## Traian Lalescu Student Contest 2011 Problem 4

Let and fixed.

1) Prove that if and only if there exists such that .

2) Prove that then there is a unique function such that .

## Traian Lalescu student contest 2011 Problem 3

For a continuous function such that prove that if is uniformly continuous, then it is bounded. Prove also that the converse of the previous statement is not true.

Traian Lalescu student contest 2011

## Traian Lalescu Student contest 2011 Problem 2

Let be a square-free positive integer, and denote by the set of its divisors. Consider , a set with the following properties:

- ;
- ;
- .

Show that there exists a positive integer such that .

Traian Lalescu student contest 2011

## Traian Lalescu contest 2009 Problem 1

Let be an increasing divergent sequence of positive real numbers and .

What can you say about the series ?

Traian Lalescu student contest 2009

## Traian Lalescu contest 2008 Problem 3

Suppose is a subset which contains the unit disk , and consider a function such that .

Consider a finite set of points with the center of gravity . Prove that for all we have the inequality .

Traian Lalescu student contest 2008

## Traian Lalescu contest 2008 Problem 1

Let . Prove that is nilpotent if and only if .

Traian Lalescu Student Contest 2008