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1. \newtheorem{definition}{Definition}[section]
2.
3.
4. \newtheorem*{problem 1}{Problem 1}
14.
15. \newtheorem{theorem}{Theorem}
16. \newtheorem{proposal}{Proposal}
17. \newtheorem{notice}{Notice}
18. \newtheorem{statement}{Statement}
19. \newtheorem{corollary}{Corollary}
20. \newtheorem{lemma}{Lemma}
21. \newtheorem{observation}{Observation}
22. \newtheorem{problem}{Problem}
23. \newtheorem{claim}{Claim}
24.
25.
26. \newcommand{\note}{\underline{Note:} }
27. \newcommand{\fact}{\underline{\textbf{Fact}:} }
28. \newcommand{\example}{\underline{Example:} }
29.
30. \usepackage{amsmath,amsthm,amssymb}
31. \usepackage[T1,T2A]{fontenc}
32. \usepackage[utf8]{inputenc}
33. \usepackage[english,russian]{babel}
34. \usepackage[all]{xy}
35. \usepackage[left=2cm,right=2cm, top=2cm,bottom=2cm,bindingoffset=0cm]{geometry}
36. \usepackage{mathtext}
37. \usepackage{amssymb}
38.
39. %\renewcommand{\Im}{\operatorname{Im}}
40. %\newcommand{\Perm}{\operatorname{S}}
41. %\newcommand{\sgn}{\operatorname{sgn}}
42. \usepackage{amsmath}
43. \newcommand{\Mat}{\operatorname{M}}
44. \usepackage{tabto}
45.
46. \renewcommand{\Re}{\mathrm{Re\:}}
47. \renewcommand{\Im}{\mathrm{Im\:}}
48. \newcommand{\Arg}{\mathrm{Arg\:}}
49. \renewcommand{\arg}{\mathrm{arg\:}}
50. \newcommand{\Mat}{\mathrm{Mat}}
51. \newcommand{\id}{\mathrm{id}}
52. \newcommand{\aut}{\mathrm{aut}}
53. \newcommand{\isom}{\xrightarrow{\sim}}
54. \newcommand{\leftisom}{\xleftarrow{\sim}}
55. \newcommand{\Hom}{\mathrm{Hom}}
56. \newcommand{\Ker}{\mathrm{Ker}\:}
57. \newcommand{\rk}{\mathrm{rk}\:}
58. \newcommand{\diag}{\mathrm{diag}}
59. \newcommand{\ort}{\mathrm{ort}}
60. \newcommand{\pr}{\mathrm{pr}}
61. \newcommand{\vol}{\mathrm{vol\:}}
62. \renewcommand{\mod}{\mathrm{\: mod\:}}
63. \DeclareMathOperator*\lowlim{\underline{lim}}
64. \DeclareMathOperator*\uplim{\overline{lim}}
65. \newcommand{\nd}{\mathbin{\&}}
66.
67. \newcommand{\X}{\mathbb{X}}
68. %\newcommand{\D}{\mathbb{D}}
69. \newcommand{\Y}{\mathbb{Y}}
70. %\newcommand{\I}{\mathbb{I}}
71. \makeatletter
72. %\DeclareRobustCommand{\I}{\operatorname{\mathd%s{I}}\@ifstar\@firstofone\@I}
73. %\newcommand{\@I}[1]{\left\{#1\right\}}
74. \makeatother
75.
76. \newcommand{\Z}{\mathbb{Z}}
77. \newcommand{\Qq}{\mathcal{Q}}
78. \newcommand{\N}{\mathbb{N}}
79. %\newcommand{\E}{\mathbb{E}} %
80. \makeatletter
81. %\DeclareRobustCommand{\E}{\operatorname{\mathd%s{E}}\@ifstar\@firstofone\@E}
82. %\newcommand{\@E}[1]{\left[#1\right]}
83. \makeatother
84.
85. \makeatletter
86. %\DeclareRobustCommand{\D}{\operatorname{\mathb%b{D}}\@ifstar\@firstofone\@D}
87. %\newcommand{\@D}[1]{\left[#1\right]}
88. \makeatother
89.
90. \makeatletter
91. %\DeclareRobustCommand{\Pr}{\operatorname{\mathds{P}}\@ifstar\@firstofone\@Pr}
92. %\newcommand{\@Pr}[1]{\left[#1\right]}
93. \makeatother
94.
95. \renewcommand{\S}{\mathbb{S}}
96. \newcommand{\Q}{\mathbb{Q}}
97. \newcommand{\R}{\mathbb{R}}
98. \newcommand{\B}{\mathbb{B}}
99. \renewcommand{\C}{\mathbb{C}}
100. \renewcommand{\L}{\mathscr{L}}
101. %\renewcommand{\P}{\mathds{P}}
102.
103.
104. \newcommand{\orthog}{\mathop{\bot}}
105. \renewcommand*\d{\mathop{}\!\mathrm{d}}
106. \renewcommand*\dd{\mathop{}\!\partial}
107.
108. %\renewcommand{\Pr}{\mathds{P}}
109. \newcommand{\pn}{\xrightarrow{\text{a. s.}}}
110. \newcommand{\pp}{\xrightarrow{\Pr}}
111. \newcommand{\pd}{\xrightarrow{d}}
112.
113.
114.
115.
116. \newcommand{\fe}{\varphi}
117. \newcommand{\e}{\varepsilon}
118. \newcommand{\ind}{\mathbin{\perp\!\!\!\perp}}
119. \newcommand{\Gauss}{\mathrm{Gauss}}
120. \newcommand{\hence}{\longrightarrow}
121. \newcommand{\bto}{\Longrightarrow}
122. \newcommand{\Bin}{\mathrm{Bin}}
123. \newcommand{\Bern}{\mathrm{Bern}}
124. \newcommand{\Geom}{\mathrm{Geom}}
125. \newcommand{\Uni}{\mathrm{U}}
126. \newcommand{\Exp}{\mathrm{Exp}}
127. \newcommand{\Ko}{\mathrm{Ko}}
128. \newcommand{\No}{\mathcal{N}}
129. \newcommand{\Pois}{\mathrm{Pois}}
130. \documentclass{report}
131. \documentclass[12pt, letterpaper]{article}
132. \documentclass[a4paper]{article}
133.
134. \usepackage[english]{babel}  % force American English hyphenation patterns
135. \usepackage[utf8x]{inputenc}  % unicode
136. \usepackage{graphicx}
137. \usepackage{wrapfig}
138. \usepackage{lipsum}  % generates filler text
139. \title{MCCME}
140. \author{mrkvmrmrkvmr }
141. \date{March 2020}
142.
143. \begin{document}
144. \textbf{7.1}
145.
146. \textbf{а} $\prod\limits_{i = 1}^{\infty} (1 - t^i) = \prod\limits_{i = 1}^{\infty}(1 + t^i + t^{2i} + \dots )$,
147. рассмотрим множитель при $t^{n}$ --- это окажется $p(n)$,
148. потому что степень $n$ может получится как раз таки $p(n)$ способами (просто раскрываем скобки).
149. \textbf{б} $\prod\limits_{i = 1}^{m} (1 - t^i) = \prod\limits_{i = 1}^{m}(1 + t^i + t^{2i} + \dots )$, рассмотрим множитель при $t^n$ --- это окажется $p(n, m)$, потому что степень $n$ может получится как раз таки $p(n, m)$ способами (просто раскрываем скобки).
150.
151. \textbf{7.2}
152.
153. \textbf{а} Из лекции известно
154. $$\sum\limits_{k = 1}^{\infty} p_k t^{k - 1}= p(t) = \frac{H(t)^\prime}{H(t)} = \frac{E(-t)^\prime}{E(-t)}$$
155. $\sum\limits_{k = 1}^{\infty} p_k t^{k - 1} \sum\limits_{k = 0}^{\infty} (-1)^k e_k(x_1, \dots , x_k) t^k = \sum\limits_{k = 0}^{\infty} (-1)^k k e_k(x_1, \dots , x_k) t^k$
156. \end{document}
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