Poisson integral

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

Pseudo sum

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

The complex sequence does not converge uniformly

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

A double Putnam 2016 series

Evaluate the series:

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

(Putnam 2016)

Solution

The group is abelian

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

Not Lebesgue integrable function

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

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

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

Solution

Identity matrix

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

Linear map and trace

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

$\mathbb{R}^2 \rightarrow \mathbb{R}$

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

Solution

A zero determinant

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

Existence of constant

Let $f:[0, 1] \rightarrow \mathbb{R}$ be a continuous function such that $f(0)=0$ and

$$\int_0^1 f(x) \, {\rm d}x = \int_0^1 x f(x) \, {\rm d}x \tag{1}$$

Prove that there exists a $c \in (0, 1)$ such that

$$\int_0^c x f(x) \, {\rm d}x = \frac{c}{2} \int_0^c f(x) \, {\rm d}x$$

Solution

A special integral

For $x \geq 1$ we define $f(x)$ as the unique number $c$ such that $c^c = x $. Evaluate the integral:

$$\mathcal{J}=\int_0^e f(x) \, {\rm d}x$$

(Euler Competition 2013)

Solution

All distances are integer

Prove that for every $n \geq 3$ there exist $n$ points on the plane , not all colinear, such that the distance between any of them is actually an integer number.

Solution

A very interesting integral

Evaluate the integral:

$$\mathcal{J} =\int_0^1 \left( \frac{1}{1-x} + \frac{1}{\ln x} \right) \, {\rm d}x$$

Solution

Least number $n$ so that embeds

Let ${\rm Gl}_2 \left(\mathbb{F_5} \right)$ be the group of invertible $2 \times 2$ matrices over $\mathbb{F}_5$ and $\mathcal{S}_n$ be the group of permutations of $n$ objects. What is the least $n \in \mathbb{N}$ such that there is an embedding of ${\rm Gl}_2 \left(\mathbb{F_5} \right)$ into $\mathcal{S}_n$ ?

Solution

Another Fibonacci series

Let $F_n$ denote the $n$ -th Fibonacci number. Evaluate the sum:

$$\mathcal{S} = \sum_{n=0}^\infty \sum_{k=0}^n \frac{F_{2k}F_{n-k}}{10^n}$$

Solution

A series involving Fibanacci

Let $F_n$ denote the $n$-th Fibonacci number with initial values $F_1=F_2=1$. Prove that:

$$\sum_{n=0}^{\infty} \arctan \frac{1}{F_{2n+1}} = \frac{\pi}{2}$$

Solution

Convergence and dyadic numbers

A real number $x$ is said to be dyadic rational provided there is an integer $k$ and a non negative integer $n$ for which $\displaystyle x=\frac{k}{2^n}$ . For each $x \in [0, 1]$ and each $n \in \mathbb{N}$ set:

$$f_n(x) = \left\{\begin{matrix} 1 &, & x =\dfrac{k}{2^n} , \; k \in \mathbb{N} \\ 0& , & \text{otherwise} \end{matrix}\right.$$

1. Prove that the dyadic numbers are dense in $\mathbb{R}$.
2. Let $f:[0, 1] \rightarrow \mathbb{R}$ be the function to which the sequence $\{f_n\}_{n \in \mathbb{N}}$ converges pointwise. Prove that $\bigintsss_0^1 f(x) \, {\rm d}x$ does not exist.
3. Show that the convergence $f_n \rightarrow f$ is not uniform.

Solution

An evaluation of integral with unknown $f$

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function in $[0, 1]$, strictly monotonic and $f(0)=1$ . If forall $x \in \mathbb{R}$ holds $f\left( f(x) \right)=x$ then evaluate the integral

$$\mathcal{J} = \int_0^1 \left(x - f(x) \right)^{2016} \, {\rm d}x$$

Solution

Integral and inequality

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a positive real valued and continuous function such that it is periodic of period $T=1$. Prove that:

$$\int_0^1 \frac{f(x)}{f \left(x + \frac{1}{2} \right)}\, {\rm d}x  \geq 1$$

Solution