id
int64 -30,985
55.9k
| text
stringlengths 5
437k
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|---|---|
7,129
|
\sqrt{x_k} = F_k \implies F_k^2 = x_k
|
-22,282
|
(y + \left(-1\right)) \cdot (7 \cdot (-1) + y) = y^2 - y \cdot 8 + 7
|
-7,740
|
\left(12 - 8 \cdot i\right)/(-4) = \dfrac{1}{-4} \cdot 12 - 8 \cdot i/(-4)
|
30,485
|
\sin(c_2 + c_1) = \sin{c_1} \cos{c_2} + \sin{c_2} \cos{c_1}
|
-17,518
|
31 = 42\times (-1) + 73
|
10,876
|
\frac12*(\sin(h - x) + \sin(x + h)) = \cos(x)*\sin(h)
|
-3,726
|
\dfrac{40}{5} \cdot \frac{1}{x^4} \cdot x^5 = \frac{1}{x^4 \cdot 5} \cdot x^5 \cdot 40
|
7,665
|
x*(-y) = -x y = -x y
|
9,049
|
\pi\times i/2 = i\times \pi\times 2/4
|
12,104
|
k\cdot d\cdot h = k\cdot d\cdot h
|
-20,787
|
(3*(-1) - 3*m)/((-2)*m)*\frac13*3 = \dfrac{1}{(-6)*m}*(-9*m + 9*(-1))
|
23,563
|
\sin(\pi/4)*\sin(0)*2 = 0
|
-1,086
|
-\frac43 \cdot \tfrac{7}{4} = \frac{\frac{1}{4} \cdot 7}{\frac{1}{4} (-3)}
|
8,008
|
\frac{3 + n}{(-1) + n} = 1 + \tfrac{4}{n + (-1)}
|
-19,538
|
\frac92\cdot 9/4 = \dfrac{\dfrac{1}{2}\cdot 9}{1/9\cdot 4}
|
15,797
|
-\frac{n}{n + (-1)} = -\frac{1}{n + (-1)}\cdot \left(n + (-1) + 1\right) = -(1 + \dfrac{1}{n + (-1)})
|
-1,408
|
\dfrac{1/5\cdot 3}{7\cdot \dfrac18} = 3/5\cdot \frac{8}{7}
|
30,303
|
\mathbb{E}\left(U_1 U_2\right) = \mathbb{E}\left(U_1\right) \mathbb{E}\left(U_2\right)
|
5,098
|
\tan{\frac{x}{2}} = \dfrac{\sin{x}}{1 + \cos{x}} = \frac{1}{\sin{x}}\cdot (1 - \cos{x})
|
18,658
|
1 - x^6 = (1 - x^3)*(1 + x^3) = (1 - x)*\left(1 + x + x^2\right)*(1 + x)*(1 - x + x^2)
|
7,708
|
\frac{4!^2}{4^8} = \frac{1}{4^4} \cdot 4! \approx 0.008789
|
15,518
|
\frac{1}{n^2 + a_n^2} \cdot ((-1) + n^2 \cdot a_n) = a_n - \frac{1 + a_n^3}{n^2 + a_n^2}
|
17,005
|
5\cdot \left(-1\right) + y^2 - 4\cdot y = (y + 1)\cdot (y + 5\cdot (-1))
|
27,368
|
\binom{m}{-i + m} = \binom{m}{i}
|
54,752
|
0.9 = 90\%
|
3,211
|
\left(7 = 8 + \left(-1\right) \Rightarrow (-1) + 8 \cdot M = 7^{1 + 2 \cdot x}\right) \Rightarrow 7^{1 + x \cdot 2} + 1 = M \cdot 8
|
3,389
|
\frac{13}{12} = \frac12 + 1/4 + \tfrac13
|
-20,814
|
(7*(-1) - 35*p)/35 = \tfrac{1}{5}*(-5*p + (-1))*\frac77
|
27,137
|
(1 + 1) \times (2 + 1) + \left(-1\right) = 5
|
10,775
|
k^2 + k + 1 = \frac{k^3 + (-1)}{\left(-1\right) + k}
|
-1,057
|
\frac{1}{5} = \tfrac{1}{5}
|
30,112
|
2\cdot \cos(\frac{\pi}{9}\cdot 4) = 2\cdot \sin\left(\tfrac{1}{18}\cdot \pi\right)
|
-26,651
|
4\cdot (-1) + t^4\cdot 49 = (2\cdot (-1) + t^2\cdot 7)\cdot (2 + 7\cdot t^2)
|
23,214
|
\frac{3900}{27405} = 1300\cdot 3/(27405)
|
2,195
|
\ln(2) = 1/2 + \dfrac{1}{6} + \dfrac{1}{30} + 1/56 + \dots \leq 1 + \dfrac14 + \frac{1}{25} + \frac{1}{49} + \dots
|
1,064
|
\frac{1}{2\cdot \lambda} + 1/\lambda = \frac{3}{2\cdot \lambda}
|
-20,470
|
-\dfrac{10}{-10}*\left(-10/7\right) = 100/(-70)
|
19,910
|
1/(h_2 h_1) = \frac{1}{h_1 h_2}
|
-12,044
|
\dfrac{3}{4} = \frac{1}{16\cdot π}\cdot p\cdot 16\cdot π = p
|
-22,242
|
(r + 3)\cdot (r + 9\cdot (-1)) = r^2 - 6\cdot r + 27\cdot (-1)
|
31,733
|
\cos\left(2 \times z\right) = \cos^2\left(z\right) - \sin^2(z) = 1 - 2 \times \sin^2(z)
|
-4,183
|
\frac{x^4}{x^4} \cdot 2/8 = \frac{1}{8 \cdot x^4} \cdot x^4 \cdot 2
|
35,337
|
|E\cdot B| = |E|\cdot |B|
|
-6,400
|
\frac{4}{k \cdot k + k\cdot 16 + 63} = \frac{4}{(k + 9)\cdot \left(7 + k\right)}
|
17,179
|
v = ((-1) + 2*y)^{1/3} \Rightarrow 2*y + \left(-1\right) = v^3
|
-1,264
|
\frac19\times 7/(\tfrac15\times (-3)) = -5/3\times 7/9
|
32,210
|
{1 + n \choose n} + 1 = {n + 2 \choose 1 + n}
|
32,863
|
\frac{x!}{i!*\left(-i + x\right)!} = \binom{x}{i}
|
-11,600
|
0 + 12 - 12 \cdot i = 12 - i \cdot 12
|
-9,435
|
t \cdot 2 \cdot 3 \cdot 11 - 2 \cdot 2 \cdot 2 \cdot 11 = t \cdot 66 + 88 (-1)
|
14,374
|
\dfrac{1}{7} + \tfrac{1}{15} = 22/105
|
-10,544
|
-\frac{96}{36\cdot s + 60} = 12/12\cdot \left(-\frac{1}{5 + s\cdot 3}\cdot 8\right)
|
-27,633
|
17 + 8 \left(-1\right) + 8 (-1) = 9 + 8 (-1) = 1
|
-10,149
|
0.01 (-64) = -\frac{64}{100} = -\dfrac{1}{25}16
|
44,658
|
2\cdot j = j + j
|
2,749
|
5 = n \implies -1 = (-1)^n
|
38,576
|
26 + 5 \cdot \left(-1\right) = 21
|
30,233
|
n + 1 + 1 = 1 + n + 1
|
15,970
|
a \cdot b = -b \cdot \left(-a\right)
|
44,140
|
\frac{3}{10} = 150/500
|
24,470
|
(1 + n)^2 = n^2 + 2*n + 1
|
-26,624
|
(-8 \cdot q + p \cdot 9)^2 = 81 \cdot p^2 - 144 \cdot p \cdot q + q^2 \cdot 64
|
9,300
|
j - l = l - j = -(j - l)
|
27,761
|
B\cdot \frac{A}{B} = \frac{B}{B}\cdot A
|
-10,321
|
\left(5\times r + 2\right)/(5\times r)\times \frac{10}{10} = \frac{r\times 50 + 20}{50\times r}
|
18,255
|
\frac{1}{C^3} = (\frac{1}{C})^2 * 1/C = \frac{\frac{1}{C^2}}{C}*1
|
19,655
|
t*((-1) + t)*\left(2*(-1) + t\right)! = t!
|
-30,239
|
(x + 2 \times (-1)) \times (10 \times (-1) + x) = 20 + x^2 - 12 \times x
|
28,338
|
z\cdot x_2 = z\cdot x_2
|
4,842
|
|x_j|\cdot |z_j| = |x_j\cdot z_j|
|
-11,634
|
-i \cdot 8 + 4 = -i \cdot 8 + 4 + 0 \left(-1\right)
|
-28,689
|
x^2 - 6*x + 13 = x^2 - 6*x + 9 + 4 = (x + 3*(-1))^2 + 4 = \left(x*\left(-3\right)\right) * \left(x*\left(-3\right)\right) + 2^2
|
29,455
|
\left[y, 0\right] = ( y, 0 + 0) = \left( y, 0\right) + ( y, 0)
|
37,129
|
\frac{1}{\tfrac1c} = \frac{1}{1/c}
|
-4,795
|
30.5\cdot 10^{3 + 1} = 10^4\cdot 30.5
|
5,223
|
(d\cdot x)^2 = \left(d\cdot x\right) \cdot \left(d\cdot x\right)
|
7,054
|
\sin(\tfrac32\cdot \pi - H) = -\cos(H)
|
423
|
(-1) + \frac{-a + x}{A - a} = \dfrac{1}{A - a}*\left(-A + x\right)
|
-2,699
|
\sqrt{3}*3 = \sqrt{3}*(4*\left(-1\right) + 5 + 2)
|
7,089
|
2 \cdot i + 1 = i^2 + 2 \cdot i + 1 - i \cdot i = (i + 1)^2 - i^2 \cdot \cdots
|
-19,272
|
\frac{5 / 3}{1/6 \cdot 7} \cdot 1 = 5/3 \cdot 6/7
|
-18,941
|
\frac79 = \frac{A_s}{9 \cdot \pi} \cdot 9 \cdot \pi = A_s
|
-25,831
|
6 + z^2\cdot 3 - 5\cdot z = \frac{1}{z + 2}\cdot \left(z^3\cdot 3 + z^2 - 4\cdot z + 12\right)
|
34,849
|
1 = -3921 \cdot 255 + 3952 \cdot 253
|
-23
|
5\cdot (-1) - 10 = -15
|
-5,962
|
\frac{5}{3 \cdot \left(9 \cdot (-1) + t\right)} = \frac{5}{27 \cdot (-1) + 3 \cdot t}
|
29,409
|
v \cdot \left(b + e\right) = b \cdot v + v \cdot e
|
29,667
|
\binom{n+r}{r}=\binom{n+r}{n}
|
29,900
|
4^2 + 10^2 + 28 \cdot 28 = 15^2 \cdot 4
|
-9,451
|
64 \cdot x \cdot x - x \cdot 48 = x \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot x - x \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3
|
21,911
|
C + \gamma + x = x + C + \gamma
|
30,782
|
\left(2/y + 1\right)^{-\dfrac{1}{2}} = (\dfrac{1}{y}\cdot (2 + y))^{-\tfrac12}
|
36,122
|
1/(d*\frac1a) = \frac{a}{d}
|
19,356
|
\left(x + 2\right)^2 = (x + 2)*\left(x + 2\right) = (x + 2)*x + (x + 2)*2 = x^2 + 2*x + 2*x + 4
|
24,784
|
\frac12 \cdot {8 \choose 4} = 35
|
24,897
|
h \times c \times g = c \times h \times g
|
10,271
|
\binom{2n}{n-1} = \frac{(2n)!}{(n-1)!(n+1)!} = \frac{n}{n+1}\binom{2n}{n}
|
42,820
|
24911296875000000000000 = 5^{18}\cdot 3^{13}\cdot 2^{12}
|
14,154
|
A^{n+1}=A^n \times A
|
-25,819
|
\frac{4 / 3}{7 \cdot 1/4}1 = \frac{4}{3} \cdot \frac{1}{7}4 = \dfrac{4}{3 \cdot 7}4 = 16/21
|
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