\input j--problems-.tex
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\competitionheader
The 13th Annual Vojt\v ech Jarn\'\i k
International Mathematical Competition
Ostrava, 2nd April 2003
Category II
\problem 1
Two real square matrices $A$~and~$B$ satisfy the conditions
$A^{2002} = B^{2003} = I$ and $AB = BA$. Prove that $A +
B + I$ is invertible. (The symbol~$I$ denotes the identity
matrix.)
\endproblem 10
\problem 2
Let $\{D_1, D_2, \ldots, D_n\}$ be a set of disks in the
Euclidean plane. (A disk is a set of points whose distance
from the given centre is less than or equal to the given
radius.) Let $a_{ij} = S(D_i \cap D_j)$ be the area of
$D_i \cap D_j$. Prove that the inequality
$$
\sum_{i=1}^n \sum_{j=1}^n a_{ij} x_i x_j \geq 0
$$
holds for any real numbers $x_1$,~$x_2$, \dots,~$x_n$.
\endproblem 10
\problem 3
Let $\{a_n\}_{n=0}^\infty$ be the sequence of real numbers
satisfying $a_0 = 0$, \ $a_1 = 1$ and
$$
a_{n+2} = a_{n+1} + {a_n \over 2^n}
$$
for every $n \geq 0$. Prove that
$$
\lim_{n \to \infty} a_n =
1 + \sum_{n=1}^\infty
{1 \over 2^{n(n-1)/2} \prod_{k=1}^n (2^k - 1)}
\,\hbox{.}
\postdisplaypenalty=10000
$$
\endproblem 10
\problem 4
Let $f, g\colon [\mkern1mu 0, 1] \to (0, +\infty)$ be two
continuous functions such that $f$~and~${g \over f}$ are
increasing. Prove that
$$
\int_{0}^{1}
{\int_{0}^{x} f(t) \,{\rm d}t \over
\int_{0}^{x} g(t) \,{\rm d}t} \,{\rm d}x \leq
2\int_{0}^{1}
{f(t) \over g(t)} \,{\rm d}t
\,\hbox{.}
\postdisplaypenalty=10000
$$
\endproblem 10
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\bye
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