\input j--problems-.tex
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\competitionheader
The 9th Annual Vojt\v ech Jarn\'\i k
International Mathematical Competition
Ostrava, 24th March 1999
Category I
\problem 1
Find the limit
$$
% The standard definition of the \big macro
% for the 10pt font uses \vbox to 8.5pt {}.
% But \fontdimen6\scriptfont0--point font
% is used in the superscript. Therefore
% 8.5pt * \fontdimen6\scriptfont0 / 10pt =
% .85\fontdimen6\scriptfont0
%
\lim_{n \to \infty}
\Biggl (\prod_{k=1}^n {k \over k+n} \Biggr )^{
\!
\mathopen {\hbox {$
\left (
\vbox to .85\fontdimen6\scriptfont0 {}
\right.
\nulldelimiterspace=0pt
\mathsurround=0pt
$}}
%
{\rm e}^{\! {1999 \over n}} - 1
%
\mathclose {\hbox {$
\left.
\vbox to .85\fontdimen6\scriptfont0 {}
\right )
\nulldelimiterspace=0pt
\mathsurround=0pt
$}}
}
\hbox{.}
\postdisplaypenalty=10000
$$
\endproblem 10
\problem 2
Find all natural numbers $n \geq 1$ such that
the implication
$$
(11 \mid a^n + b^n) \,\;\Longrightarrow\;
(11 \mid a \>\wedge\> 11 \mid b)
$$
holds for any two natural numbers $a$~and~$b$.
\endproblem 8
\problem 3
Let $A_1$,~\dots,~$A_n$ be points of an ellipsoid
with center~$O$ in~${\msbm R}^n$ such that $O A_i$,
for $i = 1$,~\dots,~$n$, are mutually orthogonal.
Prove that the distance of the point~$O$ from the
hyperplane $A_1 A_2 \ldots A_n$ does not depend on
the choice of the points $A_1$,~\dots,~$A_n$.
\endproblem 14
\problem 4
Show that the following implication holds
for any two complex numbers $x$~and~$y$:
if $x + y, \allowbreak\, x^2 + y^2, \allowbreak\,
x^3 + y^3, \allowbreak\, x^4 + y^4 \in {\msbm Z}$,
then $x^n + y^n \in {\msbm Z}$ for all natural~$n$.
\endproblem 8
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\bye
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