\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 II
\problem 1
Let $n \geq 2$ be an integer and let $x_1$,~$x_2$,
\dots,~$x_n$ be real numbers. Consider $N = {n \choose 2}$
sums $x_i + x_j$, \ $1 \leq i < j \leq n$, and denote them
by $y_1$,~$y_2$, \dots,~$y_N$ (in an arbitrary order).
For which~$n$ are the numbers $x_1$,~$x_2$, \dots,~$x_n$
uniquely determined by the numbers $y_1$,~$y_2$,
\dots,~$y_N$?
\endproblem 10
\problem 2
Let $f\colon [\mkern1mu 0, 1] \to {\msbm R}$ be a
continuous function. Define a sequence of functions
$f_n\colon [\mkern1mu 0, 1] \to {\msbm R}$ in the
following way:
$$
f_0(x) = f(x) \,, \qquad
f_{n+1}(x) = \int_0^x f_n(t) \,{\rm d}t \,, \qquad
n = 0, 1, 2, \ldots\,\hbox{.}
$$
Prove that if $f_n(1) = 0$ for all~$n$, then
$f(x) \equiv 0$.
\endproblem 10
\problem 3
Let $f\colon (0, +\infty) \to (0, +\infty)$
be a decreasing function which satisfies
$\int_0^\infty f(x) \,{\rm d}x < +\infty$.
Prove that
$\lim_{x \rightarrow +\infty} x f(x) = 0$.
\endproblem 10
\problem 4
Let $R$~be an associative non-commutative ring and let
$n > 2$ be a fixed natural number. Assume that $x^n = x$
for all $x \in R$. Prove that $xy^{n-1} = y^{n-1}x$ holds
for all $x, y \in R$.
\endproblem 10
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\bye
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