\input j--problems-.tex
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\competitionheader
The 11th Annual Vojt\v ech Jarn\'\i k
International Mathematical Competition
Ostrava, 4th April 2001
Category I
\problem 1
Let $A$~be a set of positive integers such that
for any $x, y \in A$
$$
x > y \,\;\Longrightarrow\; x - y \geq {xy \over 25}
\,\hbox{.}
$$
Find the maximal possible number of elements of the set~$A$.
\endproblem 10
\problem 2
Prove that for any prime $p \geq 5$, the number
$$
\sum_{0 < k < {2p \over 3}} {p \choose k}
$$
is divisible by~$p^2$.
\endproblem 10
\problem 3
Let $n \geq 2$ be a natural number. Prove that
$$
\prod_{k=2}^n \ln k < {\sqrt {n!} \over n} \,\hbox{.}
\postdisplaypenalty=10000
$$
\endproblem 10
\problem 4
Let $A$, $B$, $C$ be nonempty sets in~${\msbm R}^n$.
Suppose that $A$~is bounded, $C$~is closed and convex,
and $A + B \subseteq A + C$. Prove that $B \subseteq C$.
We remind that $E + F = \{\, e + f : e \in E, f \in F \,\}$
and $D \subseteq {\msbm R}^n$ is convex iff $tx + (1-t)y \in
D$ for all $x, y \in D$ and any $t \in [\mkern1mu 0, 1]$.
\endproblem 10
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\bye
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