IMC 1997

Let $a_n$ be a sequence of reals. Suppose $\sum a_n$ converges. Do these sums converge aswell?

(a) $a_1+a_2+(a_4+a_3)+(a_8+...+a_5)+(a_{16}+...+a_9)+...$

(b) ${a_1+a_2+(a_3)+(a_4)+(a_5+a_7)+(a_6+a_8)+(a_9+a_{11}+a_{13}+a_{15})+(a_{10}+a_{12}+a_{14}+a_{16})+(a_{17}+a_{19}+...}$

Let $f\in C^3(\mathbb{R})$ nonnegative function with $f(0)=f'(0)=0, f''(0)>0$ . Define $g(x)$ as follows: $$\{ \begin{array}{ccc}g(x)= (\frac{\sqrt{f(x)}}{f'(x)})' &\text{for}& x\not=0 g(x)=0 &\text{for}& x=0\end{array}$$

(a) Show that $g$ is bounded in some neighbourhood of $0$ . (b) Is the above true for $f\in C^2(\mathbb{R})$ ?

Let $M \in GL_{2n}(K)$ , represented in block form as $$M = \left[ \begin{array}{cc} A & B C & D \end{array} \right] , M^{-1} = \left[ \begin{array}{cc} E & F G & H \end{array} \right]$$ Show that $\det M.\det H=\det A$ .