SEEMOUS 2010

Let $f_0:[0,1]\to\mathbb R$ be a continuous function. Define the sequence of functions $f_n:[0,1]\to\mathbb R$ by $$f_n(x)=\int^x_0f_{n-1}(t)dt$$ for all integers $n\ge1$ .

a) Prove that the series $\sum_{n=1}^\infty f_n(x)$ is convergent for every $x\in[0,1]$ . b) Find an explicit formula for the sum of the series $\sum_{n=1}^\infty f_n(x),x\in[0,1]$ .

Inside a square consider circles such that the sum of their circumferences is twice the perimeter of the square.

a) Find the minimum number of circles having this property. b) Prove that there exist infinitely many lines which intersect at least 3 of these circles.

Denote by $\mathcal M_2(\mathbb R)$ the set of all $2\times2$ matrices with real entries. Prove that:

a) for every $A\in\mathcal M_2(\mathbb R)$ there exist $B,C\in\mathcal M_2(\mathbb R)$ such that $A=B^2+C^2$ ; b) there do not exist $B,C\in\mathcal M_2(\mathbb R)$ such that $\begin{pmatrix}0&11&0\end{pmatrix}=B^2+C^2$ and $BC=CB$ .

Suppose that $A$ and $B$ are $n\times n$ matrices with integer entries, and $\det B\ne0$ . Prove that there exists $m\in\mathbb N$ such that the product $AB^{-1}$ can be represented as $$AB^{-1}=\sum_{k=1}^mN_k^{-1},$$ where $N_k$ are $n\times n$ matrices with integer entries for all $k=1,\ldots,m$ , and $N_i\ne N_j$ for $i\ne j$ .