\input j--problems-.tex
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\competitionheader
The 12th Annual Vojt\v ech Jarn\'\i k
International Mathematical Competition
Ostrava, 10th April 2002
Category II
\problem 1
Find all complex solutions to the system
$$
\eqalign {
(a + {\rm i}c)^3 +
({\rm i}a + b)^3 +
(-b + {\rm i}c)^3 &= -6 \> \hbox{,} \cr
(a + {\rm i}c)^2 +
({\rm i}a + b)^2 +
(-b + {\rm i}c)^2 &= 6 \> \hbox{,} \cr
(1 + {\rm i})a + 2{\rm i}c &= 0 \> \hbox{.} \cr
\noalign {\nointerlineskip \kern\jot }
\omit\span\omit $\mkern-4.5mu$\hrulefill $\mkern-4.5mu$}
\postdisplaypenalty=10000
$$
\endproblem 10
\problem 2
A ring~$R$ (not necessarily commutative) contains at least
one zero divisor and the number of zero divisors is finite.
Prove that $R$~is finite.
\endproblem 10
\problem 3
Let $E$~be the set of all continuous functions
$u\colon [\mkern1mu 0, 1] \to {\msbm R}$ satisfying
$$
u^2(t) \leq
1 + 4 \int_0^t s\bigl |u(s)\bigr | \,{\rm d}s \,,
\quad
\forall t \in [\mkern1mu 0,1] \>\hbox{.}
$$
Let $\varphi\colon E \to {\msbm R}$ be defined by
$$
\varphi(u) = \int_0^1 \bigl (u^2(x)-u(x)\bigr ) \,{\rm d}x
\,\hbox{.}
$$
Prove that $\varphi$~has a maximum value and find it.
\endproblem 10
\problem 4
Prove that
$$
\lim_{n \to \infty}
n^2 \Biggl (
\int_0^1 \root n \of {1+x^n} \,{\rm d}x - 1
\Biggr ) =
{\pi^2 \over 12} \,\hbox{.}
\postdisplaypenalty=10000
$$
\endproblem 10
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\bye
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