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Part of 2010 Laurențiu Panaitopol, Tulcea

Problems(4)

Find natural function

Source: Gazeta matematică

Find the strictly monotone functions

f:\{ 0\}\cup\mathbb{N}\longrightarrow\{ 0\}\cup\mathbb{N}

that satisfy the following two properties:

\text{(i)} f(2n)=n+f(n),

for any nonnegative integers

n.

\text{(ii)} f(n)

is a perfect square if and only if

n

is a perfect square.

functionFind all functionsalgebra

Injective, but not surjective

Source: Gazeta matematică

Let be a nonnegative integer

n

\sqrt n

is not integer. Show that the function

f:\{ a+b\sqrt n | a,b\in\{ 0\}\cup\mathbb{N} , a^2-nb^2=1 \}\longrightarrow\{ 0\}\cup\mathbb{N} , f(x) =\lfloor x \rfloor

is injective and non-surjective.

functionfloor functionalgebraquadratic integers

Freshman's dream for A,B,2,4, implies nilpotence for AB

Source: Gazeta matematică

n\times n

complex matrices

A,B

satisfying the equations

(A+B)^2=A^2+B^2

(A+B)^4=A^4+B^4.

(AB)^2=0.

freshman s dreamFrobeniusnilpotence

Kind of MVT negation

Source: Gazeta matematică

Let be a real number

c

and a differentiable function

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

f(c)\neq \frac{1}{b-a}\int_a^b f(x)dx,

for any real numbers

a\neq b.

f'(c)=0.

functionreal analysisintegrationprimitivableness