\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 I
\problem 1
Differentiable functions
$f_1, \ldots, f_n\colon {\msbm R} \to {\msbm R}$
are linearly independent. Prove that there exist
at least $n-1$ linearly independent functions
among $f_1'$,~\dots,~$f_n'$.
\endproblem 10
\problem 2
Let $p > 3$ be a prime number and $n = (2^{2p} - 1) / 3$.
Show that $n$~divides $2^n-2$.
\endproblem 10
\problem 3
Positive numbers $x_1$,~\dots,~$x_n$ satisfy
$$
{1 \over 1+x_1} +
{1 \over 1+x_2} + \cdots + {1 \over 1+x_n} = 1 \,\hbox{.}
$$
Prove that
$$
\sqrt {x_1} +
\sqrt {x_2} + \cdots + \sqrt {x_n} \geq
(n - 1) \biggl ({1 \over \sqrt {x_1}} +
{1 \over \sqrt {x_2}} + \cdots +
{1 \over \sqrt {x_n}}\biggr )
\mskip1mu\hbox{.}
\postdisplaypenalty=10000
$$
\endproblem 10
\problem 4
The numbers $1$,~$2$, \dots,~$n$ are assigned to the
vertices of a regular $n$-gon in an arbitrary order.
For each edge compute the product of the two numbers
at the endpoints and sum up these products.
What is the smallest possible value of this sum?
\endproblem 10
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\bye
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