2

Part of 2012 Centers of Excellency of Suceava

Problems(4)

Find function xf(x/a)-f(a/x)=b

Source:

Find all functions

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

that verify, for any nonzero real number

x

xf(x/a)-f(a/x)=b,

a\neq 0,b

are two real numbers.
Dan Popescu

functionFind all functionsalgebra

Counting combinatorics

Source:

Find the number of unordered choices of

k

lists, each having

m

distinct ordered objects, among a number of

mn

objects.
Cătălin Țigăeru

countingcombinatorics

A not so hard limit

Source:

\lim_{n\to\infty } \frac{f(1)+(f(2))^2+\cdots +(f(n))^n}{(f(n))^n} ,

f:\mathbb{R}\longrightarrow\mathbb{R}_{>0 }

is an unbounded and nondecreasing function.
Dan Popescu

Easy verification ring theory

Source:

\left\{ X\in\mathcal{M}_2\left( \mathbb{Z}_3 \right)\left| \begin{pmatrix} 1&1\\2&2 \end{pmatrix} X\begin{pmatrix} 1&2\\2&1 \end{pmatrix} =0 \right. \right\}

is a multiplicative ring.
Cătălin Țigăeru

abstract algebramatrix ringRing Theory