\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 II
\problem 1
Find the minimal~$k$ such that every set of $k$~different
lines in~${\msbm R}^3$ contains either $3$~mutually parallel
lines or $3$~mutually intersecting lines or $3$~mutually
skew lines.
\endproblem 12
\problem 2
Let $a, b \in {\msbm R}$, \ $a \leq b$.
Assume that $f\colon [\mkern1mu a, b \mkern2.2mu] \to
[\mkern1mu a, b \mkern2.2mu]$ satisfies
$\bigl |f(x) - f(y)\bigr | \leq
\mathopen |x - y\mathclose |$
for every $x, y \in [\mkern1mu a, b \mkern2.2mu]$.
Choose an $x_1 \in [\mkern1mu a, b \mkern2.2mu]$ and define
$$
x_{n+1} = {x_n + f(x_n) \over 2} \,, \qquad
n = 1, 2, 3, \ldots\,\hbox{.}
$$
Show that $\{x_n\}_{n=1}^\infty$ converges to some fixed
point of~$f$.
\endproblem 7
\problem 3
Suppose that we have a countable set~$A$ of balls and a unit
cube in~${\msbm R}^3$. Assume that for every finite subset
$B$~of~$A$ it is possible to put all balls of~$B$ into the
cube in such a way that they have disjoint interiors. Show
that it is possible to arrange all the balls in the cube so
that all of them have pairwise disjoint interiors.
\endproblem 11
\problem 4
Let $u_1, u_2, \ldots, u_n \in
C(\mkern.5mu [\mkern1mu 0, 1]^n)$ be nonnegative and
continuous functions, and let $u_j$ do not depend on
the $j$-th variable for $j = 1$,~\dots,~$n$. Show that
$$
\Biggl (\int_{[\mkern1mu 0,1]^n} \prod_{j=1}^n u_j
\Biggr )^{\!\! n-1} \leq
\prod_{j=1}^n \int_{[\mkern1mu 0,1]^n} \! u_j^{n-1}
\hbox{.}
\postdisplaypenalty=10000
$$
\endproblem 10
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\bye
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