\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 I
\problem 1
Is there a countable set~$Y$ and an uncountable
family~${\cal F}$ of its subsets such that
for every two distinct $A, B \in {\cal F}$,
their intersection $A \cap B$ is finite?
\endproblem 10
\problem 2
Let $f\colon {\msbm N} \to {\msbm R}$ be given by
$$
f(n) = n^{{1 \over 2} \tau(n)}
$$
for $n \in {\msbm N} = \{1, 2, \ldots\}$ where $\tau(n)$
is the number of divisors of~$n$. Show that $f$~is an
injection into~${\msbm N}$.
\endproblem 10
\problem 3
Let $a_1, a_2, \ldots$ be a bounded sequence of reals.
Is it true that the fact
$$
\lim_{N \to \infty}
{1 \over N} \sum_{n=1}^N a_n = b
\qquad \hbox {and} \qquad
\lim_{N \to \infty}
{1 \over \log N} \sum_{n=1}^N {a_n \over n} = c
$$
implies $b = c$?
\endproblem 10
\problem 4
Let us choose arbitrarily $n$~vertices of a regular $2n$-gon
and colour them red. The remaining vertices are
coloured blue. We arrange all red-red distances
into a non-decreasing sequence and do the same with
the blue-blue distances. Prove that the sequences
are equal.
\endproblem 10
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\bye
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