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Traian Lalescu Student Contest 2011 Problem 4

May 25, 2011 Leave a comment

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

May 14, 2011 Leave a comment

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

Traian Lalescu Student contest 2011 Problem 2

May 14, 2011 Leave a comment

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

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Traian Lalescu contest 2009 Problem 1

May 7, 2011 2 comments

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

Traian Lalescu contest 2008 Problem 1

May 7, 2011 1 comment

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}^*.

Traian Lalescu Student Contest 2008

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