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  1. \newtheorem{definition}{Definition}[section]
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  4. \newtheorem*{problem 1}{Problem 1}
  5. \newtheorem*{task4}{Task 4}
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  15. \newtheorem{theorem}{Theorem}
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  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}
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  26. \newcommand{\note}{\underline{Note:} }
  27. \newcommand{\fact}{\underline{\textbf{Fact}:} }
  28. \newcommand{\example}{\underline{Example:} }
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  30. \usepackage{amsmath,amsthm,amssymb}
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  63. \DeclareMathOperator*\lowlim{\underline{lim}}
  64. \DeclareMathOperator*\uplim{\overline{lim}}
  65. \newcommand{\nd}{\mathbin{\&}}
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  79. %\newcommand{\E}{\mathbb{E}} %
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  81. %\DeclareRobustCommand{\E}{\operatorname{\mathd%s{E}}\@ifstar\@firstofone\@E}
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  120. \newcommand{\hence}{\longrightarrow}
  121. \newcommand{\bto}{\Longrightarrow}
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  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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