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ddrg
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math_formulas

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int64
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14,830
(2^6 + 2 \cdot 2 \cdot 8 + 2^2 \cdot 2^2 \cdot 3)/12 = 12
2,120
y \cdot y + z^2 = 25 rightarrow (25 - z \cdot z)^{1/2} = y
14,169
-7 = k + 4\cdot \left(-1\right)\Longrightarrow k = -3
-9,183
-n\cdot 2\cdot 5\cdot 13\cdot n = -n^2\cdot 130
2,617
q + q^2 + \dots + q^n = 1 + 2*q + 3*q * q + \dots + q^{n + (-1)}*n
19,809
\dfrac{1}{x} + 1/x - \dfrac{1}{x^2} = \tfrac1x \cdot 2 - \frac{1}{x^2} \lt 2/x
-2,237
\frac{1}{12}*2 = \frac{1}{12}*7 - \dfrac{5}{12}
2,814
2*\sin{\frac{\theta}{2}}*\cos{\theta/2} = \sin{\theta}
-24,985
2*2\pi = 4\pi
29,131
1 + x^2 = 1 + x + x^2 - x
16,714
\frac{57}{26} = \frac{1}{1 - x} \cdot (1 - x \cdot x) \Rightarrow 26 \cdot x^2 - 57 \cdot x + 31 = 26 \cdot \left(x + (-1)\right) \cdot (x - \frac{31}{26}) = 0
2,939
5 = |7 \times \left(-1\right) + 2|
1,299
\cos{2 \cdot x} = 2 \cdot \cos^2{x} + (-1) = \dfrac{1 - \tan^2{x}}{1 + \tan^2{x}}
16,766
\sqrt{-x} \cdot \sqrt{-x} = \sqrt{-x \cdot (-x)} = \sqrt{x^2}
522
5 - 0 \cdot 3 + 9/3 = 5 + 0 \cdot (-1) + \frac93 = 5 + 0 \cdot \left(-1\right) + 3 = 5 + 3 = 8
11,638
1^2 \cdot 1 + 4^3 = (1 + 4)\cdot (1^2 - 4 + 4^2) = 5\cdot 13
15,850
\binom{r + (-1) + l + 2*(-1)}{2*(-1) + l} = \binom{3*(-1) + r + l}{2*\left(-1\right) + l}
6,268
\cos{z} = 1 - 2\cdot \sin^2{z/2} > 1 - \tfrac{z^2}{2}
32,496
3 \cdot \left(-1\right) + 29 + 17 \cdot (-1) + 11 \cdot (-1) + 7 = 5
11,685
\frac{1}{x^2 + 5} + 4 = \frac{1}{x^2 + 5}*(1 + 4*(x^2 + 5)) = \frac{1}{x^2 + 5}*(4*x^2 + 21)
33,636
9990 - 21 \cdot 21^2 = 9990 + 9261 \cdot (-1) = 729 = 9^3
-6,011
\frac{1}{3\cdot (1 + a)}\cdot 3 = \frac{3}{a\cdot 3 + 3}
37,793
38 \cdot \left(-1\right) + 431 = 393
1,247
y + (-1) = x \Rightarrow y = x + 1
24,630
0 = A \cdot G \Rightarrow A = 0\text{ or }G = 0
11,902
h^{2^{m + 1}} + (-1) = \left(h^{2^m}\right)^2 + (-1) = (h^{2^m} + (-1)) (h^{2^m} + 1)
12,265
\frac14 = \tfrac32\cdot \dfrac16
50,152
0 \cdot 2 \cdot 3 + 2 + 3 = 5
22,112
2 \cos\left(x\right) = e^{i x} + e^{-i x} = e^{i x} + e^{-i x} = 2 \cosh\left(i x\right)
23,500
3^{i + 1} = 3\cdot 3^i > 3\cdot i^2 = i^2 + i^2 + i^2 \geq i^2 + 2\cdot i + 1
29,796
\sin{z} = \left(e^{i\cdot z} - e^{-i\cdot z}\right)/(2\cdot i) = -i\cdot \sinh{i\cdot z}
9,403
\dfrac{5}{9} = -12/27 + 1
-7,674
\frac{1}{2 + i} (2 - i\cdot 9) \frac{1}{2 - i} (-i + 2) = \frac{1}{i + 2} (2 - i\cdot 9)
10,500
z + m\cdot z = (1 + m)\cdot z
557
\cot\left(\theta\right) = 1/\tan(\theta) = \cos(\theta)/\sin(\theta)
-26,664
(2*x + q*5)*(-q*5 + x*2) = x^2*4 - q^2*25
9,058
\frac{2}{3} = \frac{1}{3} + 1/3
6,267
\frac{\partial}{\partial x} \sqrt{x + z} = \frac{\frac{dz}{dx} + 1}{2 \cdot \sqrt{z + x}}
8,089
e^1 = \left(1 + 1/x\right)^x\cdot exp(0) = (1 + 1/x)^x
30,914
\cos{y} = -2 \cdot \sin^2{\frac{y}{2}} + 1
9,640
\lim_{x \to 2} \frac{1}{x + 2} = \lim_{x \to 2} \frac{1}{x^2 + 4\cdot (-1)}\cdot (2\cdot (-1) + x)
-5,718
\frac{1}{7 + z^2 + 8\cdot z}\cdot (z\cdot 2 + z\cdot 4 + 4 + 4\cdot z + 28) = \frac{z\cdot 10 + 32}{z^2 + z\cdot 8 + 7}
7,516
\cos(\pi \cdot 2 - z_0) = \cos{z_0}
4,874
2^{\frac13*(n + 1)} = 2^{1/3}*2^{n/3} > n*2^{\frac13}
26,600
B*A = x*B' \implies B' = \frac{B*A}{x}
-7,393
8/21 = 6/7*\dfrac{1}{9}*4
-21,604
\cos\left(\pi \cdot 3/2\right) = 0
3,927
F*z = F*z^2 \Rightarrow z \in F
-953
6 = \frac{3}{1}\cdot 2
-681
(e^{i*\pi*4/3})^{13} = e^{\frac{4*i*\pi}{3}*13}
39,948
3^{\dfrac1n} = 3^{1/n}
17,484
n^3 = 1 + (n + \left(-1\right))^3 + (n + (-1)) \cdot (n + (-1)) \cdot 3 + 3 \cdot (n + \left(-1\right))
10,740
e^x = e^{x + (-1)}\times e
19,117
x b + b t + (x + b) (b + t) = x b + b t + x b + x t + b + b t = x t + b
-23,159
-2 = \dfrac{3}{2}\cdot \left(-4/3\right)
3,586
0.0625 \times \left(0.0001 + ... + 0.1111\right) = 0.0625 \times 0.8888
28,649
-s^2 + s = 1/4 - (-\dfrac{1}{2} + s)^2
31,128
10^2 + 10*26 + 26 * 26 = 28*(3^2 + 3*4 + 4^2) = 28*37
17,086
\dfrac{1}{n \cdot (\left(\frac{x}{n}\right) \cdot \left(\frac{x}{n}\right) + 1)} = \dfrac{1}{n^2 + x^2} \cdot n
-18,073
23 \times (-1) + 32 = 9
-26,552
2x^2 - 40 x + 200 = 2(x^2 - 20 x + 100) = 2\left(x + 10 \left(-1\right)\right)^2
31,105
\tfrac{1}{1 + g \cdot g} = \frac{\mathrm{d}}{\mathrm{d}g} \arctan(g)
18,601
U_n = \left(U_n^2 + U_n\right)/2 + \left(-U_n \cdot U_n + U_n\right)/2
41,281
6*3^4 = 3^5*2
10,139
\dfrac{1}{12}\times 5\times \pi = 75\times \frac{1}{180}\times \pi
2,749
5 = n rightarrow -1 = (-1)^n
11,395
x + yx = x\cdot \left(y + 1\right)
-18,702
\left(-1\right)\times 0.1841 + 0.5793 = 0.3952
3,285
(1/2)^{-2/3} + \left(-1\right) = 2^{\frac{1}{3}2} + (-1) = 4^{1/3} + (-1)
-490
\pi \cdot 27/2 - \pi \cdot 12 = 3/2 \cdot \pi
13,613
1 = \sin(x) + 2.5\Longrightarrow \sin(x) = -1.5
-19,459
\frac{5 / 3}{1/6} \cdot 1 = 5/3 \cdot \dfrac11 \cdot 6
52,085
1 + 5 + 5 = 2 + 3 + 6
29,311
\frac{\mathrm{d}e}{\mathrm{d}x} \cdot \frac{\mathrm{d}e}{\mathrm{d}u} = \frac{\mathrm{d}e}{\mathrm{d}x}
-2,344
-\dfrac{2}{13} + \frac{1}{13}\cdot 3 = 1/13
50,121
\left(1 + z \cdot z\right)\cdot (1 + z^4)\cdot (1 + z^8)\cdot ...\cdot (1 + z^{2^k})\cdot (1 + z^{2^{k + 1}}) = \frac{1 - z^{2^{k + 1}}}{1 - z^2}\cdot (1 + z^{2^{k + 1}}) = \frac{1}{1 - z \cdot z}\cdot (1 - z^{2^{k + 1 + 1}})
24,828
\frac{q_1 q_2}{q_1+q_2}=\frac 1{\frac 1 {q_1}+\frac 1 {q_2}}
18,364
Z \times Y + X \times Y + Z \times X' = Z \times X' + X \times Y
13,771
2^{\frac{1}{2}} \cdot 2^{1/2} = 2
15,082
\frac{1}{1 + e^{b\cdot x}} = \frac{1}{e^{-x\cdot b} + 1}\cdot e^{-x\cdot b}
28,764
\frac{(-1) + y^2}{(-1) + y} = y + 1
1,904
\sqrt{2} (b_2 + b_1) + g_1 + g_2 = g_2 + b_2 \sqrt{2} + g_1 + \sqrt{2} b_1
9,607
x M/K = |x M| = \max{|x|,|M|} = \max{x/K,M/K} = x/K \frac1K M
-26,514
(8 \cdot z + 9 \cdot (-1)) \cdot (8 \cdot z + 9 \cdot (-1)) = (8 \cdot z)^2 - 2 \cdot z \cdot 8 \cdot 9 + 9^2
17,627
A x^2 = x^2 A
12,142
l^{d_1 + d_2} = l^{d_1}\cdot l^{d_2}
-22,316
n^2 + 11\cdot n + 24 = (n + 8)\cdot (3 + n)
19,803
g - b = -(-g + b)
21,893
\left(y + 2(-1) + y*3 + 6(-1) = 0 \Rightarrow -8 = y*4\right) \Rightarrow y = -2
-17,890
44 \cdot (-1) + 82 = 38
30,222
p^2 + (-1) = (p + (-1))\cdot (p + 1)
31,949
\frac{\pi}{2} + 2\cdot \pi = \frac{5\cdot \pi}{2}
20,432
\sin{s}\cdot \cos{s}\cdot 2 = \sin{2\cdot s}
7,515
U^2 + 36 U^2 = 37 U^2 = 37 U U
-28,854
11 = 55*(-1) + 66
9,003
-\dfrac{1}{5} + 1 = \frac15 \cdot 4
-19,283
\frac{2}{9}*\frac89 = \frac{\dfrac{1}{9}}{9*\frac18} 2
-20,258
\frac{1}{63}\cdot (9\cdot r + 36) = \frac99\cdot \frac{1}{7}\cdot (r + 4)
24,655
7 + 4 = v + 8 \Rightarrow 3 = v
13,555
8/216 = 2/36*\frac{1}{6}*4