\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 I
\problem 1
Let $a$ and $d$ be two positive integers. Prove that
there exists a constant~$K$ such that every set of
$K$~consecutive elements of the arithmetic progression
$\{a+nd\}_{n=1}^\infty$ contains at least one number
which is not prime.
\endproblem 15
\problem 2
Find the limit
$$
\lim_{n \to \infty}
\Biggl (
{\bigl (1 + {1 \over n}\bigr )^n \over {\rm e}}
\Biggr )^{\!\! n}
\hbox{.}
\postdisplaypenalty=10000
$$
\endproblem 20
\problem 3
Give an example of a sequence of continuous functions
on~${\msbm R}$ converging pointwise to~$0$ which is not
uniformly convergent on any nonempty open set.
\endproblem 30
\problem 4-M
Prove the inequality
$$
{n\pi \over 4} - {1 \over \sqrt {8n}} \leq
{1 \over 2} +
\sum_{k=1}^{n-1}
{\textstyle \sqrt {1 - {k^2 \over n^2}}} \leq
{n\pi \over 4}
$$
for every integer $n \geq 2$.
\endproblem 35
\problem 4-I
Prove that there exists a program in standard Pascal which
prints out its own ASCII code. No disk operations are
permitted.
\endproblem 35
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\bye
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