\input j--problems-.tex
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\competitionheader
The 7th Annual Vojt\v ech Jarn\'\i k
International Mathematical Competition
Ostrava, 9th April 1997
Category I
\problem 1
Let $a$~be an odd positive integer.
Prove that if $d$~divides $(a^2 + 2)$, then
$d \equiv 1 \allowbreak\,\, ({\rm mod}\,\, 8)$ or
$d \equiv 3 \allowbreak\,\, ({\rm mod}\,\, 8)$.
\endproblem 10
\problem 2
Let $\alpha \in (0, 1]$ be a given real number and let a
real sequence $\{a_n\}_{n=1}^\infty$ satisfy the inequality
$$
a_{n+1} \leq \alpha a_n + (1 - \alpha) a_{n-1} \qquad
\hbox {for } n = 2,\!\ 3,\!\ \ldots
$$
Prove that if $\{a_n\}$ is bounded, then it must be
convergent.
\endproblem 12
\problem 3
Let $c_1$,~$c_2$, \dots,~$c_n$ be real numbers such that
$$
c^k_1 + c^k_2 + \cdots + c^k_n > 0 \qquad
\hbox {for all } k = 1,\!\ 2,\!\ \ldots
$$
Let us put
$$
f(x) = {1 \over (1 - c_1 x)(1 - c_2 x) \ldots (1 - c_n x)}
\,\hbox{.}
$$
Show that $f^{(k)}(0) > 0$ for all $k = 1$,~$2$,~\dots
\endproblem 15
\problem 4-M
Find all real numbers $a > 0$ for which the series
$$
\sum_{n=1}^\infty {a^{f(n)} \over n^2}
$$
is convergent; $f(n)$ denotes the number of 0's in the
decimal expansion of~$n$.
\endproblem 13
\problem 4-I
Let us declare
$$
\vbox {\sfcode`;=3000
\+ {\bf const} {\it N\_MAX} = 255;\cr
\+ {\bf type\/} \cleartabs& tR &= {\bf array}
[1\thinspace.\thinspace.\thinspace
{\it N\_MAX\/}] {\bf of\/} real;\cr
\+ & tN &= {\bf array}
[1\thinspace.\thinspace.\thinspace
{\it N\_MAX\/}] {\bf of\/} integer;\cr
}
$$
and let \underbar {random} be a function with no arguments
which returns real random values distributed uniformly in
$[\mkern1mu 0, 1)$.
You are to choose $K$~unique random integer numbers
($1 \leq K \leq N \leq {\it N\_MAX}$) from $1$ to~$N$
so that the probability of the choice of a number~$i$
is equal to a given~$P_i$ (if the number has not been
chosen yet; the choice is to be repeated otherwise),
$\sum_{i=1}^N P_i = 1$.
Write a procedure in Pascal that returns such $K$~integer
numbers in the first $K$~elements of the vector of the
type~\underbar {tN}. The input arguments of the procedure
are $K$,~$N$ and the vector of $P_i$'s.
\endproblem 13
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\bye
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