \left| z \right| = R\}$.

**Solution**

Evaluate the series:

$$\sum_{k=1}^{\infty}\frac{(-1)^{k-1}}{k}\sum_{n=0}^{\infty}\frac{1}{k2^n+1}$$

*(Putnam 2016)*
**Solution**

$$\sum_{k=1}^{\infty}\frac{(-1)^{k-1}}{k}\sum_{n=0}^{\infty}\frac{1}{k2^n+1}$$

Let $\mathcal{G}$ be a finite group such that $\left ( \left | \mathcal{G} \right | , 3 \right ) =1$. If for the elements $a, \beta \in \mathcal{G}$ holds that

$$\left ( a \beta \right )^3 = a^3 \beta^3$$

then prove that $\mathcal{G}$ is abelian.

**Solution**

$$\left ( a \beta \right )^3 = a^3 \beta^3$$

then prove that $\mathcal{G}$ is abelian.

Let $x \in \mathbb{R}$. Given the series:

\begin{equation} \sum_{n=2}^{\infty} \frac{\sin nx}{\ln n} \end{equation}

**Solution**

\begin{equation} \sum_{n=2}^{\infty} \frac{\sin nx}{\ln n} \end{equation}

- Prove that $(1)$ converges forall $x \in \mathbb{R}$.
- Prove that $(1)$ is not a Fourier series of a Lebesgue integrable function.

Let $A \in \mathcal{M}_3 \left( \mathbb{R} \right)$ such that $\det A =1$ and ${\rm tr} (A)= {\rm tr} (A^{-1})=0$. Prove that $A^3=\mathbb{I}_{3 \times 3}$.

**Solution**

Let $f:\mathbb{F}^{n \times n} \rightarrow \mathbb{F}$ be a linear map such that $f\left ( AB \right ) = f \left ( BA \right )$ forall $A, B \in \mathbb{F}^{n \times n}$. Prove that there exists a $\kappa \in \mathbb{F}$ such that $f\left ( A \right ) = \kappa \;{\rm tr} \left ( A \right )$ forall $A \in \mathbb{F}^{n \times n}$.

**Solution**

Prove that there does not exist an $1-1$ and continuous mapping from $\mathbb{R}^2$ to $\mathbb{R}$.

**Solution**

Let $A \in \mathcal{M}_n \left( \mathbb{C} \right)$ with $n \geq 2$ such that

$$\det \left ( A+X \right )=\det A + \det X$$

for every matrix $X \in \mathcal{M}_n \left( \mathbb{C} \right)$. Prove that $A=\mathbb{O}$.

**Solution**

$$\det \left ( A+X \right )=\det A + \det X$$

for every matrix $X \in \mathcal{M}_n \left( \mathbb{C} \right)$. Prove that $A=\mathbb{O}$.

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