batch_output.txt

y_{4} - y_{5} - y_{6} + y_{a} - y_{b} - y_{c}
$\pi(x) = \pi(y)$
$4\pi^2 a^3$
a^{1} + b^{2} + c^{3} + d^{4}
$x^2=2x+1$
$-2ab \cos\gamma$
$\frac{1+\sqrt{5}}{2}$
\cos \left( \frac{\pi}{2} + \alpha \right) = - \sin \alpha
$(a + b)^2 = a^2 +2ab + b^2$
$a = 2$
$x^3=x^2+x+1$
$\frac{a}{c}=\frac{c}{b}$
\sqrt{a^{2} + b^{2} + c^{2}}
\lim_{x \rightarrow 0} \left( 1 + x \right)^{\frac{1}{x}}
$x=1$
$3-n = 1$
$e^{ix}\neq 1$
$2^e-1$
\lim_{x \rightarrow \infty} \frac{1}{x^{2}} = 0
\lim_{x \rightarrow 0} \frac{\left( 1 - \cos x \right) \left( 1 + \cos x \right)}{x^{2} \left( 1 + \cos x \right)}
$\sum_{k=1}^{\infty} \frac{a_{k}}{k!}$
a x^{2} + b x + c = 0
e^{x^{- \left( \frac{x}{2} \right)^{n}}}
$\frac{4}{3}\pi a b^2$
$y_i = a x_i + b$
$ z = 1 + \cos 2 \theta + i \sin 2 \theta $
$\frac{1}{\sqrt{x^2-1}}$
$\log_ca$
$1 + \sqrt{2}$
$a^2 + 2ab + b^2 - 2ab$
$\sin(1)$
$\frac{\sqrt{3}}{2}+\frac{1}{2}i$
\sum^{n}_{k = 1} \frac{k + 1}{a_{k} a_{k + 1}}
\lim_{x \rightarrow 2} \frac{x - \sqrt{3 x - 2}}{\sqrt{x + 2} - 2}
$-\frac{1}{x^2}$
$2^{n-1}$
$63 - 187 \neq -169$
$\frac{x}{c}$
$\frac{1}{b+1}$
$(a-d)^2+4bc$
$\frac{b-a}{2^{n+1}}$
$i_1 i_2=i_3$
x^{i} + x^{i + 1} + x^{i + 2}
$\frac{5}{2} \sqrt{5+2\sqrt{5}}$
a^{1} + b^{2} + c^{3} + d^{4}
$1 + 1 = 3$
$\frac{1}{24}\pi^4$
$2^{-1}$
$x_1 = -2 + i$
\log_{b} a = \frac{1}{\log_{a} b}
x = 2 n \pi \pm \frac{\pi}{3}
x^{i} + x^{i + 1} + x^{i + 2}
$a k+n$
$(n+1)^2$
$(x_n)$
\lim_{x \rightarrow \frac{\pi}{2} - 0} \tan x = \infty
$\cos(\pi) = - 1$
x^{2} + y^{2} = z^{2}
\sum^{n}_{k = 1} k = \frac{1}{2} \left( n^{2} + n \right)
\left( y + 1 \right)^{2} = y^{2} + 2 y + 1
$2i\pi$
$ ( A - B ) \times C = ( A \times C ) - ( B \times C ) $
$\log_ca$
$\frac{1}{n(n+1)(n+2)}$
$x = \frac{2}{3}$
$ \frac{a^{2}+b^{2}} {a + b} + \frac{ b^{2}+c^{2}} {b+c} +\frac{ c^{2}+a^{2}} {c+a} \geq a + b +c $
\log_{b} a = \frac{1}{\log_{a} b}
$3+2\sqrt{2}$
$\frac {1 + \frac { 1}{4}x}{1 - \frac { 3}{4}x+ \frac { 1}{4}x^2 - \frac 1{24}x^3}$
\log_{b} a = \frac{1}{\log_{a} b}
$2^{k - 1} + k$
$222_1$
$4\pi^2 a^3$
$i^4 = 1$
\sqrt{2} ( \frac{1}{\sqrt{2}} \sin x + \frac{1}{\sqrt{2}} \cos x )
$(-i)^2 = -1$
$\frac{\sqrt{3}}{3}$
a x^{2} + b x + c = 0
$2a = b$
$\frac{\pi}{2}$
$2(2-1)=(1+1)(2-1)$
$\sum_{k=1}^{\infty} \frac{a_{k}}{k!}$
$4_12_12$
\log_{b} a = \frac{1}{\log_{a} b}
- 1 \leq \cos \theta \leq 1
$1+i$
$178 - 64 + 21 \leq 93$
$y^2=x^3+ax+b$
\sum^{n}_{k = 1} \frac{k + 1}{a_{k} a_{k + 1}}
$(B\times C)$
$\frac{-\sqrt{3}}{2}+\frac{1}{2}i$
\sin \alpha = \sqrt{1 - \cos^{2} \alpha}
$2^{43}$
$i^4 = 1$
\lim_{x \rightarrow \infty} \frac{3 x - 2}{x + 3}
$x^{(k)}$
\lim_{x \rightarrow 2} \frac{x - \sqrt{3 x - 2}}{\sqrt{x + 2} - 2}
$\frac{1 + \sqrt{5}}{2}$
\lim_{x \rightarrow 2} \frac{x - \sqrt{3 x - 2}}{\sqrt{x + 2} - 2}
$e^{inx}$
x^{n} + y^{n} = z^{n}
x^{i} + x^{i + 1} + x^{i + 2}
$x = a$
$d = 1$
\frac{z - z - z - z - z}{y - y - y - y - y}
\left( a^{2} + b^{2} \right) \left( c^{2} + d^{2} \right) \geq \left( a c + b d \right)^{2}
$k_2 = k_1$
\int \frac{x^{2} - e^{2 x}}{x - e^{x}} d x
$2\sin(x)$
\lim_{x \rightarrow 0} \left( 1 + x \right)^{\frac{1}{x}}
$(n - 1)$
\int \frac{x^{2} - e^{2 x}}{x - e^{x}} d x
$\sin x=\frac{e^{ix}-e^{-ix}}{2i}$
$n(2n+1)$
\sum^{n}_{k = 1} \frac{k + 1}{a_{k} a_{k + 1}}
$i_1 + i_4 = i_2 + i_3$
$(a + bi) (a - bi) = a^2 + b^2$
$\cos (a - b) = \cos a \cos b + \sin a \sin b$
$(171 \times 137) \times 83 = 1944441$
$(2 - 3i)$
\left( y + 1 \right)^{2} = y^{2} + 2 y + 1
\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{5}}}}}
\sin \left( \alpha + \beta \right) \sin \left( \alpha - \beta \right) = \sin^{2} \alpha - \sin^{2} \beta
\sqrt{2} ( \frac{1}{\sqrt{2}} \sin x + \frac{1}{\sqrt{2}} \cos x )
$\frac{n}{d}$
$y = b$
$\frac{4}{3}\pi$
$y = x^n$
$n_1+\cdots+n_{k}=n$
\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{5}}}}}
$e^{i\pi}= -1$
$x = \sum_{i=1}^k a_i e_i$
$- \sin x$
\frac{z - z - z - z - z}{y - y - y - y - y}
$\sin(\frac x2)$
$e^{i\pi}= -1$
$(38 \times (28 - 126)) - 70 \leq -3793$
$y=ax^2+bx+c$
x^{n} + y^{n} = z^{n}
$x + x^2$
$x^3 + 3(2 - \sqrt{3})x^2 - 3x - 2 + \sqrt{3}$
$\log_aa = 1$
$\frac{n(n+1)}{2}$
\lim_{x \rightarrow \infty} \frac{3 x - 2}{x + 3}
\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{5}}}}}
a^{1} + b^{2} + c^{3} + d^{4}
$2^{2^1}+1$
\lim_{x \rightarrow \frac{\pi}{2} - 0} \tan x = \infty
$(a-b)^2 = a^2 - 2ab + b^2$
$2^{e} - 1$
a_{i} + a_{j} + a_{k} = b_{i}
x = 2 n \pi \pm \frac{\pi}{3}
$a_n = 1$
\sin \left( \alpha + \beta \right) \sin \left( \alpha - \beta \right) = \sin^{2} \alpha - \sin^{2} \beta
$\sin(\frac{\pi}{3})$
$c = \sum_{i=1}^{n} x_i a_i$
\sum^{n}_{k = 1} \frac{k + 1}{a_{k} a_{k + 1}}
$\frac{1}{2}\pi^2$
$b_i(k)$
\frac{z - z - z - z - z}{y - y - y - y - y}
\lim_{x \rightarrow \infty} \frac{1}{x^{2}} = 0
$a_3=1$
$1 - \frac {1 }{ 3} = \frac {2 }{ 3}$
$e_3 = 15$
\lim_{x \rightarrow \infty} \frac{1}{x^{2}} = 0
\frac{a + b + c + d + e}{a + b + c + d}
$2^{2^{2^2}}$
$\cos(\pi) = -1$
$\frac{\sqrt{4}}{2}$
$a^2 + b^2 + c^2 + d^2 = 1$
$a + b$
${2 -\frac{\log(\sqrt{2})}{\log(2)}}$
$\frac{n^2}{2}+\frac{n}{2}+\frac{n^2}{2}-\frac{n}{2}$
$\phi_j(k)$
\left( y + 1 \right)^{2} = y^{2} + 2 y + 1
$n = 1$
$e_2 = 21$
$anx^{n-1}$
\lim_{x \rightarrow \infty} \frac{3 x - 2}{x + 3}
$n^{(1)}$
a x^{2} + b x + c = 0
$k=n$
\lim_{x \rightarrow \frac{\pi}{2} - 0} \tan x = \infty
$\sum_{k=1}^{\infty} \frac{a_{k}}{k!}$
$a^2 + b^2 + c^2$
$45 - 98 + 170 - 125 + 76 \neq 77$
$\frac{\pi-3}{\sqrt{2}}$
$2 = \frac {(1 + 3) }{ 2}$
e^{x^{- \left( \frac{x}{2} \right)^{n}}}
$\frac{n(n-3)}{2}$
y_{4} - y_{5} - y_{6} + y_{a} - y_{b} - y_{c}
x^{2} + y^{2} = z^{2}
\sqrt{2} \left( \frac{1}{\sqrt{2}} \sin x + \frac{1}{\sqrt{2}} \cos x \right)
$n=b-a+1$
$x^3+ax+b$
$-\sqrt{y}$
$\gamma_{12}$
$e^{-x}$
$y =x^2$
$(A \times B)$
$a^{b^{c}}$
\left( a^{2} + b^{2} \right) \left( c^{2} + d^{2} \right) \geq \left( a c + b d \right)^{2}
$(a_{n})$
\sum^{n}_{k = 1} k = \frac{1}{2} \left( n^{2} + n \right)
\sqrt{a^{2} + b^{2} + c^{2}}
a^{1} + b^{2} + c^{3} + d^{4}
$ax^2 = c$
$(2i)^n$
$\frac{-1 + i\sqrt{3}}{2}$
\left( y + 1 \right)^{2} = y^{2} + 2 y + 1
$(a+b)^2 = a^2 + 2ab + b^2$
\lim_{x \rightarrow \infty} \frac{1}{x^{2}} = 0
x = 2 n \pi \pm \frac{\pi}{3}
\sin \alpha = \sqrt{1 - \cos^{2} \alpha}
$\sqrt{-1}$
$1 + a = 1$
\int \frac{x^{2} - e^{2 x}}{x - e^{x}} d x
$y = x^2 + \frac1x$
$(B+C)=A$
$ F ( b ) - F ( a )$
$n(n+2)$
$\frac{5}{2}$
$1 + \frac 1{1 + \frac { 1}{1}}$
$(a+b)^2=a^2+2ab+b^2$
$y^{n-1}$
\sqrt{2} \left( \frac{1}{\sqrt{2}} \sin x + \frac{1}{\sqrt{2}} \cos x \right)
$y=x^2+5x-3$
$(a+b)^2=a^2+b^2+2ab$
\lim_{x \rightarrow 0} \frac{\left( 1 - \cos x \right) \left( 1 + \cos x \right)}{x^{2} \left( 1 + \cos x \right)}
$\frac{\sqrt{3}}{2}+\frac{1}{2}i$
$\frac{1}{d^2}$
\cos \left( \frac{\pi}{2} + \alpha \right) = - \sin \alpha
- 1 \leq \cos \theta \leq 1
$(a+b)(a-b)$
$y^2=x^3-n^2x$
$2^{2^{2^2}}$
$x+k$
$i = \sqrt{-1}$
\lim_{x \rightarrow \frac{\pi}{2} - 0} \tan x = \infty
$\frac cb$
$(100 \div 198) \times (172 + 102 + 2) \geq 138$
$\sqrt {14}$
$2 = 1$
$\frac 1{1 - x + \frac {x^2}2}$
\frac{a + b + c + d + e}{a + b + c + d}
$x^4-x^2$
$\sum\frac{(-1)^n}n$
$\frac{-1}{n+1}$
$\cos(\pi) = - 1$
$a b + b a$
\cos \left( \frac{\pi}{2} + \alpha \right) = - \sin \alpha
$\sqrt{\sqrt{n^2}}+1$
$x = \sum_{i=1}^k a_i e_i$
\lim_{x \rightarrow 2} \frac{x - \sqrt{3 x - 2}}{\sqrt{x + 2} - 2}
\sin \left( \alpha + \beta \right) \sin \left( \alpha - \beta \right) = \sin^{2} \alpha - \sin^{2} \beta
\sin \left( \alpha + \beta \right) \sin \left( \alpha - \beta \right) = \sin^{2} \alpha - \sin^{2} \beta
$8 \times 122 \geq 975$
$x_n=b$
$\sqrt{a + b\sqrt {c}} = d + e\sqrt {c}$
$y=-x^3-5x^2+\frac{2}{3}x-1$
$\frac{n+2}{2}$
$(b_n + a_n)$
$132$
$3 + 2 = \log_2 32$
$2^{\sqrt{2}}$
y_{4} - y_{5} - y_{6} + y_{a} - y_{b} - y_{c}
$\sin(x)=\sin(y)$
$2^{-1}$
$ a \lt x \leq b $
$a = \sqrt{5}b$
$n+i$
a_{i} + a_{j} + a_{k} = b_{i}
$(2^n - 1)^2 - 2$
$a^2 + 2ab + b^2 - 2ab$
\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{5}}}}}
$(a + b)^2 = a^2 +2ab + b^2$
x^{n} + y^{n} = z^{n}
$x+2$
a x^{2} + b x + c = 0
e^{x^{- \left( \frac{x}{2} \right)^{n}}}
$i^2 = -1$
\frac{z - z - z - z - z}{y - y - y - y - y}
\int \frac{x^{2} - e^{2 x}}{x - e^{x}} d x
$(c_n)$
x^{n} + y^{n} = z^{n}
$ x + \frac{x^3}{3!} + \frac{x^5}{5!}+\ldots$
$y_{1} = 55$
$2 = \sqrt{2+2}$
$c^2 = \frac{1}{2}d^2$
$k_y = k \sin \theta \sin \phi$
$a(e-a^{-1}(a-x))=x$
$\sqrt{(5-x)d}$
$x_4 = \sqrt{2}$
\lim_{x \rightarrow 0} \left( 1 + x \right)^{\frac{1}{x}}
\frac{a + b + c + d + e}{a + b + c + d}
$n(1)$
\sin \alpha = \sqrt{1 - \cos^{2} \alpha}
$4c = 2\sqrt{2} d$
\sin \alpha = \sqrt{1 - \cos^{2} \alpha}
\left( a^{2} + b^{2} \right) \left( c^{2} + d^{2} \right) \geq \left( a c + b d \right)^{2}
\lim_{x \rightarrow 0} \frac{\left( 1 - \cos x \right) \left( 1 + \cos x \right)}{x^{2} \left( 1 + \cos x \right)}
$x = \frac{1}{3}$
x = 2 n \pi \pm \frac{\pi}{3}
$\frac{\frac {2}{1}}{\frac {3}{2}}=\frac{4}{3}$
$179 - 158 - 4 - 130 = 155$
$y^{(k)}$
$x^2=ax+b$
$((173 \div 142) - 23) + ((8 + 180) \div 129) \geq -20$
$((151 + 38) \times (11 \times 111)) - 98 \leq 230671$
\sum^{n}_{k = 1} k = \frac{1}{2} \left( n^{2} + n \right)
- 1 \leq \cos \theta \leq 1
$x=x_{c}+y$
$\sum_{i=1}^n (2i-1) = n^2$
$x \neq y$
$\sqrt{1+x^2}$
- 1 \leq \cos \theta \leq 1
$2^2 - 1$
\left( a^{2} + b^{2} \right) \left( c^{2} + d^{2} \right) \geq \left( a c + b d \right)^{2}
$b = ak$
$i_{1}=4$
x^{2} + y^{2} = z^{2}
x^{2} + y^{2} = z^{2}
\sqrt{a^{2} + b^{2} + c^{2}}
e^{x^{- \left( \frac{x}{2} \right)^{n}}}
y_{4} - y_{5} - y_{6} + y_{a} - y_{b} - y_{c}
a_{i} + a_{j} + a_{k} = b_{i}
a_{i} + a_{j} + a_{k} = b_{i}
$2=1+1$
$i^2 = -1$
\sqrt{a^{2} + b^{2} + c^{2}}
\lim_{x \rightarrow \infty} \frac{3 x - 2}{x + 3}
$\frac{2\pi}{3}$
$cn = 1$
\lim_{x \rightarrow 0} \frac{\left( 1 - \cos x \right) \left( 1 + \cos x \right)}{x^{2} \left( 1 + \cos x \right)}
$x^3=x+1$
$y = \frac{d}{a}$
$y = x^4+b_1x^3+b_2x^2+b_3x+b_4$
\lim_{x \rightarrow 0} \left( 1 + x \right)^{\frac{1}{x}}
$\frac{3^{\frac{1}{2}}}{2}$
$2+\sqrt{2}$
$c = \sqrt{a^2-b^2}$
$i_1 + i_2$
\cos \left( \frac{\pi}{2} + \alpha \right) = - \sin \alpha
\frac{a + b + c + d + e}{a + b + c + d}
x^{i} + x^{i + 1} + x^{i + 2}
\sum^{n}_{k = 1} k = \frac{1}{2} \left( n^{2} + n \right)
$ax^2+bxy+cy^2$
$a^2+b^2+c^2=1$