MathDB

Fractional part property

Source:

12/13/2019

Let be two real numbers

x,y,

and a natural number

n_0

such that

\{ n_0x \} = \{ n_0y \}

and

\{ (n_0+1)x \} = \{ (n_0+1)y \} ,

where

\{\}

denotes the fractional part. Show that

\{ nx \} =\{ ny \} ,

for any natural number

n.

Ovidiu Pop

algebraequationsFloorfractional part

function in C(2)

Source: Nicolae Coculescu contest

11/19/2013

Let

w\in \mathbb{C}\setminus \mathbb{R}

,

|w|\neq 1

. Prove that

f\colon \mathbb{C} \to \mathbb{C}

, given by

f(z)= z+w\overline{z}

, is a bijection, and find its inverse.

functionalgebra proposedalgebra

Integer part of expression dependent on the sides of a triangle

Source:

12/13/2019

Calculate

\left\lfloor \frac{(a^2+b^2+c^2)(a+b+c)}{a^3+b^3+c^3} \right\rfloor ,

where

a,b,c

are the lengths of the side of a triangle.
Costel Anghel

geometryalgebraFloorinequalitiesthree variable inequality

f(x+y)+f(x-y)=f(x)+f(y) +f(f(x+y))

Source:

12/13/2019

Find all functions

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

satisfying the equation

f(x+y)+f(x-y)=f(x)+f(y) +f(f(x+y)) ,

for any rational numbers

x,y.

Mihai Onucu Drîmbe

functionFind all functionsalgebra

Matrices A s.t. tr(A)=tr(A²)=0

Source:

12/13/2019

Let

\mathbb{K}

be a field and let be a matrix

M\in\mathcal{M}_3(\mathbb{K} )

having the property that

\text{tr} (A) =\text{tr} (A^2) =0 .

Show that there is a

\mu\in \mathbb{K}

such that

A^3=\mu A

or

A^3=\mu I.

Cristinel Mortici

linear algebramatrix

Interesting abelian subgroup of the complex numbers

Source:

12/13/2019

Let be the set

G=\{ (u,v)\in \mathbb{C}^2| u\neq 0 \}

and a function

\varphi :\mathbb{C}\setminus\{ 0\}\longrightarrow\mathbb{C}\setminus\{ 0\}

having the property that the operation

*:G^2\longrightarrow G

defined as

(a,b)*(c,d)=(ac,bc+d\varphi (a))

is associative.
a) Show that

(G,*)

is a group. b) Describe

\varphi ,

knowing that

(G,*)

is a commutative group.
Marius Perianu

functiongroup theoryabstract algebracomplex numbers

1

Problems(6)