Koło matematyczne I LO w Białymstoku

 
Zadania PDF.

Źródło zadań w texu.

 
%        File: mecz_mat.tex
%     Created: Fri Dec 17 01:00 PM 2010 C
% Last Change: Fri Dec 17 01:00 PM 2010 C
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% THEOREMS -------------------------------------------------------
\newtheorem{thm}{Twierdzenie}[section]
\newtheorem{cor}[thm]{Wniosek}
\newtheorem{lem}[thm]{Lemat}
\newtheorem{defn}[thm]{Definicja}
\newtheorem{tozs}[thm]{Tożsamość}
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\begin{document}
\large
\renewcommand{\thefigure}{}
\begin{figure}
\section{Powodzenia!}
\end{figure}
 
%\subsection{międzynarodowe}
\begin{enumerate}
    \item Найти наибольшее количество разных перестановок из $\{\sigma,
        \sigma^2, \sigma^3, \dots\}$, если $\sigma$~--- перестановка
        12-элементного множества ($\sigma^k$ это $k$--кратная композиция
        $\sigma$).
    \item
        \begin{CJK}{UTF8}{gbsn}
            计算
        \end{CJK}
        $\frac{2}{1\cdot 2\cdot 3}
        +
        \frac{2}{2\cdot 3\cdot 4}
        +\dots+
        \frac{2}{2009\cdot 2010\cdot 2011}$.
    \item Bestem alle positive heltal $n$, således at $5^{(n-1)!} - 1$ er delelig med
        $n$.
    \item
        Prove that in any triangle the following inequality holds: $pR \geq
        2S$, where $p, R, S$ are respectively the half of circumference of the
        triangle (the semiperimeter), the radius of the circumcircle and the
        area of the triangle.
    \item
        Es wurde solches konvexe Sechseck $ABCDEF$ gegeben, dass für allen
        Vierecken $ABCD$, $CDEF$, $EFAB$ ein Umkreis existiert. Zeigen, dass für das
        Sechseck $ABCDEF$ auch ein Umkreis existiert.
    \item Sia $f(x)$ un polinomio a~coefficienti interi tale che $f(3) = 5$. Se un
        intero $n$ ha
        la propriet\`a che $f(n^3) = 15$, quali sono i~possibili valori di $n$?
 
\end{enumerate}
 
\end{document}