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-rw-r--r--Makefile14
-rw-r--r--alg/alg-1.tex76
-rw-r--r--calc/sol0113.tex (renamed from sol0113.tex)0
-rw-r--r--calc/sol0120.tex (renamed from sol0120.tex)0
-rw-r--r--calc/sol0127.tex (renamed from sol0127.tex)0
-rw-r--r--calc/sol0203.tex (renamed from sol0203.tex)0
-rw-r--r--calc/sol0210.tex (renamed from sol0210.tex)0
-rw-r--r--cat.pdfbin0 -> 27160 bytes
-rw-r--r--cat.svg706
-rw-r--r--dm/dm-14.tex (renamed from dm-14.tex)0
-rw-r--r--dm/dm-15.tex (renamed from dm-15.tex)0
-rw-r--r--dm/dm-16.tex (renamed from dm-16.tex)0
-rw-r--r--dm/dm-17.tex (renamed from dm-17.tex)0
-rw-r--r--dm/dm-18.tex (renamed from dm-18.tex)0
-rw-r--r--dm/dm-19.tex (renamed from dm-19.tex)2
-rw-r--r--intro.sty94
-rw-r--r--intro.tex59
l---------last.tex1
-rw-r--r--mathshit.sty12
19 files changed, 897 insertions, 67 deletions
diff --git a/Makefile b/Makefile
new file mode 100644
index 0000000..3b7b246
--- /dev/null
+++ b/Makefile
@@ -0,0 +1,14 @@
+MV ?=mv
+RM ?=rm -rf
+RMRECURSIVE ?=$(RM) -r
+
+
+all:
+ true
+
+clean:
+ $(MV) cat.pdf cat.pdf.bak
+ $(RM) *.log *.synctex.gz *.pdf
+ $(MV) cat.pdf.bak cat.pdf
+ $(RMRECURSIVE) $(shell fd -I '[a-f0-9]{32}')
+
diff --git a/alg/alg-1.tex b/alg/alg-1.tex
new file mode 100644
index 0000000..a9906b8
--- /dev/null
+++ b/alg/alg-1.tex
@@ -0,0 +1,76 @@
+\documentclass[11pt]{article}
+\usepackage[x11names, svgnames, rgb]{xcolor}
+\usepackage{../intro}
+\usepackage{cancel}
+
+\lhead{\color{gray} \small \textit{Шарафатдинов Камиль 192}}
+\rhead{\color{gray} \small \textit{алг-1}}
+
+\title{Алгебра 1}
+
+% -- Here bet dragons --
+\begin{document}
+\maketitle
+\drawcat{50pt}
+\clearpage
+
+\dmquestion{1}
+
+ Докажем, что если $\ord(g) = m$, то
+ \[
+ n \divby m \Leftrightarrow g^n = e
+ \]
+
+ \begin{itemize}[leftmargin=1in]
+ \item [$\Rightarrow$] \leavevmode \\
+
+ \[
+ g^n = \br{g^m}^{\frac n m} = e^{\frac n m} = e
+ \]
+
+ \item [$\Leftarrow$] \leavevmode \\
+
+ Пусть это не так и $\exists\ n \ : \ g^n = e$, но $n$ не делится на $m$. Тогда
+ \[
+ n = mq + r, \ \ 0 \leqslant r < m
+ \]
+ \[
+ e = g^n = g^{mq + r} = g^{mq} g^r = e^q g^r = g^r
+ \]
+
+ Но так как $r < m$, значит, $\ord(g) = r < m$. Противоречие.
+ \end{itemize}
+
+ Тогда понятно, что $\ord(g^k) = \frac{\text{НОК}(m, k)}{k}$
+
+\dmquestion{2}
+
+ Пусть $G = \cycle{g}$ и надо доказать, что подгруппа $H \subseteq G$ - циклическая.
+ Неформально, я хочу показать, что $H$ порождается $g^{\gcd(\{d \ | \ g^d \in H\})}$.
+
+ Введем новую операцию для удобства
+ \[
+ \deg(x) \coloneqq \min \{d \ | \ x = g^d, d \geqslant 0\}
+ \]
+
+ Возьмем какой-то элемент $h_0 \in H$ так, что $h_0 \neq e$.
+ Построим последовательность $\{h_i\}$ хитрым образом, докажем, что она конечна и последний ее элемент $h_n$ порождает $H$.
+
+ Пусть $\cycle{h_i} \neq H$ (иначе $H = \cycle{h_i}$ и все доказано), тогда возьмем какой-нибудь элемент $h$ из $H$, но не из $\cycle{h_i}$.
+
+ Так как уравнение
+ \[
+ x \deg(h) + y \deg(h_i) = \gcd(\deg(h), \deg(h_i))
+ \]
+ всегда имеет решение $(x, y)$ в целых числах, то $g^{\gcd(\deg(h),\ \deg(h_i))}$ обязан лежать в $H$.
+
+ Тогда пусть по построению $h_{i + 1} = g^{\gcd(\deg(h),\ \deg(h_i))}$. Заметим, что $\cycle{h_{i + 1}} \in H$, поскольку $H$ - группа.
+
+ Очевидно, что $\gcd(\deg(h), \ \deg(h_i)) \leqslant \deg(h_i)$. Но так как $h \notin \cycle{h_i}$, то $\deg(h)\ \cancel{\divby}\deg(h_i)$, а значит $\gcd(\deg(h), \ \deg(h_i)) < \deg(h_i)$.
+
+ Значит, $h_i > h_{i + 1}$, поэтому $|\{h_n\}| < \infty$. Значит, в какой-то момент перестанет выполняться предположение о том, что $\cycle{h_i} \neq H$, что и требовалось.
+
+\dmquestion{4}
+
+
+\end{document}
diff --git a/sol0113.tex b/calc/sol0113.tex
index 06ac8d6..06ac8d6 100644
--- a/sol0113.tex
+++ b/calc/sol0113.tex
diff --git a/sol0120.tex b/calc/sol0120.tex
index f073973..f073973 100644
--- a/sol0120.tex
+++ b/calc/sol0120.tex
diff --git a/sol0127.tex b/calc/sol0127.tex
index 1098687..1098687 100644
--- a/sol0127.tex
+++ b/calc/sol0127.tex
diff --git a/sol0203.tex b/calc/sol0203.tex
index f568637..f568637 100644
--- a/sol0203.tex
+++ b/calc/sol0203.tex
diff --git a/sol0210.tex b/calc/sol0210.tex
index bb4ad94..bb4ad94 100644
--- a/sol0210.tex
+++ b/calc/sol0210.tex
diff --git a/cat.pdf b/cat.pdf
new file mode 100644
index 0000000..1b58542
--- /dev/null
+++ b/cat.pdf
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diff --git a/cat.svg b/cat.svg
new file mode 100644
index 0000000..5284eba
--- /dev/null
+++ b/cat.svg
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+ c-2.272-3.064-4.397-6.238-6.333-9.525c-2.077-3.525-3.935-7.179-5.536-10.944c-1.63-3.834-2.992-7.783-4.067-11.809
+ c-1.06-3.974-1.841-8.025-2.293-12.113c-0.441-3.988-0.562-8.01-0.32-12.016c0.226-3.74,0.751-7.459,1.453-11.138
+ c0.743-3.891,1.67-7.734,2.618-11.579c1.035-4.196,2.096-8.388,3.053-12.603c0.937-4.131,1.784-8.285,2.468-12.466
+ c0.664-4.056,1.194-8.135,1.51-12.234c0.153-1.979,0.26-3.961,0.311-5.945c0.054-2.062-0.058-4.138,0.092-6.193
+ c0.152-2.079-0.008-4.178-0.413-6.221c-0.37-1.863-0.525-3.769-0.833-5.646c-0.63-3.836-1.532-7.629-2.746-11.322
+ c-1.261-3.836-2.842-7.612-4.89-11.097c-0.441-0.751-1.01-1.469-1.395-2.244c-0.425-0.855-0.751-1.699-1.308-2.491
+ c-0.555-0.789-1.208-1.508-1.929-2.149c-0.371-0.33-0.761-0.639-1.167-0.925C198.621,150.544,197.962,149.947,197.633,149.947z"/>
+ </g>
+</g>
+</svg>
diff --git a/dm-14.tex b/dm/dm-14.tex
index 313e693..313e693 100644
--- a/dm-14.tex
+++ b/dm/dm-14.tex
diff --git a/dm-15.tex b/dm/dm-15.tex
index db0de91..db0de91 100644
--- a/dm-15.tex
+++ b/dm/dm-15.tex
diff --git a/dm-16.tex b/dm/dm-16.tex
index 054f0a2..054f0a2 100644
--- a/dm-16.tex
+++ b/dm/dm-16.tex
diff --git a/dm-17.tex b/dm/dm-17.tex
index 8d5e101..8d5e101 100644
--- a/dm-17.tex
+++ b/dm/dm-17.tex
diff --git a/dm-18.tex b/dm/dm-18.tex
index e764814..e764814 100644
--- a/dm-18.tex
+++ b/dm/dm-18.tex
diff --git a/dm-19.tex b/dm/dm-19.tex
index 12aa975..6d77d4b 100644
--- a/dm-19.tex
+++ b/dm/dm-19.tex
@@ -26,7 +26,7 @@
что каждые две отличаются не более чем на 1.
Пусть мы умеем, запросив значение в этих числах, понимать в какой части
(или в каких двух подряд идущих частях) лежит максимум.
- Тогда за 2 хода мы умеем уменьшать рассматриваемый массив в $\dfrac{3}{2}$ раза, а значит, мы сведем задачу к поиску максимума на массиве из менее чем трех элементов, а значит, решим задачу за менее чем $4\log_3 n$ ходов.
+ Тогда за 2 хода мы умеем уменьшать рассматриваемый массив в $\frac{3}{2}$ раза, а значит, мы сведем задачу к поиску максимума на массиве из менее чем трех элементов, а значит, решим задачу за менее чем $4\log_3 n$ ходов.
\dmquestion{2}
diff --git a/intro.sty b/intro.sty
new file mode 100644
index 0000000..2c9efac
--- /dev/null
+++ b/intro.sty
@@ -0,0 +1,94 @@
+\NeedsTeXFormat{LaTeX2e}
+\ProvidesPackage{intro}[2020/04/04 Math tasks intro]
+
+\usepackage[no-math]{fontspec}
+\setmainfont{Gentium Plus}
+\setmonofont{Inconsolata LGC}
+\newfontfamily\slab{Roboto Slab}
+\newfontfamily\roboto{Roboto}
+
+\setlength\headheight{13.6pt}
+
+\usepackage{
+ amsmath, amssymb, mathtools, relsize,
+ graphicx, xcolor,
+ fancyhdr,
+ hyperref,
+ enumerate,
+ framed,
+ lastpage
+}
+\usepackage{datetime2}
+\usepackage[shortlabels]{enumitem}
+\usepackage{mathshit}
+
+
+\flushbottom
+
+\renewcommand{\baselinestretch}{1.1}
+
+\usepackage{geometry}
+\geometry{
+ b5paper,
+ left=0.4in, right=0.4in,
+ top=0.5in, bottom=0.6in,
+ headsep=.1in
+}
+
+\newcommand{\question}[2]{
+ \def\FrameCommand{
+ \fboxsep=\FrameSep\fcolorbox{gray}{white}
+ }
+ \MakeFramed{
+ \advance\hsize-\width \FrameRestore
+ }
+ \noindent {\small \roboto \textbf{#1}}
+ #2
+ \endMakeFramed
+}
+
+\newcommand{\dmquestion}[1]{
+ \begin{center}
+ {
+ \color{darkgray}
+ \framebox[3em]{
+ \color{black}
+ \texttt{#1}
+ }
+ }
+ \end{center}
+}
+
+\setlength\FrameSep{0.75em}
+\setlength\OuterFrameSep{\partopsep}
+
+\hypersetup{colorlinks=true, linkcolor=magenta}
+
+% -- Left/right header text and footer (to appear on every page) --
+\pagestyle{fancy}
+\renewcommand{\footrulewidth}{0pt}
+\renewcommand{\headrulewidth}{0pt}
+
+% clear first page
+\fancypagestyle{plain}
+{
+ \fancyhf{}
+ \renewcommand{\headrulewidth}{0pt}
+ \renewcommand{\footrulewidth}{0pt}
+}
+
+\cfoot{}
+\rfoot{\footnotesize \texttt{\thepage}}
+
+\author{Шарафатдинов Камиль \and БПМИ192}
+\date{билд \qquad \small\texttt{\DTMnow+0300}}
+
+\setlength{\leftskip}{24pt}
+\setlength{\rightskip}{24pt}
+
+% simplified inkscape svg insertion
+\newcommand{\drawcat}[1]{
+ \begin{figure}
+ \includegraphics[width=#1,page=1]{cat.pdf}
+ \end{figure}
+}
diff --git a/intro.tex b/intro.tex
deleted file mode 100644
index 2281906..0000000
--- a/intro.tex
+++ /dev/null
@@ -1,59 +0,0 @@
-\usepackage[sfdefault,condensed,scaled=0.9]{roboto}
-%\usepackage[T2A]{fontenc}
-%\usepackage{inconsolata}
-%\setmonofont[Scale=0.9]{Inconsolata}
-
-\setlength\headheight{13.6pt}
-
-\usepackage{
- amsmath, amssymb, mathtools, relsize,
- graphicx, xcolor,
- fancyhdr, hyperref, enumerate
-}
-\usepackage[shortlabels]{enumitem}
-\usepackage{mathshit}
-
-\flushbottom
-
-\renewcommand{\baselinestretch}{1.1}
-
-\usepackage{geometry}
-\geometry{
- b5paper,
- left=0.4in, right=0.4in,
- top=0.5in, bottom=0.6in,
- headsep=.1in
-}
-
-\newenvironment{cframed}[1][gray]
-{
- \def\FrameCommand{
- \fboxsep=\FrameSep\fcolorbox{#1}{white}
- }
- \MakeFramed{
- \advance\hsize-\width \FrameRestore
- }
-}
-{\endMakeFramed}
-
-\newcommand{\question}[2]{
- \bigskip\bigskip
- \begin{cframed}
- \noindent \textbf{#1}
- #2
- \end{cframed}
-}
-
-
-\setlength\FrameSep{0.75em}
-\setlength\OuterFrameSep{\partopsep}
-
-\hypersetup{colorlinks=true, linkcolor=magenta}
-
-% -- Left/right header text and footer (to appear on every page) --
-\pagestyle{fancy}
-\renewcommand{\footrulewidth}{0.4pt}
-\renewcommand{\headrulewidth}{0.4pt}
-
-\cfoot{}
-\rfoot{\thepage}
diff --git a/last.tex b/last.tex
new file mode 120000
index 0000000..d90caf2
--- /dev/null
+++ b/last.tex
@@ -0,0 +1 @@
+alg/alg-1.tex \ No newline at end of file
diff --git a/mathshit.sty b/mathshit.sty
index 5a9cd02..b4ca786 100644
--- a/mathshit.sty
+++ b/mathshit.sty
@@ -1,11 +1,13 @@
\NeedsTeXFormat{LaTeX2e}
\ProvidesPackage{mathshit}[2020/04/04 Math Shit]
-\RequirePackage{framed}
-
\DeclareMathOperator{\dif}{d \!}
+\DeclareMathOperator{\ord}{ord}
+\DeclareMathOperator{\lcm}{lcm}
+
\newcommand{\norm}[1]{\left\lVert#1\right\rVert}
\newcommand{\dist}[2]{\norm{#1 - #2}}
+\newcommand{\cycle}[1]{\langle #1 \rangle}
\makeatletter
\newenvironment{sqcases}{%
@@ -22,11 +24,7 @@
\makeatother
\DeclareRobustCommand{\divby}{%
- \mathrel{\vbox{\baselineskip.65ex\lineskiplimit0pt\hbox{.}\hbox{.}\hbox{.}}}%
-}
-
-\newcommand{\dmquestion}[1]{
- \begin{center} \textbf{#1} \end{center}
+ \mathrel{\vbox{\baselineskip.5ex\lineskiplimit0pt\hbox{$.$}\hbox{$.$}\hbox{$.$}}}%
}
\newcommand{\br}[1]{\left( #1 \right)}
generated by cgit v1.2.1-18-gbd029 (git 2.20.0) at 2025-07-30 17:26:20 +0000