\input j--problems-.tex
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\competitionheader
The 10th Annual Vojt\v ech Jarn\'\i k
International Mathematical Competition
Ostrava, 5th April 2000
Category II
\problem 1
Let $p$~be a prime of the form $p = 4n - 1$ where $n$~is
a positive integer. Prove that
$$
\prod_{k=1}^p (k^2 + 1) \equiv 4 \pmod p \,\hbox{.}
\postdisplaypenalty=10000
$$
\endproblem 10
\problem 2
If we write the sequence AAABABBB along the perimeter of
a circle, then every word of the length~$3$ consisting of
letters A~and~B (i.e.\ AAA, AAB, ABA, BAB, ABB, BBB, BBA,
BAA) occurs exactly once on the perimeter. Decide whether
it is possible to write a sequence of letters from a
$k$-element alphabet along the perimeter of a circle in such
a way that every word of the length~$l$ (i.e.\ an ordered
$l$-tuple of letters) occurs exactly once on the perimeter.
\endproblem 10
\problem 3
Let $m$,~$n$ be positive integers and let
$x \in [\mkern1mu 0, 1]$. Prove that
$$
(1 - x^n)^m +
\bigl (1 - (1 - x)^m\bigr )^{\mskip-1mu n} \geq 1
\,\hbox{.}
\postdisplaypenalty=10000
$$
\endproblem 10
\problem 4
Let $\cal B$~be a family of open balls in~${\msbm R}^n$
and $c < \lambda\bigl (\bigcup {\cal B}\bigr )$ where
$\lambda$~is the $n$-dimensional Lebesgue measure. Show
that there exists a finite family of pairwise disjoint
balls $\{U_i\}_{i=1}^k \subseteq {\cal B}$ such that
$$
\sum_{j=1}^k \lambda(U_j) > {c \over 3^n} \,\hbox{.}
\postdisplaypenalty=10000
$$
\endproblem 10
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\bye
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