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differ diff --git a/mat/domace_naloge/3/dokument.synctex.gz b/mat/domace_naloge/3/dokument.synctex.gz new file mode 100644 index 0000000..c2d13fe Binary files /dev/null and b/mat/domace_naloge/3/dokument.synctex.gz differ diff --git a/mat/domace_naloge/3/dokument.tex b/mat/domace_naloge/3/dokument.tex new file mode 100644 index 0000000..fba3ab3 --- /dev/null +++ b/mat/domace_naloge/3/dokument.tex @@ -0,0 +1,246 @@ +% !TeX encoding = UTF-8 +% !TeX spellcheck = sl_SI +\documentclass[]{article} +\usepackage[utf8]{inputenc} +\usepackage{etoolbox} +\usepackage[hidelinks]{hyperref} +\usepackage[a4paper, total={7in, 10in}]{geometry} +\usepackage{graphicx} +\usepackage{hologo} +\usepackage{amssymb} +\usepackage{pgfplots} +\usepackage[inline]{enumitem} +\usepackage{amsmath} +\usepackage{siunitx} +\usepackage{multicol} +\usepackage{tabularcalc} +\usepackage{amsmath} +\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}{0} +\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}{3} +\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{10. 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 10. septembra 2020, in njihove rešitve, ki sem jih spisal sam. Naloge obsegajo snov \textit{\snovdn}in so iz učbenika (stran 15). Kjer je bilo potrebno izbrati neke poljubne naloge, sem jih vedno izbral naključno. +\end{abstract} +\tableofcontents + +\section{Učbenik \textit{Matematika 2}: Stran 15} % 1, 2, 3, 4 +\begin{enumerate}[label=\textbf{\arabic*.}] + \item Zapiši vsaj pet dvojic sosednjih kotov z vrhovi v ogliščih pravokotnika, katerih kraki ležijo na nosilkah stranic ali diagonal pravokotnika, tako da kota v nobeni dvojici nista sokota. % 1. naloga + + $$\measuredangle(AB, AC), \measuredangle(AC, AD), \measuredangle(BA, BD), \measuredangle(BD, BC), \measuredangle(CB, CA)$$ + + \item Pet poltrakov ima skupno izhodišče $A$. Poltraki oblikujejo pet kotov tako, da je prvi soseden drugemu, drugi tretjemu ... peti prvemu. Velikosti teh kotov so v razmerju $1:2:3:4:5$. Izračunaj velikost posameznega kota. Ali ležita katera izmed poltrakov na isti premici? + + $$ \ang{360} = \alpha + \beta + \chi + \delta + \epsilon= 1\alpha + 2\alpha + 3\alpha + 4\alpha + 5\alpha = 15\alpha \rightarrow \alpha = 24 $$ + + \newcommand\alfast{24} + + Velikosti kotov so: $\alfast; \pgfmathparse{\alfast*2}\ang{\pgfmathresult}; \pgfmathparse{\alfast*3}\ang{\pgfmathresult}; \pgfmathparse{\alfast*4}\ang{\pgfmathresult}; \pgfmathparse{\alfast*5}\ang{\pgfmathresult}$. + \begin{center} + \begin{tikzpicture} + % \tkzInit[xmin=-6,ymin=-4,xmax=6,ymax=6] + \tkzDefPoint(0,0){A} + \tkzDefPoint(0:3){H} + \tkzDefPoint(24:3){J} + \tkzDefPoint((48+24):3){K} + \tkzDefPoint((72+48+24):3){L} + \tkzDefPoint((96+72+48+24):3){M} + % \tkzDefPoint((120+96+72+48+24):3){M} + % \tkzDefLine[mediator](A,B) + % \tkzDefLine[mediator](C,D) + \tkzDrawLine[](A,H) % \tkzLabelLine[](A,H){$h$} % add = 1 and 1 + \tkzDrawLine[](A,J) % \tkzLabelLine[](A,J){$j$} % add = 1 and 1 + \tkzDrawLine[](A,K) % \tkzLabelLine[](A,K){$k$} % add = 1 and 1 + \tkzDrawLine[](A,L) % \tkzLabelLine[](A,L){$l$} + \tkzDrawLine[](A,M) % \tkzLabelLine[](A,M){$m$} + \tkzFillAngle[fill=gray!10](H,A,J) \tkzLabelAngle[pos=2,draw,circle,fill=gray!10](H,A,J){$\alpha$} % size = 2 cm + \tkzFillAngle[fill=cyan!10](J,A,K) \tkzLabelAngle[pos=2,draw,circle,fill=cyan!10](J,A,K){$\beta$} + \tkzFillAngle[fill=yellow!10](K,A,L) \tkzLabelAngle[pos=2,draw,circle,fill=yellow!10](K,A,L){$\chi$} + \tkzFillAngle[fill=magenta!10](L,A,M) \tkzLabelAngle[pos=2,draw,circle,fill=magenta!10](L,A,M){$\delta$} + \tkzFillAngle[fill=white!10](M,A,H) \tkzLabelAngle[pos=2,draw,circle,fill=white!10](M,A,H){$\epsilon$} + \tkzMarkAngles[mark = none](H,A,J J,A,K K,A,L L,A,M M,A,H) + + % \tkzDefLine[bisector](S,C,B) + % \tkzGetPoint{b} + % \tkzDrawLine[color=black](C,b) + % \tkzLabelLine[right](C,b){$b$} + % \tkzDefPointOnLine[pos=1.2](A,B) % dodano v 2018! + % \tkzGetPoint{P} + % \tkzInterLL(A,C)(B,D) \tkzGetPoint{I} + + % \tkzDrawPolygon (A,B,C,D) + % \tkzDrawSegments(A,C B,D) + % \tkzMarkRightAngles[fill=Maroon!20,size=.3,opacity=.5](D,A,B A,B,C B,C,D C,D,A) + \tkzDrawPoints(A) + + % \tkzMarkSegments[mark=s||](I,A I,B I,C I,D) + % \tkzLabelPoints(C,D) + % \tkzLabelPoints[above=6pt](I) + \tkzLabelPoints[](A) + \end{tikzpicture} + \end{center} + + \newpage + + \item Dan je konveksen osemkotnik $ABCDEFGH$. + + \begin{enumerate}[label=\textbf{\xslalph*)}] + \item Katera oglišča so notranje točke $\measuredangle DAG$, katera so zunanje in katera robne? + + Notranji sta F in E, robni sta G in D, zunanje pa so C, B in H. + + \item Katera oglišča so notranje točke $\measuredangle GAD$, katera so zunanje in katera robne? + + Notranje so H, B in C, robni sta G in D, zunanji pa sta E in F. + + \item Največ koliko oglišč je zunanjih točk kota, katerega vrh je v oglišču osemkotnika in katerega kraka ležita na nosilkah stranic ali diagonal osemkotnika? Nariši ustrezno sliko. + + Največ sedem. Primer: $\measuredangle HGH = \alpha$ + + \begin{tikzpicture} + \tkzDefPoint(0,0){O} + \tkzDefPoint(5,0){A} + \tkzDrawCircle[color = gray!10](O,A) + % osemkotnik: 360/8=45 + \foreach[count=\i] \ANG in {45,45*2,45*3,45*4,45*5,45*6,45*7,45*8} { + \tkzDefPoint({5*cos(\ANG*pi/180)},{5*sin(\ANG*pi/180)}){P\i} + } + \tkzDrawPolygon[](P1,P2,P3,P4,P5,P6,P7,P8) % ultra thick, color = red!50!Gold + \tkzDrawPoints[](P1,P2,P3,P4,P5,P6,P7,P8) % size = 15 + % \tkzLabelPoints(P1,P2,P3,P4,P5,P6,P7,P8) + \tkzLabelPoint[above right](P1){$A$} + \tkzLabelPoint[above](P2){$B$} + \tkzLabelPoint[above left](P3){$C$} + \tkzLabelPoint[left](P4){$D$} + \tkzLabelPoint[below left](P5){$E$} + \tkzLabelPoint[below](P6){$F$} + \tkzLabelPoint(P7){$G$} + \tkzLabelPoint[right](P8){$H$} + \tkzDrawLine[add = .5 and 1.5](P7,P8) \tkzLabelLine[pos = 2.4, above left](P7,P8){$k_1$} % add = 1 and 1 + \tkzDrawLine[add = .5 and 1.5](P7,P8) \tkzLabelLine[pos = 2, below right](P7,P8){$k_2$} % add = 1 and 1 + \tkzFillAngle[fill=cyan!10](P8,P7,P8) \tkzLabelAngle[pos=2](P8,P7,P8){$\alpha$} % draw,circle,fill=cyan!10 + \end{tikzpicture} + \end{enumerate} + +% \lstinputlisting[language=c]{d=nx.c} + \item \begin{enumerate}[label=\textbf{\xslalph*)}]\item Ali je lahko oglišč, ki so zunanje točke kota, katerega vrh je v oglišču dvajsetkotnika in katerega kraka ležita na nosilkah stranic ali diagonal tega dvajsetkotnika, prav toliko kot oglišč, ki so notranje točke tega kota? + + Ne. Če sta kraka ista poltraka, je bodisi notranjost bodisi zunanjost kota $\emptyset$, če pa gresta kraka skozi dve različni oglišči, pa nam na razpolago ostane le še sedemnajst oglišč, kar pa ni deljivo z dve. + + \item Kakšno je naravno število $n$, če je oglišč, ki so zunanje točke kota, katerega vrh je v oglišču $n$-kotnika in katerega kraka ležita na nosilkah stranic ali diagonal tega $n$-kotnika, prav toliko kot oglišč, ki so notranje točke tega kota? + + Liho. Da lahko število oglišč razdelimo na dva dela (notranja in zunanja oglišča), mora le--to biti sodo. Če gresta kraka skozi isto oglišče, je bodisi notranjost bodisi zunanjost kota $\emptyset$. Če gresta skozi dve različni oglišči, pa nam po odštevanju teh dveh oglišč in oglišča z vrhom kota od oglišč $n$ kotnika, kjer je $n$ sodo število, ostane zgolj liho število kotov, kar pa ni deljivo na dva dela. + \end{enumerate} + +\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/3/dokument.toc b/mat/domace_naloge/3/dokument.toc new file mode 100644 index 0000000..b40bf1f --- /dev/null +++ b/mat/domace_naloge/3/dokument.toc @@ -0,0 +1,2 @@ +\contentsline {section}{\numberline {1}Učbenik \textit {Matematika 2}: Stran 15}{1}{section.1}% +\contentsline {section}{\numberline {2}Zaključek}{2}{section.2}% -- cgit v1.2.3