From 4c5c55c0e7c5ba6231a347c63d86923e0e5f000e Mon Sep 17 00:00:00 2001 From: sijanec Date: Wed, 9 Sep 2020 21:35:07 +0200 Subject: 2. dn @ mat --- mat/domace_naloge/2/a.out | Bin 0 -> 8392 bytes mat/domace_naloge/2/d=nx.c | 13 + mat/domace_naloge/2/dokument.aux | 22 + mat/domace_naloge/2/dokument.log | 1421 +++++++++++++++++++++++++++++++ mat/domace_naloge/2/dokument.out | 3 + mat/domace_naloge/2/dokument.pdf | Bin 0 -> 196173 bytes mat/domace_naloge/2/dokument.synctex.gz | Bin 0 -> 26585 bytes mat/domace_naloge/2/dokument.tex | 170 ++++ mat/domace_naloge/2/dokument.toc | 3 + 9 files changed, 1632 insertions(+) create mode 100755 mat/domace_naloge/2/a.out create mode 100644 mat/domace_naloge/2/d=nx.c create mode 100644 mat/domace_naloge/2/dokument.aux create mode 100644 mat/domace_naloge/2/dokument.log create mode 100644 mat/domace_naloge/2/dokument.out create mode 100644 mat/domace_naloge/2/dokument.pdf create mode 100644 mat/domace_naloge/2/dokument.synctex.gz create mode 100644 mat/domace_naloge/2/dokument.tex create mode 100644 mat/domace_naloge/2/dokument.toc diff --git a/mat/domace_naloge/2/a.out b/mat/domace_naloge/2/a.out new file mode 100755 index 0000000..0fd74aa Binary files /dev/null and b/mat/domace_naloge/2/a.out differ diff --git a/mat/domace_naloge/2/d=nx.c b/mat/domace_naloge/2/d=nx.c new file mode 100644 index 0000000..8f91387 --- /dev/null +++ b/mat/domace_naloge/2/d=nx.c @@ -0,0 +1,13 @@ +#include +long long unsigned int d(long long unsigned int n) { return (n*(n-3))/2; } +long long unsigned int x(long long unsigned int n) { return (n/2)-(3/2); } +int main() { + long long unsigned int n = 3; + while (n < 420696969) { + if (d(n) != n*x(n)) { + fprintf(stdout, "%lld-kotnik! d(n)=%lld, x(n)=%lld, n*x=%lld\n", n, d(n), x(n), n*x(n)); + } + n = n+2; + } +} + diff --git a/mat/domace_naloge/2/dokument.aux b/mat/domace_naloge/2/dokument.aux new file mode 100644 index 0000000..448b47d --- /dev/null +++ b/mat/domace_naloge/2/dokument.aux @@ -0,0 +1,22 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+\usepackage{float} +\usepackage{tasks} +\usepackage{filecontents} +\usepackage{textcomp} +\usepackage{tkz-euclide} +\usepackage{listings} +\usetikzlibrary{calc} %% not really needed............. idk. +%\usetikzlibrary{external} +% \usetkzobj{all} % tkz-euclide > 3.02 tega ne potrebuje več (: +%\tikzexternalize +\usepackage{xcolor} +\sisetup{output-decimal-marker = {,}, quotient-mode=fraction,per-mode=fraction} % per-mode=symbol +\newcommand\ddfrac[2]{\frac{\displaystyle #1}{\displaystyle #2}} +\newcommand{\functionSamples}{100} % fix to fancier value upon release, keep low during development +\newcommand{\razhroscevanje}{1} +\definecolor{codegreen}{rgb}{0,0.6,0} +\definecolor{codegray}{rgb}{0.5,0.5,0.5} +\definecolor{codepurple}{rgb}{0.58,0,0.82} +\definecolor{backcolour}{rgb}{0.95,0.95,0.92} + +\lstdefinestyle{mystyle}{ + backgroundcolor=\color{backcolour}, + commentstyle=\color{codegreen}, + keywordstyle=\color{magenta}, + numberstyle=\tiny\color{codegray}, + stringstyle=\color{codepurple}, + basicstyle=\ttfamily\footnotesize, + breakatwhitespace=false, + breaklines=true, + captionpos=b, + keepspaces=true, + numbers=left, + numbersep=5pt, + showspaces=false, + showstringspaces=false, + showtabs=false, + tabsize=2 +} + +\lstset{style=mystyle} +\def\@maketitle{% + \newpage + \null + \vskip 2em% + \begin{center}% + \let \footnote \thanks + {\LARGE \@title \par}% + \vskip 1.5em% + {\large + \lineskip .5em% + \begin{tabular}[t]{c}% <------ + \@author% <------ Authors + \end{tabular}\par}% <------ + \vskip 1em% + {\large \@date}% + \end{center}% + \par + \vskip 1.5em} +%opening +\newcommand{\stevilkadn}{2} +\newcommand{\snovdn}{Konveksne množice } +\makeatletter +\newcommand{\xslalph}[1]{\expandafter\@xslalph\csname c@#1\endcsname} +\newcommand{\@xslalph}[1]{% + \ifcase#1\or a\or b\or c\or \v{c}\or d\or e\or f\or g\or h\or i% + \or j\or k\or l\or m\or n\or o\or p\or r\or s\or \v{s}% + \or t\or u\or v\or z\or \v{z} + \else\@ctrerr\fi% +} +\AddEnumerateCounter{\xslalph}{\@xslalph}{m} +\makeatother +\newcommand\gauss[2]{1/(#2*sqrt(2*pi))*exp(-((x-#1)^2)/(2*#2^2))} % Gauss function, parameters mu and sigma +\newcommand*\textfrac[2]{ + \frac{\text{#1}}{\text{#2}} +} +\renewcommand\abstractname{Povzetek} +\date{8. september 2020} +\renewcommand\contentsname{Kazalo vsebine} +\renewcommand\figurename{Slika} +\renewcommand\abstractname{Povzetek} +\newcommand{\iic}{I\textsuperscript{2}C } +% \patchcmd{\thebibliography}{\section*{\refname}}{}{}{} +\title{% + \snovdn --- \stevilkadn. domača naloga \\ + \large Matematika, Gimnazija Bežigrad} +\author{\begin{tabular}{rl} + \textbf{Profesor:} & prof. Vilko Domajnko \\ + \textbf{Avtor:} & Anton Luka Šijanec, 2. a +% \textbf{Avtor:} & Anton Luka Šijanec \\ & Member 2 \\ & Member 3 +\end{tabular}} + + +% \everymath{\displaystyle} % https://tex.stackexchange.com/a/32847/212260 +\begin{document} +\maketitle + +\begin{abstract} + Ta dokument obsega naloge, naročene dijakom 8. septembra 2020, in njihove rešitve, ki sem jih spisal sam. Naloge obsegajo snov \textit{\snovdn}in so iz učbenika (stran 11). Kjer je bilo potrebno izbrati neke poljubne naloge, sem jih vedno izbral naključno. +\end{abstract} +\tableofcontents + +\section{Učbenik \textit{Matematika 2}: Stran 11} % 5, 7, 8, 6 +\begin{enumerate}[label=\textbf{\arabic*.}] + \setcounter{enumi}{4} + \item Kateri konveksni večkotnik ima 14 diagonal? % 5. naloga + + Število diagonal za konveksni $n$-kotnik izračunamo s funkcijo \texttt{ int d(int n) \{ return (n*(n-3))/2; \} } oziroma matematično $d=\frac{n(n-3)}{2}$. Če torej rešimo $14=\frac{n(n-3)}{2}$ za $n$, dobimo $n=7 \vee n=-4$. Mnogokotniki pa imajo lahko zgolj več kot dva kota, torej je odgovor sedemkotnik. + + \item Ali obstaja večkotnik, katerega število stranic je večkratnik števila diagonal? + + $n=\frac{n(n-3)}{2}$ rešimo za $n$ in dobimo $n=0 \vee n=5$. Zaradi enakega pravila, kot je opisano zgoraj, pa je edini možen rezultat petkotnik. + + \item Jaka je ugotovil, da je število diagonal nekega večkotnika enako večkratniku števila stranic tega večkotnika. Lina je trdila, da je takih večkotnikov nešteto. Ali ima prav? + + Najprej bi rad omenil, da je to vprašanje kratkomalo čudno, saj ne vemo, na koga se nanaša zadnji stavek. Odgovor bom napisal, kot da se nanaša na Lino. + + Enačbo $nx=\frac{n(n-3)}{2}$ lahko pretvorimo v $x=\frac{n}{2} - \frac{3}{2}$. Z dobljeno enačbo lahko za poljuben konveksen $n$-kotnik izračunamo faktor množenja, s katerim moramo pomnožiti $n$, da dobimo število diagonal. Torej ima Lina prav. + + Seveda je to dokaj logično, ker lahko dobimo vsako število, s tem da neko število zmnožimo z nekim drugim. Če pa imamo v mislih zgolj cela števila kot večkratnike, pa Lina nima prav, saj bo faktor množenja celo število samo, če je število stranic mnogokotnika liho. + + Slednjo trditev lahko približno preverimo s programom tako, da poizkusimo vse mnogokotnike od trikotnika do 420696969-kotnika, ki imajo liho število stranic. + + \lstinputlisting[language=c]{d=nx.c} + + \item[\textbf{8.*}] Ali je presek dveh poljubnih konveksnih množic konveksna množica? Ali je unija dveh poljubnih konveksnih množic konveksna množica? + + Presek je, ker: + + $$ M = \text{konveksna} \wedge N = \text{konveksna} \wedge A, B \text{(poljubni)} \in M \cap N \rightarrow$$ + $$ AB \subset M \wedge AB \subset N \rightarrow $$ + $$ AB \subset M \cap N = \text{konveksna}$$ + + Unija ni, ker, če sta $M$ in $N$ polravnini iste ravnine, + + $$ M = \text{polravnina} \wedge N = \text{polravnina} \wedge M \cup N \not= R \wedge A \in M \wedge B \in N \wedge A \cap B = \emptyset \rightarrow $$ + $$ AB \setminus ( M \cup N ) \not= \emptyset $$ +\end{enumerate} + +\section{Zaključek} +Ta dokument je informativne narave in se lahko še spreminja. Najnovejša različica, torej PDFji in +\hologo{LaTeX} +izvorna koda, zgodovina sprememb in prejšnje različice so na voljo mojem šolskem Git repozitoriju na +\url{https://github.com/sijanec/sola-gimb-2} v mapi +\href{https://github.com/sijanec/sola-gimb-2/tree/master/mat/domace_naloge/\stevilkadn/}{/mat/domace\_naloge/ \stevilkadn/}. Povezava za ogled zadnje različice tega dokumenta v PDF obliki je \url{http://razor.arnes.si/~asija3/files/sola/gimb/2/mat/domace_naloge/\stevilkadn/dokument.pdf} in/ali \url{https://github.com/sijanec/sola-gimb-2/raw/master/mat/domace_naloge/\stevilkadn/dokument.pdf}. + +\if\razhroscevanje1 +\section{Razhroščevalne informacije} +Te informacije so generirane, ker je omogočeno razhroščevanje. Prej objavo dokumenta izklopite razhroščevanje. To naredite tako, da nastavite ukaz \texttt{razhroscevanje} na 0 v začetku dokumenta. + +Grafi imajo natančnost \functionSamples\space točk na graf. + +Konec generiranja dokumenta: \input|"date -Ins" +\fi +% \item $$$$ +\end{document} diff --git a/mat/domace_naloge/2/dokument.toc b/mat/domace_naloge/2/dokument.toc new file mode 100644 index 0000000..9f953f1 --- /dev/null +++ b/mat/domace_naloge/2/dokument.toc @@ -0,0 +1,3 @@ +\contentsline {section}{\numberline {1}Učbenik \textit {Matematika 2}: Stran 11}{1}{section.1}% +\contentsline {section}{\numberline {2}Zaključek}{2}{section.2}% +\contentsline {section}{\numberline {3}Razhroščevalne informacije}{2}{section.3}% -- cgit v1.2.3