### Archive

Posts Tagged ‘traian lalescu’

## Traian Lalescu Student Contest 2011 Problem 4

Let $D=(0,\infty)\times (0,\infty),\ u \in C^1(D)$ and $\varepsilon>0$ fixed.

1) Prove that $x \frac{\partial u}{\partial x}(x,y)+y\frac{\partial u}{\partial y}(x,y)=u(x,y),\ \forall (x,y) \in D$ if and only if there exists $\phi \in C^1(0,\infty)$ such that $u(x,y)=x\phi(y/x),\ \forall (x,y) \in D$.

2) Prove that $\left|x \frac{\partial u}{\partial x}(x,y)+y\frac{\partial u}{\partial y}(x,y)-u(x,y)\right|\leq \varepsilon,\ \forall (x,y) \in D$ then there is a unique function $\phi \in C^1(0,\infty)$ such that $\left|u(x,y)-x\phi(y/x),\right| \leq \varepsilon \ \forall (x,y) \in D$.

## Traian Lalescu student contest 2011 Problem 3

For a continuous function $f : [0,\infty) \to [0,\infty)$ such that $\displaystyle \int_0^\infty f(x)dx <\infty$ prove that if $f$ is uniformly continuous, then it is bounded. Prove also that the converse of the previous statement is not true.

Traian Lalescu student contest 2011

Categories: Analysis, Problem Solving

## Traian Lalescu Student contest 2011 Problem 2

Let $n \geq 2$ be a square-free positive integer, and denote by $D_n$ the set of its divisors. Consider $D \subset D_n$, a set with the following properties:

• $1 \in D$;
• $x \in D \Rightarrow n/x \in D$;
• $x,y \in D \Rightarrow \gcd(x,y) \in D$.

Show that there exists a positive integer $k$ such that $|D|=2^k$.

Traian Lalescu student contest 2011

## Traian Lalescu contest 2009 Problem 1

Let $(x_n)$ be an increasing divergent sequence of positive real numbers and $\alpha \leq 1$.

What can you say about the series $\displaystyle \sum_{n=1}^\infty \left(\frac{x_n-x_{n-1}}{x_n}\right)^\alpha$?

Traian Lalescu student contest 2009

## Traian Lalescu contest 2008 Problem 3

Suppose $U\subset \Bbb{R}^2$ is a subset which contains the unit disk $D$, and consider a function $f \in C^1(D)$ such that $\displaystyle \left|\frac{\partial f}{\partial x}(P)\right|\leq 1, \left|\frac{\partial f}{\partial y}(P)\right|\leq 1,\ \forall P \in D$.

Consider a finite set of points $M_1,M_2,...,M_n \in D$ with the center of gravity $O$. Prove that for all $P \in D$ we have the inequality $\displaystyle \left|f(P)-\sum_{i=1}^n f(M_i)\right| \leq 2$.

Traian Lalescu student contest 2008

Let $A \in \mathcal{M}_{n \times n}(\Bbb{C})$. Prove that $A$ is nilpotent if and only if $tr(A^k)=0,\ \forall k \in \Bbb{N}^*$.