\nopagenumbers
\noindent {\bf 11th Vietnamese Mathematics Olympiad for College Students 2003}
\vskip 25pt
\noindent {\bf A1}. A is the matrix $$\left( \matrix{a & b & 0 & 0\cr b & a & -1 & 0\cr 0 & -1 & a & b\cr 0 & 0 & b & a\cr}\right)$$ where $a, b$ are real with $a>|b|$. Show that the eigenvalues of $A$ are positive reals.
\vskip 12pt
\noindent {\bf A2}. B is the matrix $$\left( \matrix{a & b^{\alpha \over \beta} & c^{\alpha \over \gamma}\cr b^{\beta \over \alpha} & d & p^{\beta \over \gamma}\cr c^{\gamma \over \alpha} & p^{\gamma \over \beta} & q\cr} \right)$$ where $a, b, c, d, p, q$ are reals and $\alpha,\beta,\gamma$ are non-zero reals. Show that $B$ has real eigenvalues. (You may assume that real symmetric matrices always have real eigenvalues).
\vskip 12pt
\noindent {\bf A3}.  $d_k$ is the $k\times k$ determinant $$\left| \matrix{0 & 1 & 1 & \ldots & 1 & 1 & 1\cr 1 & 0 & x & \ldots & x & x & x\cr 1 & x & 0 & \ldots & x & x & x\cr \ldots \cr 1 & x & x & \ldots & x & 0 & x\cr 1 & x & x & \ldots & x & x & 0\cr}\right|$$ Find $d_2+d_3+\ldots +d_n$.
\vskip 12pt
\noindent {\bf A4}. $P$ and $Q$ are two square matrices of order $n$ such that $PQ = QP$ and $P^r = Q^s = 0$ for some positive integers $r, s$. $I_n$ is the unit matrix of order $n$. Show that $I_n+P+Q$ and $I_n-P-Q$ are inverse.
\vskip 12pt
\noindent {\bf A5}. $A$ is a square matrix such that $A^{2003}=0$. Prove that {\it rank}$(A)=$ {\it rank}$(A+A^2+A^3+ \cdots +A^n)$.
\vskip 12pt
\noindent {\bf A6}. $A$ is the matrix $$\left(\matrix{1+x_1 & 1 & 1 & 1\cr 1 & 1+x_2 & 1 & 1\cr 1 & 1 & 1+x_3 & 1\cr 1 & 1 & 1 & 1+x_4\cr}\right)$$ where $x_1,x_2,x_3,x_4$ are the roots of the polynomial $f(x)=x^4-x+1$. Find $det\ A$.
\vskip 12pt
\noindent {\bf A7}. Let $P(x)$ be a polynomial of order $n>1$ with real coefficients and $m$ real roots. Show that $Q(x)=(x^2+1)P(x)+P'(x)$ has at least $m$ real roots.
\vskip 12pt
\noindent {\bf B1}. Find all continuous functions $f\colon R\to R$ such that $f(x+2002)(f(x)+\sqrt{2003})=-2004$ for all $x$.
\vskip 12pt
\noindent {\bf B2}. Find all continuous functions $f\colon[0,1]\to R$ which are differentiable on $(0,1)$ and satisfy $f(0)=f(1)=1$, and $2003 f'(x)+2004 f(x)\ge 2004$ for all $x$ in $(0,1)$.
\vskip 12pt
\noindent {\bf B3}. Given $a<b$, and a differentiable function $f\colon[a,b]\to R$ such that $f(a)=-{b-a\over 2}, f(b)={b-a\over 2}$ and $f({a+b\over 2})\not= 0$, prove that there are three distinct numbers $c_1,c_2,c_3\in (a,b)$ such that $f'(c_1)f'(c_2)f'(c_3)=1$.
\vskip 12pt
\noindent {\bf B4}. Let $x_k={1\over 2!}+{2\over 3!}+{3\over 4!} + \cdots + {k \over (k+1)!}$. Find $\lim_{n\to \infty} \root n \of {x_1^n+x_2^n+\cdots +x_{2003}^n}$.
\vskip 12pt
\noindent {\bf B5}. $f\colon [0,{\pi \over 2}]\to R$ is continuous, and $f(0)>0, \int_0^{\pi \over 2}f(t) dt\le 1$. Prove that $f(z)=sin z$ for some $z\in (0,{\pi \over 2})$.
\vskip 12pt
\noindent {\bf B6}. Given $a<b$, the continuous functions $f,g\colon [a,b]\to [a,b]$ satisfy $f\bigl( g(x)\bigr) =g\bigl( f(x)\bigr)$ for all $x$. $f$ is monotonic. Show that $f(z) = g(z) = z$ for some $z\in [a,b]$.
\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
