\nopagenumbers
\noindent {\bf 9th Vietnamese Mathematics Olympiad for College Students 2001}
\vskip 25pt
\noindent {\bf A1}. $A$ is the matrix $\left( \matrix{1 & 0 & 1\cr 0 & 1 & 2\cr 0 & 0 & -1\cr}\right)$. Find all $3\times 3$ matrices $B$ such that $AB+BA=0$.
\vskip 12pt
\noindent {\bf A2}. $A, B$ are real square matrices such that $A^{2001}=0$ and $AB=A+B$. Show that $det\ B =0$.
\vskip 12pt
\noindent {\bf A3}. $a,b,c,d$ are reals such that $ax^2+(b+c)x+d+e=0$ has a root $x\ge 1$. Show that $ay^4+by^3+cy^2+dy+e=0$ also has a root $y\ge 1$.
\vskip 12pt
\noindent {\bf A4}. $a_1,a_2,\ldots ,a_n$ are points in $R^n$ (so each $a_i$ has $n$ coordinates). $A$ is the matrix whose $i,j$ element is $a_ia_j$. Show that $det\ A$ is non-negative, and that the eigenvalues of $A$ are all non-negative.
\vskip 12pt
\noindent {\bf A5}. Every element of an $n\times n$ matrix is an even integer. Show that none of its eigenvalues are odd integers.
\vskip 12pt
\noindent {\bf A6}. $A$ is the $n\times n$ matrix $$\left( \matrix{a+b & ab & 0 & 0 & \ldots & 0 & 0\cr 1 & a+b & ab & 0 & \ldots & 0 & 0\cr 0 & 1 & a+b & ab & \ldots & 0 & 0\cr \ldots \cr 0 & 0 & 0 & 0 & \ldots & a+b & ab\cr 0 & 0 & 0 & 0 & \ldots & 1 & a+b\cr}\right)$$ where $a,b$ are reals. Find $det\ A$.
\vskip 12pt
\noindent {\bf B1}. The function $f(x)$ has $f''(x)>0$ for all $x>0$ and the graph is asymptotic to $y=ax+b$ as $x\to \infty$. Show that $f(x)-ax-b$ has negative derivative for all $x>0$, and that $f(x)>ax+b$ for all $x>0$.
\vskip 12pt
\noindent {\bf B2}. $p,q$ are reals such that $p>0,q<0,p+q<1$. The non-negative real sequence $a_1,a_2, a_3,\ldots$ satisfies $a_{n+2}\le pa_{n+1}+qa_n$ for all $n$. Show that the sequence converges and find its limit.
\vskip 12pt
\noindent {\bf B3}. Let $g(x)=(1+x)(1+x^2)\cdots (1+x^{2001})$, and $f(x)={x^{2000}\over g(x)}$. Show that $\int_x^1 f(t) dt=f(x)$ for some $x\in (0,1)$.
\vskip 12pt
\noindent {\bf B4}. A function $f\colon R\to R$ has a second derivative everywhere and $f(x)+f''(x)\ge 0$ for all $x$. Show that $f(x)+f(x+\pi)\ge 0$ for all $x$.
\vskip 12pt
\noindent {\bf B5}. $f(x)$ is defined for $x\ge 1$ and $f(1)\not= 0$. It satisfies $f(x+1) = 2001f(x)^2+f(x)$ for all $x$. Find $$\lim_{n\to \infty} \left({f(1) \over f(2)}+{f(2) \over f(3)}+ \cdots + {f(n) \over f(n+1)}\right)$$.
\vskip 12pt
\noindent {\bf B6}. $f\colon [a,b]\to R$ is a differentiable function such that $f(x)^2+f'(x)^2>0$ for all $x\in [a,b]$. Show that $f$ only has finitely many distinct roots.
\vskip 20pt
$R$ is the reals. $[a,b]$ represents the closed interval, and $(a,b)$ represents the open interval.
\vskip 20pt
\noindent \copyright John Scholes

\noindent jscholes@kalva.demon.co.uk

\noindent 23 August 2003

\bye
