\input j--problems-.tex
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\competitionheader
The 8th Annual Vojt\v ech Jarn\'\i k
International Mathematical Competition
Ostrava, 1st April 1998
Category II
\problem 1
Let $H$~be a complex Hilbert space.
Let $T\colon H \to H$ be a bounded linear operator such that
$\bigl |(Tx,x)\bigr | \leq \mathopen \| x \mathclose \|^2$
for each $x \in H$. Assume that $\mu \in {\msbm C}$, \
$\mathopen |\mu\mathclose | = 1$, is an eigenvalue
with the corresponding eigenspace $E = \{\, \phi \in H :
T\phi = \mu \phi \,\}$. Prove that the orthogonal
complement $E^\perp = \{\, x \in H :
\forall \phi \in E\colon (x,\phi) = 0 \,\}$ of~$E$ is
$T$-invariant, i.e., $T(E^\perp) \subseteq E^\perp$.
\endproblem 15
\problem 2
Decide whether there is a member in the arithmetic sequence
$\{a_n\}_{n=1}^\infty$ whose first member is $a_1 = 1998$
and the common difference $d = 131$ which is a palindrome
(palindrome is a number such that its decimal expansion is
symmetric, e.g., 7, 33, 433334, 2135312 and so on).
\endproblem 25
\problem 3
Show that all complex roots of the polynomial
$P(z) = a_0 z^n + a_1 z^{n-1} + \cdots + a_{n-1} z + a_n$,
where $0 < a_0 < \cdots < a_n$, satisfy
$\mathopen |z\mathclose | > 1$.
\endproblem 25
\problem 4-M
A function $f\colon {\msbm R} \to {\msbm R}$ has the
property that for every $x, y \in {\msbm R}$ there
exists a real number~$t$ (depending on $x$~and~$y$)
such that $0 < t < 1$ and
$$
f \bigl (tx + (1 - t)y \bigr ) = t f(x) + (1 - t) f(y)
\,\hbox{.}
$$
Does it imply that
$$
f \Bigl ({x + y \over 2} \Bigr ) = {f(x) + f(y) \over 2}
$$
for every $x, y \in {\msbm R}$?
\endproblem 35
\problem 4-I
Let us consider a first-order language~$L$ with a
ternary predicate ${\rm Plus}$. Hence (well-formed)
formulas of~$L$ are built of symbols for variables,
logical connectives, quantifiers, brackets, and the
predicate symbol ${\rm Plus}$.
$$
(\exists x_1)(\forall x_2)\colon
\bigl ({\rm Plus}(x_2,x_1,x_2) \wedge
(\forall x_3)\colon \neg {\rm Plus}(x_1,x_3,x_3) \bigr )
$$
is an example of such a formula. Recall that a formula is
{\it closed\/} iff each variable symbol occurs within the
scope of a quantifier.
Show that there exists an algorithm which decides whether
or not a given closed formula of~$L$ is true for the
set~${\msbm N}$ of natural numbers ($\{0, 1, 2, \ldots \}$)
where ${\rm Plus}(x,y,z)$ is interpreted as $x + y = z$.
\endproblem 35
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