VTRMC 2004

Let $I$ denote the $2\times2$ identity matrix $\begin{pmatrix}1&00&1\end{pmatrix}$ and let $$M=\begin{pmatrix}I&AB&C\end{pmatrix},\enspace N=\begin{pmatrix}I&BA&C\end{pmatrix}$$ where $A,B,C$ are arbitrary $2\times2$ matrices which entries in $\mathbb R$ , the real numbers. Thus $M$ and $N$ are $4\times4$ matrices with entries in $\mathbb R$ . Is it true that $M$ is invertible (i.e. there is a $4\times4$ matrix $X$ such that $MX=XM=I$ ) implies $N$ is invertible? Justify your answer.

A sequence of integers $\{f(n)\}$ for $n=0,1,2,\ldots$ is defined as follows: $f(0)=0$ and for $n>0$ , $$\begin{matrix}f(n)=&f(n-1)+3,&\text{if }n=0\text{ or }1\pmod6,&f(n-1)+1,&\text{if }n=2\text{ or }5\pmod6,&f(n-1)+2,&\text{if }n=3\text{ or }4\pmod6.\end{matrix}$$ Derive an explicit formula for $f(n)$ when $n\equiv0\pmod6$ , showing all necessary details in your derivation.

Let $f(x)=\int^x_0\sin(t^2-t+x)dt$ . Compute $f''(x)+f(x)$ and deduce that $f^{(12)}(0)+f^{(10)}(0)=0$ .

An enormous party has an infinite number of people. Each two people either know or don't know each other. Given a positive integer $n$ , prove there are $n$ people in the party such that either they all know each other, or nobody knows each other (so the first possibility means that if $A$ and $B$ are any two of the $n$ people, then $A$ knows $B$ , whereas the second possibility means that if $A$ and $B$ are any two of the $n$ people, then $A$ does not know $B$ ).

Let $\{a_n\}$ be a sequence of positive real numbers such that $\lim_{n\to\infty}a_n=0$ . Prove that $\sum^\infty_{n=1}\left|1-\frac{a_{n+1}}{a_n}\right|$ is divergent.