MathDB

a²+b²+c²<=2

Source: G.M. 10/2011

11/25/2019

Let be three real numbers

a,b,c\in [0,1]

satisfying the condition

ab+bc+ca=1.

Prove that

a^2+b^2+c^2\le 2,

and determine the cases in which equality is attained.

inequalitiesalgebra

Find f(m+n)-f(n)+f(m)+2mn

Source:

11/25/2019

Find the functions

f:\mathbb{Z}\longrightarrow\mathbb{Z}_{\ge 0}

that satisfy the following two conditions:
\text{(a)} f(m+n)=f(n)+f(m)+2mn, \forall m,n\in\mathbb{Z}

\text{(b)} f(f(1))-f(1)

is a perfect square
Marin Ionescu

functionFind all functionsalgebra

A^m=B^n implies [A,B]=0

Source:

11/25/2019

Let be two

2\times 2

real matrices

A,B

having the property that all their natural powers are not real multiples of the identity. Prove that if some natural power of

A

is equal to some natural power of

B,

then,

A,B

commute. Is the converse statement true?
Cosmin Nitu

linear algebra

N. o pairs (x,y) s.t. x²=y³ in a group of order 2011

Source:

11/25/2019

Find the number of pairs of elements, from a group of order

2011,

such that the square of the first element of the pair is equal to the cube of the second element.
Teodor Radu

group theory

1