\input j--problems-.tex
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\competitionheader
The 7th Annual Vojt\v ech Jarn\'\i k
International Mathematical Competition
Ostrava, 9th April 1997
Category II
\problem 1
Decide whether it is possible to cover the 3-dimensional
Euclidean space with lines which are pairwise skew
(i.e.\ not coplanar).
\endproblem 12
\problem 2
Let $f\colon {\msbm C} \to {\msbm C}$ be a holomorphic
function with the property that $\bigl |f(z)\bigr | = 1$ for
all $z \in {\msbm C}$ such that $\mathopen |z\mathclose | =
1$. Prove that there exist a $\theta \in {\msbm R}$ and a
$k \in \{0, 1, 2, \ldots\}$ so that
$$
f(z) = {\rm e}^{{\rm i}\theta} z^k
$$
for all $z \in {\msbm C}$.
\endproblem 10
\problem 3
Let $u \in C^2\bigl (\overline D\bigr )$, $u = 0$
on~$\partial D$ where $D$~is the open unit ball
in~${\msbm R}^3$. Prove that the following inequality
holds for all $\varepsilon > 0$:
$$
\int_D \mathopen |\nabla u\mathclose |^2 \,{\rm d} V \leq
\varepsilon \int_D (\Delta u)^2 \,{\rm d} V +
{1 \over 4\varepsilon} \int_D u^2 \,{\rm d} V
\mskip1.5mu\hbox{.}
$$
(We recall that $\nabla u = \bigl [{\partial u \over
\partial x}, {\partial u \over \partial y},
{\partial u \over \partial z}\bigr ]$ and $\Delta u =
{\partial^2 u \over \partial x^2} + {\partial^2 u \over
\partial y^2} + {\partial^2 u \over \partial z^2}$ are
gradient and Laplacian respectively.)
\endproblem 13
\problem 4-M
Prove that
$$
\sum_{n=1}^\infty {n^2 \over (7n)!} =
{1 \over 7^3} \sum_{k=1}^2 \sum_{j=0}^6
{\rm e}^{\cos (2\pi j / 7)} \cdot
\cos \biggl ({2k\pi j \over 7} +
\sin {2\pi j \over 7}\biggr )
\mskip1mu\hbox{.}
\postdisplaypenalty=10000
$$
\endproblem 15
\problem 4-I
Problem ${\it Div}_3$ is specified as follows:
$$
\vbox {\narrower \narrower
\setbox0 = \hbox {\it Question:\enspace}
\parindent = \wd0
\item {\it Instance:} a finite string of symbols 0~and~1.
\item {\it Question:} is the given string a binary code
of a number divisible by~3?
}
$$
(It is obvious that there is a program which solves problem
${\it Div}_3$, i.e., it outputs the right answer {\it yes}
or {\it no} for any string of 0's and 1's.)
But you should show that there is no program ${\it
Gen\mathchar"742D\allowbreak
Test\mathchar"742D\allowbreak
Data}$ with the specification:
$$
\vbox {\narrower \narrower
\setbox0 = \hbox {\it Question:\enspace}
\parindent = \wd0
\item {\it Input:} any program~$P$.
\item {\it Output:} a finite set $D(P)$ of strings of 0's
and 1's such that the program~$P$ solves problem
${\it Div}_3$ iff the program~$P$ outputs the
correct answer for all inputs from $D(P)$.
}
$$
{\it Remark.} You can use the Recursion Theorem, which can
be expressed in the following form:
{\sl For any program ${\it Transf}$ which transforms
programs in some way (i.e., for any given program~$P$,
it constructs some other program $P' = {\it Transf}(P)$),
there exists a program~$P_0$ whose input\slash output
behaviour is not affected by the transformation (i.e.,
$P_0$ and ${\it Transf}(P_0)$ yield the same outputs for
the same inputs).}
\endproblem 15
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\bye
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