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