\input j--problems-.tex
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\competitionheader
The 13th Annual Vojt\v ech Jarn\'\i k
International Mathematical Competition
Ostrava, 2nd April 2003
Category I
\problem 1
Let $d(k)$ denote the number of all natural divisors of a
natural number~$k$. Prove that for any natural number~$n_0$
the sequence $\bigl \{d(n^2 + 1)\bigr \}_{n=n_0}^\infty$ is
not strictly monotone.
\endproblem 10
\problem 2
Let $A = (a_{ij})$ be an $m \times n$ real matrix with at
least one non-zero element. For each $i \in \{1, \ldots,
m\}$, let $R_i = \sum_{j=1}^n a_{ij}$ be the sum of the
$i$-th row of the matrix~$A$, and for each $j \in \{1,
\ldots, n\}$, let $C_j = \sum_{i=1}^m a_{ij}$ be the sum
of the $j$-th column of the matrix~$A$. Prove that there
exist indices $k \in \{1, \ldots, m\}$ and $l \in \{1,
\ldots, n\}$ such that
$$
\tabskip = \centering
\halign to \displaywidth {\tabskip = 0pt
\hfil $#$&${}#$\hfil \qquad &
\hfil $#$&${}#$\hfil \qquad &
\hfil $#$&${}#$\hfil \tabskip = \centering \cr
%
a_{kl} &> 0 \,\hbox{,} &
R_{k} &\geq 0 \,\hbox{,} &
C_{l} &\geq 0 \,\hbox{,} \cr
%
\noalign {\noindent or }
%
a_{kl} &< 0 \,\hbox{,} &
R_{k} &\leq 0 \,\hbox{,} &
C_{l} &\leq 0 \,\hbox{.} \cr }
\postdisplaypenalty=10000
$$
\endproblem 10
\problem 3
Find the limit
$$
\lim_{n \to \infty}
\sqrt {1 + 2\sqrt {1 + 3\sqrt
{\cdots + (n-1)\sqrt{1+n}\,}\,}\,}
\hbox{.}
\postdisplaypenalty=10000
$$
\endproblem 10
\problem 4
Let $A$~and~$B$ be complex hermitian $2 \times 2$ matrices
having the pairs of eigenvalues $(\alpha_1, \alpha_2)$ and
$(\beta_1, \beta_2)$, respectively. Determine all possible
pairs of eigenvalues $(\gamma_1, \gamma_2)$ of the matrix
$C = A + B$. (We recall that a matrix $A = (a_{ij})$ is
hermitian if and only if $a_{ij} = \overline a_{ji}$ for
all $i$~and~$j$.)
\endproblem 10
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\bye
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