MathDB

Hermite-like identity

Source:

12/14/2019

Let be a natural number

n\ge 2.

Find the real numbers

a

that satisfy the equation

\lfloor nx \rfloor =\sum_{k=1}^{n} \lfloor x+(k-1)a \rfloor ,

for any real numbers

x.

Marius Perianu

floor functionalgebraequations

Sequence defined with powers of the order $ 1/n $

Source:

12/14/2019

Let bet a sequence

\left( a_n \right)_{n\ge 1}

with

a_1=1

and defined as

a_n=\sqrt[n]{1+na_{n-1}} .

Show that

\left( a_n \right)_{n\ge 1}

is convergent and determine its limit.
Florian Dumitrel

real analysisSequenceslimits

trigonometric and ln equation

Source:

12/14/2019

Solve in the real numbers the equation:

\cos^2 \frac{(x-2)\pi }{4} +\cos\frac{(x-2)\pi }{3} =\log_3 (x^2-4x+6)

Gheorghe Mihai

quadraticsmonotonyequationsalgebratrigonometrylogarithms

Nice primitivability result

Source:

12/14/2019

Consider a function

f:\mathbb{R}\longrightarrow\mathbb{R}

that admits bounded primitives. Prove that the function

g:\mathbb{R}\longrightarrow\mathbb{R}

defined as

f(x)=\left\{ \begin{matrix} x, & &emsp; x\le 0 \\ f(1/x)\cdot\ln x ,& &emsp; x>0 \end{matrix}\right.

admits primitives.
Florian Dumitrel

functionprimitivesreal analysis

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