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\documentclass{article}
\usepackage{amsmath,amssymb}

\begin{document}

Step 1
$\left[ \frac{n}{2} \right]$

Step 2
\[ 
   \sum_{i = 1}^{ \left[ \frac{n}{2} \right] } 
\]

Step 3
\[ 
   x_{i, i + 1}^{i^{2}} \qquad \left[ \frac{i + 3}{3} \right] 
\]

Step 4
\[
   \binom{ x_{i,i + 1}^{i^{2}} }{ \left[ \frac{i + 3}{3} \right] } 
\]

Step 5
$\sqrt{ \mu(i)^{ \frac{3}{2} } (i^{2} - 1) }$

Step 6
$\sqrt[3]{ \rho(i) - 2 }$  $\sqrt[3]{ \rho(i) - 1 }$

Step 7
\[
   \frac{ \sqrt{ \mu(i)^{ \frac{3}{2}} (i^{2} -1) } } 
        { \sqrt[3]{\rho(i) - 2} + \sqrt[3]{\rho(i) - 1} } 
\]

Step 8
\[
   \sum_{i = 1}^{ \left[ \frac{n}{2} \right] }
      \binom{ x_{i, i + 1}^{i^{2}} }
            { \left[ \frac{i + 3}{3} \right] }
      \frac{ \sqrt{ \mu(i)^{ \frac{3}{2}} (i^{2} - 1) } } 
           { \sqrt[3]{\rho(i) - 2} + \sqrt[3]{\rho(i) - 1} }
\]

\[\sum_{i=1}^{\left[\frac{n}{2}\right]}\binom{x_{i,i+1}^{i^{2}}}
{\left[\frac{i+3}{3}\right]}\frac{\sqrt{\mu(i)^{\frac{3}
{2}}(i^{2}-1)}}{\sqrt[3]{\rho(i)-2}+\sqrt[3]{\rho(i)-1}}\] 

\[\sum_{i=1}^{\left[\frac{n}{2}\right]}\binom{x_{i,i+1}^{i^{2}}}
{\left[\frac{i+3}{3}\right]}\frac{\sqrt{\mu(i)^{\frac{3}
{2}}}(i^{2}-1)}}{\sqrt[3]{\rho(i)-2}+\sqrt[3]{\rho(i)-1}}\] 


Formula 1 
\[
   x \mapsto \{\, c \in C \mid c \leq x \,\} 
\]

Formula 2
\[ 
   \left| \bigcup (\, I_{j} \mid j \in J \,) \right| 
    < \mathfrak{m}
\]

Formula 3
\[ 
   A = \{\, x \in X \mid x \in X_{i} 
            \mbox{\quad for some } i \in I \,\} 
\]

Formula 4
\[
   \langle a_{1}, a_{2} \rangle \leq \langle a'_{1}, a'_{2}\rangle
    \qquad \mbox{if{f}} \qquad a_{1} < a'_{1} \quad  \mbox{or} 
    \quad a_{1} = a'_{1} \mbox{ and } a_{2} \leq a'_{2} 
\]

Formula 5
\[ 
   \Gamma_{u'} = \{\, \gamma \mid \gamma < 2\chi, 
    \ B_{\alpha} \nsubseteq u', \ B_{\gamma} \subseteq u' \,\} 
\]

Formula 6
\[
   A = B^{2} \times \mathbb{Z}
\]

Formula 7
\[
   \left( \bigvee (\, s_{i} \mid i \in I \,) \right)^{c} = 
    \bigwedge (\, s_{i}^{c} \mid i \in I \,) 
\]

Formula 8
\[ 
   y \vee \bigvee (\, [B_{\gamma}] \mid \gamma 
    \in \Gamma \,) \equiv z \vee \bigvee (\, [B_{\gamma}] 
    \mid \gamma \in \Gamma \,) \pmod{ \Phi^{x} } 
\]

Formula 9
\[ 
   f(\mathbf{x}) = \bigvee\nolimits_{\!\mathfrak{m}} 
    \left(\, 
       \bigwedge\nolimits_{\mathfrak{m}}
       (\, x_{j} \mid j \in I_{i} \,) \mid i < \aleph_{\alpha} 
    \,\right)
\]

Formula 10
\[
   \left. F(x) \right|_{a}^{b} = F(b) - F(a)
\]

Formula 11
\[
  u \underset{\alpha}{+} v \overset{1}{\thicksim} w 
    \overset{2}{\thicksim} z 
\]

Formula 12
\[
   f(x) \overset{ \text{def} }{=} x^{2} - 1
\]

Formula 13
\[
   \overbrace{a + b + \cdots + z}^{n}
\]

Formula 14
\[
   \begin{vmatrix} 
      a + b + c & uv\\ 
      a + b & c + d 
   \end{vmatrix}
   = 7
\]

\[
   \begin{Vmatrix} 
      a + b + c & uv\\ 
      a + b & c + d 
   \end{Vmatrix} 
   = 7
\]

Formula 15
\[
   \sum_{j \in \mathbf{N}} b_{ij} \hat{y}_{j} = 
   \sum_{j \in \mathbf{N}} b^{(\lambda)}_{ij} \hat{y}_{j} + 
   (b_{ii} - \lambda_{i}) \hat{y}_{i} \hat{y} 
\]

Formula 16
\[
   \left( \prod^n_{\, j = 1} \hat x_{j} \right) H_{c} = 
    \frac{1}{2} \hat k_{ij} \det \hat{ \mathbf{K} }(i|i)
\]

\[
   \biggl( \prod^n_{\, j = 1} \hat x_{j} \biggr) H_{c} = 
    \frac{1}{2} \hat{k}_{ij} \det \widehat{ \mathbf{K} }(i|i) 
\]

Formula 17
\[
   \det \mathbf{K} (t = 1, t_{1}, \dots, t_{n}) = 
    \sum_{I \in \mathbf{n} }(-1)^{|I|}
    \prod_{i \in I} t_{i} 
    \prod_{j \in I} (D_{j} + \lambda_{j} t_{j}) 
    \det \mathbf{A}^{(\lambda)} (\overline{I} | \overline{I}) = 0 
\]

Formula 18
\[
   \lim_{(v, v') \to (0, 0)} 
    \frac{H(z + v) - H(z + v') - BH(z)(v - v')}
         {\| v - v' \|} = 0 
\]

Formula 19
\[
   \int_{\mathcal{D}} | \overline{\partial u} |^{2} 
    \Phi_{0}(z) e^{\alpha |z|^2} \geq 
    c_{4} \alpha \int_{\mathcal{D}} |u|^{2} \Phi_{0} 
    e^{\alpha |z|^{2}} + c_{5} \delta^{-2} \int_{A} 
    |u|^{2} \Phi_{0} e^{\alpha |z|^{2}} 
\]

Formula 20
\[ 
   \mathbf{A} = 
   \begin{pmatrix}
      \dfrac{\varphi \cdot X_{n, 1}}
            {\varphi_{1} \times \varepsilon_{1}} 
      & (x + \varepsilon_{2})^{2} & \cdots
      & (x + \varepsilon_{n - 1})^{n - 1} 
      & (x + \varepsilon_{n})^{n}\\
      \dfrac{\varphi \cdot X_{n, 1}}
            {\varphi_{2} \times \varepsilon_{1}} 
      & \dfrac{\varphi \cdot X_{n, 2}}
              {\varphi_{2} \times \varepsilon_{2}} 
      & \cdots & (x + \varepsilon_{n - 1})^{n - 1} 
      & (x + \varepsilon_{n})^{n}\\
      \hdotsfor{5}\\
      \dfrac{\varphi \cdot X_{n, 1}}
            {\varphi_{n} \times \varepsilon_{1}} 
      & \dfrac{\varphi \cdot X_{n, 2}}
              {\varphi_{n} \times \varepsilon_{2}} 
      & \cdots & \dfrac{\varphi \cdot X_{n, n - 1}}
                       {\varphi_{n} \times \varepsilon_{n - 1}} 
      & \dfrac{\varphi\cdot X_{n, n}}
              {\varphi_{n} \times \varepsilon_{n}}
   \end{pmatrix} 
    + \mathbf{I}_{n}
\]

\end{document}

