id topic_id thread_id type formula q_1 A.1 3062860 title "

c

" q_2 A.1 3062860 title "

f (x) = \frac{x^{2} + x + c}{x^{2} + 2 x + c}

" q_3 A.1 3062860 title "

[1, \frac{1}{3}]

" q_4 A.1 3062860 question "

f (x) = \frac{x^{2} + x + c}{x^{2} + 2 x + c}

" q_5 A.1 3062860 question "

c

" q_6 A.1 3062860 question "

f (x)

" q_7 A.1 3062860 question "

[1, \frac{1}{3}]

" q_8 A.2 3069579 title "

f^{'} (x) = f (x + 1)

" q_9 A.2 3069579 question "

\frac{d f}{d x} = f (x + 1)

" q_10 A.3 3073397 title "

\sqrt{5}

" q_11 A.3 3073397 title "

10^{10}

" q_12 A.3 3073397 question "

\sqrt{5}

" q_13 A.3 3073397 question "

10^{10}

" q_14 A.3 3073397 question "

\sqrt{5}

" q_15 A.3 3073397 question "

x

" q_16 A.3 3073397 question "

f (x)

" q_17 A.3 3073397 question "

a

" q_18 A.3 3073397 question "

b

" q_19 A.3 3073397 question "

[a, b]

" q_20 A.3 3073397 question "

f (a)

" q_21 A.3 3073397 question "

f (b)

" q_22 A.4 3083227 question "

\sum_{k = 0}^{n} (\frac{n}{k}) k

" q_23 A.4 3083227 question "

n 2^{n - 1}

" q_24 A.5 3111785 question "

P ((2) | (1)) = \frac{P ((2) \cap (1))}{P ((1))} = \frac{P ((2)) P ((1))}{P ((1))} = P ((2))

" q_25 A.6 3116896 question "

5^{133} mod 8 .

" q_26 A.6 3116896 question "

5^{n} mod 8 = 5

" q_27 A.6 3116896 question "

n

" q_28 A.6 3116896 question "

5^{133} mod 8 = 5

" q_29 A.7 3122632 title "

\frac{11^{10} - 1}{100}

" q_30 A.7 3122632 question "

11^{10} - 1

" q_31 A.7 3122632 question "

100

" q_32 A.7 3122632 question "

x

" q_33 A.7 3122632 question "

11^{10} - 1 = x (mod 100)

" q_34 A.7 3122632 question "

11^{2} \equiv 21 (mod 100)

" q_35 A.7 3122632 question "

{(11^{2})}^{2} \equiv {(21)}^{2} (mod 100)

" q_36 A.7 3122632 question "

11^{4} \equiv 441 (mod 100)

" q_37 A.7 3122632 question "

11^{4} \equiv 41 (mod 100)

" q_38 A.7 3122632 question "

{(11^{4})}^{2} \equiv {(41)}^{2} (mod 100)

" q_39 A.7 3122632 question "

11^{8} \equiv 1681 (mod 100)

" q_40 A.7 3122632 question "

11^{8} \equiv 81 (mod 100)

" q_41 A.7 3122632 question "

11^{8} \times 11^{2} \equiv (81 \times 21) (mod 100)

" q_42 A.7 3122632 question "

11^{10} \equiv 1701 (mod 100) ⟹ 11^{10} \equiv 1 (mod 100)

" q_43 A.7 3122632 question "

11^{10} - 1 \equiv (1 - 1) (mod 100) ⟹ 11^{10} - 1 \equiv 0 (mod 100)

" q_44 A.7 3122632 question "

x

" q_45 A.7 3122632 question "

x = 0

" q_46 A.7 3122632 question "

11^{10} - 1

" q_47 A.7 3122632 question "

100

" q_48 A.8 3142655 title "

{lim}_{n \to \infty} \sqrt[n]{\frac{{(27)}^{n} {(n!)}^{3}}{(3 n)!}}

" q_49 A.8 3142655 question "

{lim}_{n \to \infty} \sqrt[n]{\frac{{(27)}^{n} {(n!)}^{3}}{(3 n)!}}

" q_50 A.8 3142655 question "

l = {lim}_{n \to \infty} {(\frac{{(27)}^{n} {(n!)}^{3}}{(3 n)!})}^{\frac{1}{n}}

" q_51 A.8 3142655 question "

\ln (l) = {lim}_{n \to \infty} \frac{1}{n} [n \ln (27) + 3 \ln (n!) - \ln ((3 n)!)]

" q_52 A.9 3145414 question "

\sum_{n = 0}^{N} n x^{n}

" q_53 A.9 3145414 question "

\sum_{i = 0}^{n} i^{2} = \frac{(n^{2} + n) (2 n + 1)}{6}

" q_54 A.10 3152814 title "

\int_{0}^{\infty} \frac{\sin (x)}{x^{a}}

" q_55 A.10 3152814 question "

\int_{0}^{\infty} \frac{\sin (x)}{x^{a}}

" q_56 A.11 3158634 question "

R^{3}

" q_57 A.11 3158634 question "

u = (a, b, c)

" q_58 A.11 3158634 question "

v = (d, e, f)

" q_59 A.11 3158634 question "

u \times v = | \begin{matrix} i^{^} & a m p; j^{^} & a m p; k^{^} \\ a & a m p; b & a m p; c \\ d & a m p; e & a m p; f \end{matrix} |

" q_60 A.11 3158634 question "

R^{2}

" q_61 A.11 3158634 question "

R^{3}

" q_62 A.11 3158634 question "

u = (a, b)

" q_63 A.11 3158634 question "

v = (d, e)

" q_64 A.11 3158634 question "

R^{2}

" q_65 A.11 3158634 question "

R^{3}

" q_66 A.11 3158634 question "

Φ (u, v) = (f (u), g (v))

" q_67 A.11 3158634 question "

R^{2}

" q_68 A.11 3158634 question "

D = | \frac{\partial Φ}{\partial u} \times \frac{\partial Φ}{\partial v} |

" q_69 A.11 3158634 question "

R^{3}

" q_70 A.11 3158634 question "

R^{2}

" q_71 A.11 3158634 question "

{\int\int}_{V} f (x, y) d x d y = {\int\int}_{Q} f (Φ (u, v) | \frac{\partial Φ}{\partial u} \times \frac{\partial Φ}{\partial v} |

" q_72 A.11 3158634 question "

Φ (u, v) = (2 u \cos (v), u \sin (v))

" q_73 A.12 3159517 question "

{(1 + i \sqrt{3})}^{1 / 2}

" q_74 A.13 3169607 title "

\int_{a}^{b} f (x) d x + \int_{f (a)}^{f (b)} f^{1} (x) d x qvar_F o r m u l a # q_{7} 5 : +∫ab⁢f(x)dx∫⁢f(a)⁢f(b)⁢f-1(x)dxqvar_Formula#q76:⁢-⁢bf(b)⁢af(a)Formula#q77:⁢=y+⁢xy′⁢12(y′)2Formula#q78:⁢=⁢12y′(+⁢2xy′)yFormula#q79:⁢=+x2ytFormula#q80:⁢t′=+⁢2x⁢y′(-t′⁢2x)t′=-⁢2t⁢2x2Formula#q81:⁢∑=i1n⁢ix-i1Formula#q82:⁢+1⁢2x⁢3x2⁢4x3⁢5x4...⁢nx-n1...Formula#q83:⁢≠x1,⁢|x|<1Formula#q84:⁢SnFormula#q85:⁢=S2+1xx2x3x4...Formula#q86:⁢=⁢d(S2)⁢dxS1Formula#q87:⁢⁢|x|<1Formula#q88:⁢S2Formula#q89:⁢=-1xn-1x1-1xFormula#q90:⁢S1=⁢d(S2)⁢dx=⁢d(1-1x)⁢dx=1(-1x)2Formula#q91:⁢∫01⁢⁢ln(+1x)ln(-1x)+1xdxFormula#q92:⁢∫01⁢⁢ln(+1x)ln(-1x)+1xdxFormula#q93:=⁢I(a,b)∫01⁢⁢ln(-1⁢ax)ln(+1⁢bx)+1xdx

" q_94 A.16 3200398 question "

\frac{d^{2} I (a, b)}{d a d b}

" q_95 A.17 3206132 title "

\int_{x = 0}^{\infty} \frac{\sin (x)}{x}

" q_96 A.17 3206132 title "

\frac{e^{i z}}{z}

" q_97 A.17 3206132 question "

\int_{x = 0}^{\infty} \frac{\sin (x)}{x}

" q_98 A.17 3206132 question "

f (z) = \frac{e^{i z}}{z}

" q_99 A.17 3206132 question "

Γ = γ_{1} + γ_{R} + γ_{2} + γ_{ε}

" q_100 A.17 3206132 question "

γ_{1} (t) = t, t \in [i ε, i R]

" q_101 A.17 3206132 question "

γ_{R} (t) = R e^{i t}, t \in [\frac{π}{2}, \frac{π}{2}]

" q_102 A.17 3206132 question "

γ_{2} (t) = t, t \in [i R, i ε]

" q_103 A.17 3206132 question "

γ_{ε} (t) = ε e^{i t}, t \in [\frac{π}{2}, \frac{π}{2}]

" q_104 A.17 3206132 question "

\frac{\sin (x)}{x}

" q_105 A.17 3206132 question "

\int_{γ_{ε}} f = i π

" q_106 A.17 3206132 question "

ε \to 0

" q_107 A.17 3206132 question "

\int_{γ_{R}} f = 0

" q_108 A.17 3206132 question "

R \to \infty

" q_109 A.18 3208619 title "

{lim}_{n \to \infty} \frac{{[(n + 1) (n + 2) \dots (n + n)]}^{1 / n}}{n}

" q_110 A.18 3208619 question "

{lim}_{n \to \infty} \frac{{[(n + 1) (n + 2) \dots (n + n)]}^{\frac{1}{n}}}{n}

" q_111 A.18 3208619 question "

\log (2) + n \log (\frac{n}{n + 1})

" q_112 A.18 3208619 question "

\frac{2}{e}

" q_113 A.18 3208619 question "

\frac{4}{e}

" q_114 A.18 3208619 question "

{lim}_{n \to \infty} \frac{n \log (n) + \sum_{k = 1}^{n} \log (k + n)}{n} = {lim}_{n \to \infty} ((n + 1) \log (n + 1) + \sum_{k = 1}^{n + 1} \log (k + n)) - (n \log (n) + \sum_{k = 1}^{n} \log (k + n)) = {lim}_{n \to \infty} \log (\frac{2 n + 1}{n + 1}) - n \log (1 + \frac{1}{n}) = \log (2) - 1

" q_115 A.18 3208619 question "

2 / e

" q_116 A.19 3213405 title "

p^{4} - 1

" q_117 A.19 3213405 question "

p^{4} - 1

" q_118 A.19 3213405 question "

7^{4} - 1

" q_119 A.19 3213405 question "

2^{4} 352

" q_120 A.19 3213405 question "

11^{4} - 1

" q_121 A.19 3213405 question "

2^{4} 3561

" q_122 A.19 3213405 question "

2^{4} 35

" q_123 A.19 3213405 question "

p^{4} - 1

" q_124 A.19 3213405 question "

(p^{2} + 1) (p - 1) (p + 1)

" q_125 A.19 3213405 question "

2^{4}

" q_126 A.19 3213405 question "

16 n + x

" q_127 A.20 3214309 title "

n \in N ∖ {41}

" q_128 A.20 3214309 title "

φ (n) = 40

" q_129 A.20 3214309 question "

n \in N

" q_130 A.20 3214309 question "

φ (n) = 40

" q_131 A.20 3214309 question "

φ

" q_132 A.20 3214309 question "

n = 41

" q_133 A.20 3214309 question "

n^{'} s

" q_134 A.21 3214535 title "

9^{9^{9^{... 9}}}

" q_135 A.21 3214535 question "

9^{9^{9^{... 9}}}

" q_136 A.21 3214535 question "

9^{9^{9^{... 9}}} \equiv x (mod 100)

" q_137 A.21 3214535 question "

0 \leq x \leq 100

" q_138 A.21 3214535 question "

9^{9^{9^{... 9}}} (nine 9s) = 9^{a}

" q_139 A.21 3214535 question "

9^{a} (mod 100)

" q_140 A.21 3214535 question "

a (mod φ (100))

" q_141 A.21 3214535 question "

φ (100) = 40

" q_142 A.21 3214535 question "

a = b (mod 40)

" q_143 A.21 3214535 question "

9^{9^{9^{... 9}}} (eight 9s) = 9^{b}

" q_144 A.21 3214535 question "

9^{b} (mod 40)

" q_145 A.21 3214535 question "

b (mod φ (40))

" q_146 A.21 3214535 question "

φ (40) = 16

" q_147 A.21 3214535 question "

b = c (mod 16)

" q_148 A.21 3214535 question "

9^{9^{9^{... 9}}} (seven 9s) = 9^{c}

" q_149 A.21 3214535 question "

9^{c} (m o d 16)

" q_150 A.21 3214535 question "

c (mod p h i (16))

" q_151 A.21 3214535 question "

φ (16) = 8

" q_152 A.21 3214535 question "

c (mod 8)

" q_153 A.21 3214535 question "

9 = 1 (mod 8)

" q_154 A.21 3214535 question "

c = 1 (mod 8)

" q_155 A.22 3221158 title "

d, d + 1, d + 2 ... = N

" q_156 A.22 3221158 title "

N

" q_157 A.22 3221158 question "

d, d + 1, d + 2 ...

" q_158 A.22 3221158 question "

N

" q_159 A.22 3221158 question "

N

" q_160 A.22 3221158 question "

d

" q_161 A.22 3221158 question "

N = 30

" q_162 A.22 3221158 question "

A n s = 3

" q_163 A.22 3221158 question "

d_{1} = 4; d_{2} = 6; d_{3} = 8

" q_164 A.22 3221158 question "

d_{1} : 4 + 5 + 6 + 7 + 8 = 30

" q_165 A.22 3221158 question "

(\sum (d + n) - \sum (d - 1))

" q_166 A.23 3222198 question "

2^{7} 3^{8} 5^{2} 7^{11}

" q_167 A.23 3222198 question "

2^{3} 3^{4} 5

" q_168 A.23 3222198 question "

x

" q_169 A.23 3222198 question "

x = 2^{3} 3^{4} 5 x

" q_170 A.23 3222198 question "

x

" q_171 A.24 3226443 title "

\sqrt{2 i - 1} qvar_F o r m u l a # q_{1} 72 : ⁢-⁢2i1qvar_Formula#q173:⁢=-⁢2i1(+a⁢bi)2Formula#q174:⁢-+a2⁢2abib2Formula#q175:⁢=-a2b2-1Formula#q176:⁢=⁢2ab2Formula#q177:⁢=a2b-2Formula#q178:⁢=-b-2b2-1Formula#q179:⁢=+-b41-1Formula#q180:⁢=b42Formula#q181:⁢=b42Formula#q182:⁢-⁢2i1Formula#q183:⁢=⁢P(+x21)+(⁢P(x))21Formula#q184:⁢⁢P(0)Formula#q185:⁢=⁢P(+x21)+(⁢P(x))21Formula#q186:⁢⁢P(x)Formula#q187:=⁢P(0)0

" q_188 A.25 3227177 question "

P (1) = 1

" q_189 A.25 3227177 question "

P (2) = 2

" q_190 A.25 3227177 question "

P (5) = 5

" q_191 A.25 3227177 question "

P (26) = 26

" q_192 A.25 3227177 question "

P (677) = 677

" q_193 A.25 3227177 question "

P (x) = x

" q_194 A.25 3227177 question "

y = P (x)

" q_195 A.25 3227177 question "

y = x

" q_196 A.25 3227177 question "

P (0) = 2

" q_197 A.25 3227177 question "

P (x) = {(x^{2} + 1)}^{2} + 1

" q_198 A.25 3227177 question "

P (0) = 3

" q_199 A.25 3227177 question "

{lim}_{x \to \infty} \log_{x} P (x)

" q_200 A.25 3227177 question "

P (x)

" q_201 A.25 3227177 question "

P (0)

" q_202 A.25 3227177 question "

P (x)

" q_203 A.26 3229015 question "

\int_{0}^{\infty} \frac{\sin (x)}{x} d x .

" q_204 A.26 3229015 question "

\int_{0}^{x} \frac{s i n x}{x} d x

" q_205 A.26 3229015 question "

x - \frac{x^{3}}{3 \times 3!} + \frac{x^{5}}{5 \times 5!} - \frac{x^{7}}{7 \times 7!} + \dots = \sum_{n = 0}^{\infty} {(1)}^{n} \frac{x^{(2 n + 1)}}{(2 n + 1) \times (2 n + 1)!}

" q_206 A.26 3229015 question "

\int_{0}^{\infty} \frac{\sin (x)}{x} d x = {lim}_{x \to \infty} \sum_{n = 0}^{\infty} {(1)}^{n} \frac{x^{(2 n + 1)}}{(2 n + 1) \times (2 n + 1)!}

" q_207 A.27 3229839 title "

e^{3 i π / 2}

" q_208 A.27 3229839 question "

e^{π i} = 1

" q_209 A.27 3229839 question "

e^{3 π i / 2}

" q_210 A.27 3229839 question "

{(e^{π i})}^{3 / 2}

" q_211 A.27 3229839 question "

{(\sqrt{1})}^{3} = i^{3} = i

" q_212 A.27 3229839 question "

\sqrt{{(1)}^{3}} = \sqrt{1} = i

" q_213 A.28 3230730 title "

\sin (18^{\circ}) = \frac{a + \sqrt{b}}{c}

" q_214 A.28 3230730 title "

a + b + c

" q_215 A.28 3230730 question "

\sin (18) = \frac{a + \sqrt{b}}{c}

" q_216 A.28 3230730 question "

a + b + c

" q_218 A.28 3230730 question "

\sin (18)

" q_219 A.28 3230730 question "

x

" q_220 A.28 3230730 question "

18

" q_221 A.28 3230730 question "

y

" q_222 A.28 3230730 question "

z

" q_223 A.28 3230730 question "

\frac{x}{z}

" q_224 A.28 3230730 question "

x = a + \sqrt{b}, z = c

" q_225 A.28 3230730 question "

y

" q_226 A.28 3230730 question "

y = \sqrt{c^{2} - {(a + \sqrt{b})}^{2}}

" q_227 A.28 3230730 question "

\cos (18) = \frac{y}{z} = \frac{\sqrt{c^{2} - {(a + \sqrt{b})}^{2}}}{c}

" q_228 A.28 3230730 question "

b = {(c \sin (18) - a)}^{2} = c^{2} \sin^{2} (18) - 2 a c \sin (18) + a^{2}

" q_229 A.28 3230730 question "

\sin (18) = \frac{1 + \sqrt{5}}{4}

" q_230 A.28 3230730 question "

A, B, C

" q_231 A.28 3230730 question "

A \sin {(18)}^{2} + B \sin (18) + C = 0

" q_232 A.28 3230730 question "

\sin (18)

" q_233 A.28 3230730 question "

A x^{2} + B x + C

" q_234 A.28 3230730 question "

a = B, b = B^{2} - 4 A C, c = 2 A

" q_235 A.28 3230730 question "

s i n (18) = (1 + \sqrt{5}) / 4

" q_236 A.29 3239787 question "

3 + 2 i

" q_237 A.29 3239787 question "

4 i

" q_238 A.29 3239787 question "

i = \sqrt{1}

" q_239 A.29 3239787 question "

\frac{5 i}{\infty} = 0

" q_240 A.29 3239787 question "

\frac{5}{\infty} \frac{\sqrt{1}}{\infty} = 00 = 0,

" q_241 A.29 3239787 question "

\frac{5 x}{\infty} = 0

" q_242 A.29 3239787 question "

\frac{5}{\infty} \frac{x}{\infty} = 00 = 0 .

" q_243 A.30 3161841 title "

a^{3} + b^{3} + c^{3} - 3 a b c

" q_244 A.30 3161841 question "

a = \sum_{n = 0}^{\infty} \frac{x^{3 n}}{(3 n)!} b = \sum_{n = 1}^{\infty} \frac{x^{3 n - 2}}{(3 n - 2)!} c = \sum_{n = 1}^{\infty} \frac{x^{3 n - 1}}{(3 n - 1)!}

" q_245 A.30 3161841 question "

a^{3} + b^{3} + c^{3} - 3 a b c

" q_246 A.30 3161841 question "

(a) 1

" q_247 A.30 3161841 question "

(b) 0

" q_248 A.30 3161841 question "

(c) - 1

" q_249 A.30 3161841 question "

(d) - 2

" q_250 A.30 3161841 question "

a, b

" q_251 A.30 3161841 question "

c

" q_252 A.30 3161841 question "

e^{x}

" q_262 A.32 3103759 question "

E m p t y (x) \Leftrightarrow \exists̸ y (y \in x)

" q_263 A.32 3103759 question "

E m p t y (x) \Leftrightarrow \exists̸ y (y \in x)

" q_264 A.33 3170416 question "

f (t, x)

" q_265 A.33 3170416 question "

f

" q_266 A.33 3170416 question "

\frac{\partial^{3} f}{\partial t^{3}}

" q_267 A.33 3170416 question "

\frac{\partial^{3} f}{\partial x^{3}}

" q_268 A.34 3194414 question "

a ↑ b

" q_269 A.34 3194414 question "

a, b & g t; 0

" q_270 A.34 3194414 question "

a ↑ b

" q_271 A.34 3194414 question "

\begin{matrix} a b & a m p; = a b \\ a ↓ b & a m p; = a + b \\ a ↓ ↓ b & a m p; = see below \end{matrix} ⋮

" q_272 A.34 3194414 question "

a ↓ ↓ ↓ b

" q_273 A.34 3194414 question "

a b

" q_274 A.34 3194414 question "

a ↓^{n} b

" q_275 A.34 3194414 question "

n \geq 3

" q_276 A.34 3194414 question "

a ↓ ↓ b = a + 1

" q_277 A.34 3194414 question "

b + 1

" q_278 A.34 3194414 question "

a ↑^{n} b

" q_279 A.34 3194414 question "

n \leq 0

" q_280 A.34 3194414 question "

a ↓ ↓ b = b + 1

" q_281 A.34 3194414 question "

a ↓ b = a + b - 1

" q_282 A.34 3194414 question "

4 ↓ 3 = 4 ↓ ↓ (4 ↓ ↓ 4) = 4 ↓ ↓ (4 + 1) = (4 + 1) + 1 = 6 = 4 + 3 - 1

" q_283 A.34 3194414 question "

\begin{matrix} a ↓ ↓ b = {\begin{matrix} b + 1 & a m p; if a & l t; b \\ b + 2 & a m p; if a = b \end{matrix} \end{matrix} .

" q_284 A.34 3194414 question "

a & g t; b

" q_285 A.34 3194414 question "

a ↓ b

" q_286 A.34 3194414 question "

a ↓ ↓ ↓ b

" q_287 A.34 3194414 question "

a & g t; b

" q_288 A.34 3194414 question "

a ↓ ↓ b = b + 1 + ⌊ \frac{a}{b} ⌋,

" q_289 A.34 3194414 question "

a ↓ ↓ ↓ b

" q_290 A.35 3196128 question "

\int e^{x^{2}} d x

" q_291 A.35 3196128 question "

\int e^{2 x} d x

" q_292 A.36 3197023 question "

\neg P

" q_293 A.36 3197023 question "

t r u e

" q_294 A.36 3197023 question "

\neg P \to A_{1} \to ... \to A_{n} \to P

" q_295 A.36 3197023 question "

\neg P \to P \Leftrightarrow \neg (\neg P) \lor P \Leftrightarrow P

" q_296 A.36 3197023 question "

\neg P

" q_297 A.36 3197023 question "

P

" q_298 A.36 3197023 question "

\neg P

" q_299 A.36 3197023 question "

\neg P

" q_300 A.37 3213320 title "

f \circ g = g \circ f

" q_301 A.37 3213320 title "

f^{1} ≠ g

" q_302 A.37 3213320 title "

f ≠ id ≠ g

" q_303 A.37 3213320 question "

f

" q_304 A.37 3213320 question "

g

" q_305 A.37 3213320 question "

f \circ g = g \circ f qvar_F o r m u l a # q_{3} 06 : =⁢f(x)⁢2x F o r m u l a # q_{3} 07 : =⁢g(x)⁢3x F o r m u l a # q_{3} 08 : f F o r m u l a # q_{3} 09 : g F o r m u l a # q_{3} 10 : f F o r m u l a # q_{3} 11 : :f→∖R{1}R F o r m u l a # q_{3} 12 : =⁢f(x)/⁢2x(-1x) F o r m u l a # q_{3} 13 : g F o r m u l a # q_{3} 14 : ⁢g(x)=∘f-1gf=⁢g(⁢2x-1x)+2⁢g(⁢2x-1x), F o r m u l a # q_{3} 15 : g0 F o r m u l a # q_{3} 16 : ↦g+n1∘f-1gnf F o r m u l a # q_{3} 17 : g0 F o r m u l a # q_{3} 18 : a,b F o r m u l a # q_{3} 19 : :q,r=a+⁢bqr F o r m u l a # q_{3} 20 : ⁢Choose F o r m u l a # q_{3} 21 : q:qb<=a

" q_322 A.39 3222554 question "

2018^{2019}

" q_323 A.39 3222554 question "

2019^{2018}

" q_324 A.39 3222554 question "

2019 \log (2018)

" q_325 A.39 3222554 question "

2018 \log (2019)

" q_326 A.39 3222554 question "

\log (2019 \gt \log (2018))

" q_327 A.39 3222554 question "

2019^{2018}

" q_328 A.40 3226910 question "

a_{1} x_{1} + a_{2} x_{2} + a_{3} x_{3} + ... + a_{n} x_{n} =

" q_329 A.40 3226910 question "

a (x) y + b (x) y^{'} + c (x) y "" + d (x) y^{'''} + ... + q (x) = 0

" q_330 A.40 3226910 question "

f (x + y) = f (x) + f (y)

" q_331 A.40 3226910 question "

f (c x) = c f (x)

" q_332 A.40 3226910 question "

f

" q_333 A.41 3241161 question "

m

" q_334 A.41 3241161 question "

n

" q_335 A.41 3241161 question "

n \leq m

" q_336 A.41 3241161 question "

\sum_{r = 1}^{n} {(1)}^{(n - r)} (\frac{n}{r}) {(r)}^{m}

" q_337 A.42 3244132 question "

x^{2} + 1 = 0

" q_338 A.42 3244132 question "

\sqrt{1}

" q_339 A.43 3058291 title "

\sum_{n \geq 1} \frac{1}{n^{2} + 1} = \frac{π \coth (π) - 1}{2}

" q_340 A.43 3058291 question "

\sum_{n \geq 1} \frac{1}{n^{2} + 1} = \frac{π \coth (π) - 1}{2}

" q_341 A.43 3058291 question "

S = \sum_{n \geq 1} \frac{1}{n^{2} + 1}

" q_342 A.43 3058291 question "

e^{x} = \frac{\sinh (π)}{π} + \frac{2 \sinh (π)}{π} \sum_{n \geq 1} \frac{{(1)}^{n}}{n^{2} + 1} (\cos (n x) - n \sin (n x))

" q_343 A.43 3058291 question "

x = π

" q_344 A.43 3058291 question "

e^{π} = \frac{\sinh (π)}{π} + \frac{2 \sinh (π)}{π} \sum_{n \geq 1} \frac{{(1)}^{n}}{n^{2} + 1} (\cos (n π) - n \sin (n π))

" q_345 A.43 3058291 question "

e^{π} = \frac{\sinh (π)}{π} + \frac{2 \sinh (π)}{π} \sum_{n \geq 1} \frac{{(1)}^{n}}{n^{2} + 1} ({(1)}^{n} - n 0)

" q_346 A.43 3058291 question "

e^{π} = \frac{\sinh (π)}{π} + \frac{2 \sinh (π)}{π} S

" q_347 A.43 3058291 question "

S = \frac{π e^{π}}{2 \sinh (π)} - \frac{1}{2}

" q_348 A.44 3059606 title "

A, B \in M_{2 \times 2} (Q)

" q_349 A.44 3059606 title "

A B

" q_350 A.44 3059606 question "

G

" q_351 A.44 3059606 question "

(M_{2 \times 2} (Q), \times)

" q_352 A.44 3059606 question "

A = (\begin{matrix} 0 & a m p; 1 \\ 1 & a m p; 0 \end{matrix})

" q_353 A.44 3059606 question "

A

" q_354 A.44 3059606 question "

4

" q_355 A.44 3059606 question "

B = (\begin{matrix} 0 & a m p; 1 \\ 1 & a m p; 1 \end{matrix})

" q_356 A.44 3059606 question "

B

" q_357 A.44 3059606 question "

3

" q_358 A.44 3059606 question "

A B

" q_359 A.44 3059606 question "

G

" q_360 A.45 3060385 title "

\sin (x), \sin (2 x), \sin (3 x), ..., \sin (n x)

" q_361 A.45 3060385 title "

R

" q_362 A.45 3060385 question "

\sin (x), \sin (2 x), \sin (3 x), ..., \sin (n x)

" q_363 A.45 3060385 question "

R

" q_364 A.45 3060385 question "

0

" q_365 A.45 3060385 question "

W ≠ 0

" q_366 A.45 3060385 question "

| \begin{matrix} (1) \sin (x) & a m p; (1) \sin (2 x) & a m p; (1) \sin (3 x) & a m p; ... & a m p; (1) \sin (n x) \\ (1) \cos (x) & a m p; (2) \cos (2 x) & a m p; (3) \cos (3 x) & a m p; ... & a m p; (n) \cos (n x) \\ {(1)}^{2} \sin (x) & a m p; {(2)}^{2} \sin (2 x) & a m p; {(3)}^{2} \sin (3 x) & a m p; ... & a m p; {(n)}^{2} \sin (n x) \\ {(1)}^{3} \cos (x) & a m p; {(2)}^{3} \cos (2 x) & a m p; {(3)}^{3} \cos (3 x) & a m p; ... & a m p; {(n)}^{3} \cos (n x) \end{matrix} |

" q_367 A.45 3060385 question "

W ≠ 0

" q_368 A.46 3073455 title "

f

" q_369 A.46 3073455 title "

[0, 1]

" q_370 A.46 3073455 title "

\int x^{k} f (x) d x = 0

" q_371 A.46 3073455 title "

f = 0 a.e.

" q_372 A.46 3073455 question "

f

" q_373 A.46 3073455 question "

[0, 1]

" q_374 A.46 3073455 question "

\int_{0}^{1} x^{k} f (x) d x = 0

" q_375 A.46 3073455 question "

k = {0, 1, 2, 3, ...}

" q_376 A.46 3073455 question "

f = 0

" q_377 A.46 3073455 question "

\int_{0}^{1} f^{2} (x) d x = 0

" q_378 A.46 3073455 question "

ε \gt 0

" q_379 A.46 3073455 question "

\int_{0}^{1} f^{2} (x) d x = \int_{0}^{1} (f (x) - \sum_{n = 0}^{N} a_{n} x^{n}) f (x), d x + \int_{0}^{1} \sum_{n = 0}^{N} f (x) a_{n} x^{n} d x \leq ε M

" q_380 A.46 3073455 question "

M = {sup}_{x \in [0, 1]} | f (x) |

" q_381 A.46 3073455 question "

f

" q_382 A.46 3073455 question "

f

" q_383 A.46 3073455 question "

f

" q_384 A.46 3073455 question "

[0, 1]

" q_385 A.46 3073455 question "

\int_{0}^{1} x^{k} f (x) d x = 0

" q_386 A.46 3073455 question "

k = {0, 1, 2, 3, ...}

" q_387 A.46 3073455 question "

f = 0

" q_388 A.46 3073455 question "

0

" q_389 A.46 3073455 question "

f

" q_390 A.46 3073455 question "

[0, 1]

" q_391 A.46 3073455 question "

\int_{0}^{1} x^{k} f (x) d x = 0

" q_392 A.46 3073455 question "

k = {0, 1, 2, 3, ...}

" q_393 A.46 3073455 question "

f = 0

" q_394 A.46 3073455 question "

0

" q_395 A.46 3073455 question "

f = 0 a.e.

" q_396 A.46 3073455 question "

f

" q_397 A.46 3073455 question "

0

" q_398 A.46 3073455 question "

\int_{0}^{1} f (x) \cos (2 π n x) d x \leq \int_{0}^{1} | f (x) | | \cos (2 π n x) - \sum_{n = 0}^{N} a_{n} x^{n} | \leq ε {|| f ||}_{L^{1}}

" q_399 A.47 3082635 title "

p

" q_400 A.47 3082635 title "

0 < r < p - 1

" q_401 A.47 3082635 title "

q

" q_402 A.47 3082635 title "

r q \equiv 1 mod p

" q_403 A.47 3082635 question "

p

" q_404 A.47 3082635 question "

0 & l t; r & l t; p - 1

" q_405 A.47 3082635 question "

q

" q_406 A.47 3082635 question "

r q \equiv 1 mod p

" q_407 A.47 3082635 question "

r q - 1 = k p

" q_408 A.47 3082635 question "

k

" q_409 A.47 3082635 question "

q r - k p = 1

" q_410 A.47 3082635 question "

r

" q_411 A.47 3082635 question "

\sum_{r = 1}^{p - 2} r = \frac{(p - 2) (p - 1)}{2} = p \frac{p - 3}{2} + 1 \equiv 1 mod p

" q_412 A.47 3082635 question "

r

" q_413 A.47 3082635 question "

p

" q_414 A.47 3082635 question "

(p - 1)! \equiv 1 mod p

" q_415 A.47 3082635 question "

r

" q_416 A.47 3082635 question "

2

" q_417 A.47 3082635 question "

(p - 1)! 2 \equiv 1 mod p

" q_418 A.47 3082635 question "

(p + 1)

" q_419 A.47 3082635 question "

(p + 1) (p - 1)! = (p + 1) mod p

" q_420 A.47 3082635 question "

(p + 1) (p - 1)! = 1 mod p

" q_421 A.47 3082635 question "

r

" q_422 A.47 3082635 question "

q

" q_423 A.48 3090622 title "

x, y \geq 0

" q_424 A.48 3090622 title "

{(x + y)}^{k} \geq x^{k} + y^{k}

" q_425 A.48 3090622 title "

k \geq 1

" q_426 A.48 3090622 question "

x, y \geq 0

" q_427 A.48 3090622 question "

{(x + y)}^{k} \geq x^{k} + y^{k}

" q_428 A.48 3090622 question "

k \in R_{\geq 1}

" q_429 A.48 3090622 question "

x \leq x + y

" q_430 A.48 3090622 question "

x^{k} \leq {(x + y)}^{k}

" q_431 A.48 3090622 question "

k \geq 1

" q_432 A.48 3090622 question "

y^{k} \leq {(x + y)}^{k}

" q_433 A.48 3090622 question "

x^{k} + y^{k} \leq 2 {(x + y)}^{k}

" q_434 A.48 3090622 question "

x, y \geq 0

" q_435 A.48 3090622 question "

f (k) = {(x + y)}^{k}

" q_436 A.48 3090622 question "

g (k) = x^{k} + y^{k}

" q_437 A.48 3090622 question "

{(x + y)}^{r} \geq x^{r} + y^{r}

" q_438 A.48 3090622 question "

r \in Z^{+}

" q_439 A.48 3090622 question "

f

" q_440 A.48 3090622 question "

g

" q_441 A.48 3090622 question "

k = 1

" q_442 A.48 3090622 question "

{(x + y)}^{r} & g t; x^{r} + y^{r}

" q_443 A.48 3090622 question "

r \geq 2

" q_444 A.48 3090622 question "

m \in̸ Z

" q_445 A.48 3090622 question "

f (m) = g (m)

" q_446 A.48 3090622 question "

k = 1

" q_447 A.48 3090622 question "

f

" q_448 A.48 3090622 question "

g

" q_449 A.48 3090622 question "

{(x + y)}^{r} & g t; x^{r} + y^{r}

" q_450 A.48 3090622 question "

r \geq 2

" q_451 A.48 3090622 question "

f

" q_452 A.48 3090622 question "

g

" q_453 A.48 3090622 question "

k = 1

" q_454 A.49 3095922 question "

(\frac{2 n}{n}) = \sum_{k = 0}^{n} {(\frac{n}{k})}^{2}

" q_455 A.49 3095922 question "

{(1 + x)}^{2 n} = {[{(1 + x)}^{n}]}^{2}

" q_456 A.49 3095922 question "

x^{n}

" q_457 A.49 3095922 question "

(\frac{2 n}{n})

" q_458 A.49 3095922 question "

\sum_{k = 0}^{n} {(\frac{n}{k})}^{2}

" q_459 A.50 3096123 title "

\sum \frac{1}{n^{2 + \cos (n)}}

" q_460 A.50 3096123 question "

\sum \frac{1}{n^{2 + \cos (n)}}

" q_461 A.50 3096123 question "

" q_462 A.50 3096123 question "

ε

" q_463 A.50 3096123 question "

A_{ε}

" q_464 A.50 3096123 question "

n, | 2 + \cos (n) | \leq 1 + ε

" q_465 A.50 3096123 question "

(n \in A_{ε} \Leftrightarrow 1 \leq \cos (n) \leq 1 + ε)

" q_466 A.50 3096123 question "

A_{ε}

" q_467 A.50 3096123 question "

" q_468 A.51 3098562 question "

\sum_{r = 0}^{n} (\frac{n + r}{r}) \frac{1}{2^{r}} = 2^{n}

" q_469 A.51 3098562 question "

(\frac{n}{n}) + (\frac{n + 1}{1}) \frac{1}{2} + (\frac{n + 2}{2}) \frac{1}{2^{2}} + (\frac{n + 3}{3}) \frac{1}{2^{3}} + \dots + (\frac{n + n}{n}) \frac{1}{2^{n}}

" q_470 A.51 3098562 question "

(\frac{n}{n}) + (\frac{n + 1}{n}) \frac{1}{2} + (\frac{n + 2}{n}) \frac{1}{2^{2}} + (\frac{n + 3}{n}) \frac{1}{2^{3}} + \dots + (\frac{n + n}{n}) \frac{1}{2^{n}} .

" q_471 A.51 3098562 question "

x^{n}

" q_472 A.51 3098562 question "

{(1 + x)}^{n} + {(1 + x)}^{n + 1} \frac{1}{2} + {(1 + x)}^{n + 2} \frac{1}{2^{2}} + \dots \dots + {(1 + x)}^{2 n} \frac{1}{2^{n}} .

" q_473 A.52 3104091 title "

\forall n \in N

" q_474 A.52 3104091 title "

\exists m \in N

" q_475 A.52 3104091 title "

m > n

" q_476 A.52 3104091 title "

m

" q_477 A.52 3104091 question "

n_{1}, n_{2}, ..., n_{k} \in N

" q_478 A.52 3104091 question "

2

" q_479 A.52 3104091 question "

n = n_{1} n_{2} ... n_{k} + 1

" q_480 A.52 3104091 question "

n_{1}, n_{2}, ..., n_{k}

" q_481 A.52 3104091 question "

a, b \in N

" q_482 A.52 3104091 question "

q

" q_483 A.52 3104091 question "

r

" q_484 A.52 3104091 question "

Z^{+}

" q_485 A.52 3104091 question "

a = q b + r

" q_486 A.52 3104091 question "

0 \leq r & l t; q

" q_487 A.52 3104091 question "

k \in N

" q_488 A.52 3104091 question "

n_{1}, n_{2}, ..., n_{k} \geq 1

" q_489 A.52 3104091 question "

\forall i

" q_490 A.52 3104091 question "

n_{i} ∤ n = n_{1} n_{2} ... n_{k} + 1

" q_491 A.52 3104091 question "

\exists n \in N

" q_492 A.52 3104091 question "

\forall m \in N

" q_493 A.52 3104091 question "

m \leq n

" q_494 A.52 3104091 question "

m

" q_495 A.53 3105233 question "

g \in G

" q_496 A.53 3105233 question "

g h = 1

" q_497 A.53 3105233 question "

h g = 1

" q_498 A.53 3105233 question "

A B = 1 \Rightarrow̸ B A = 1

" q_499 A.53 3105233 question "

A

" q_500 A.53 3105233 question "

B

" q_501 A.53 3105233 question "

A B = 1 \Rightarrow B A = 1

" q_502 A.54 3110618 title "

P (N) = (S | S \subseteq N)

" q_503 A.54 3110618 question "

P (N) = (S | S \subseteq N)

" q_504 A.55 3111869 title "

\frac{1}{\sqrt{1}} = \sqrt{1}

" q_505 A.55 3111869 question "

\sqrt{1}

" q_506 A.55 3111869 question "

{(\sqrt{1})}^{2} = 1

" q_507 A.55 3111869 question "

\frac{1}{\sqrt{1}} = \sqrt{\frac{1}{1}} = \sqrt{1} .

" q_508 A.55 3111869 question "

\frac{1}{\sqrt{1}} \frac{\sqrt{1}}{\sqrt{1}} = \frac{\sqrt{1}}{1} = \sqrt{1}

" q_509 A.56 3114034 question "

S

" q_510 A.56 3114034 question "

\exists p (p is prime \to \forall x (x is prime))

" q_511 A.56 3114034 question "

S

" q_512 A.57 3116030 question "

B

" q_513 A.57 3116030 question "

R^{n}

" q_514 A.57 3116030 question "

f : B \to R^{m}

" q_515 A.57 3116030 question "

f^{1}

" q_516 A.57 3116030 question "

f (B)

" q_517 A.57 3116030 question "

f (X)

" q_518 A.57 3116030 question "

X

" q_519 A.57 3116030 question "

X

" q_520 A.57 3116030 question "

R^{n}

" q_521 A.57 3116030 question "

f : X \to R^{m}

" q_522 A.57 3116030 question "

f^{1} : f (X) \mapsto X

" q_523 A.57 3116030 question "

f (X)

" q_524 A.58 3116495 title "

3 \arcsin (\frac{1}{4}) + \arccos (\frac{11}{16}) = \frac{π}{2}

" q_525 A.58 3116495 question "

3 \arcsin (\frac{1}{4}) + \arccos (\frac{11}{16}) = \frac{π}{2}

" q_526 A.59 3116965 title "

\sum_{d | n} φ (d) = n

" q_527 A.59 3116965 question "

φ (n)

" q_528 A.59 3116965 question "

\sum_{d | n} φ (d) = n

" q_529 A.59 3116965 question "

n = \prod_{k = 1}^{m} {p_{k}}^{α_{k}}

" q_530 A.59 3116965 question "

d = \prod_{k = 1}^{m} {p_{k}}^{β_{k}}

" q_531 A.59 3116965 question "

0 \leq β_{k} \leq α_{k}

" q_532 A.59 3116965 question "

\sum_{d | n} φ (d)

" q_533 A.60 3124615 question "

a_{n} = (1 - \frac{1}{\sqrt{2}}) ... (1 - \frac{1}{\sqrt{n + 1}})

" q_534 A.60 3124615 question "

n \geq 1

" q_535 A.60 3124615 question "

{lim}_{n \to \infty} a_{n}

" q_536 A.60 3124615 question "

1

" q_537 A.60 3124615 question "

\frac{1}{\sqrt{π}}

" q_538 A.60 3124615 question "

0

" q_539 A.60 3124615 question "

a_{n}

" q_540 A.60 3124615 question "

(1 - \frac{1}{\sqrt{1 + 1}}) (1 - \frac{1}{\sqrt{2 + 1}}) (1 - \frac{1}{\sqrt{3 + 1}}) (1 - \frac{1}{\sqrt{4 + 1}}) (1 - \frac{1}{\sqrt{5 + 1}}) (1 - \frac{1}{\sqrt{6 + 1}}) (1 - \frac{1}{\sqrt{7 + 1}}) (1 - \frac{1}{\sqrt{8 + 1}}) ... ... ... (1 - \frac{1}{\sqrt{n + 1}})

" q_541 A.60 3124615 question "

(0.293) (0.423) (0.5) (0.553) (0.622) (0.647) (0.667) ... .

" q_542 A.60 3124615 question "

(D)

" q_543 A.60 3124615 question "

(\frac{\sqrt{2} - 1}{\sqrt{2}}) (\frac{\sqrt{3} - 1}{\sqrt{3}}) (\frac{\sqrt{4} - 1}{\sqrt{4}}) ... ... . (\frac{\sqrt{(n + 1)} - 1}{\sqrt{n + 1}})

" q_544 A.60 3124615 question "

(\frac{(\sqrt{2} - 1) (\sqrt{3} - 1) (\sqrt{4} - 1) ... ... . (\sqrt{n + 1} - 1)}{\sqrt{(n + 1)!}})

" q_545 A.60 3124615 question "

a_{n}

" q_546 A.60 3124615 question "

n

" q_547 A.61 3132664 title "

i, j \in N

" q_548 A.61 3132664 title "

n = 3 i + 5 j

" q_549 A.61 3132664 title "

n \geq 8

" q_550 A.61 3132664 question "

i, j \in N

" q_551 A.61 3132664 question "

n = 3 i + 5 j

" q_552 A.61 3132664 question "

n \geq 8

" q_553 A.61 3132664 question "

n = 8 ⟹ 8 = 31 + 51

" q_554 A.61 3132664 question "

n = 9 ⟹ 9 = 33 + 50

" q_555 A.61 3132664 question "

n = 10 ⟹ 10 = 30 + 52

" q_556 A.61 3132664 question "

n = h

" q_557 A.61 3132664 question "

n = h + 1

" q_558 A.61 3132664 question "

k + 1 = 3 i + 5 j

" q_559 A.62 3134352 question "

Q

" q_560 A.62 3134352 question "

Z

" q_561 A.62 3134352 question "

| Q | = | Z |

" q_562 A.62 3134352 question "

Q

" q_563 A.62 3134352 question "

Z \times Z

" q_564 A.62 3134352 question "

| Z \times Z | = | Z |

" q_565 A.63 3149293 title "

\gcd

" q_567 A.63 3149293 title "

2

" q_568 A.63 3149293 question "

n_{1}, n_{2}

" q_569 A.63 3149293 question "

lcm (n_{1}, n_{2}) = \frac{n_{1} n_{2}}{\gcd (n_{1}, n_{2})}

" q_570 A.63 3149293 question "

r

" q_571 A.63 3149293 question "

n_{1}, n_{2}, n_{3}, ..., n_{r}

" q_572 A.63 3149293 question "

lcm (n_{1}, n_{2}, n_{3}, ..., n_{r}) = \frac{\prod_{i = 1}^{r} n_{i}}{g c d (n_{1}, n 2, n_{3}, ..., n_{r})}

" q_573 A.64 3152146 question "

f : [a, b] \to R

" q_574 A.64 3152146 question "

f ([a, b]) \subset [a, b]

" q_575 A.64 3152146 question "

c \in [a, b]

" q_576 A.64 3152146 question "

f (c) = c

" q_577 A.64 3152146 question "

f (a) = a

" q_578 A.64 3152146 question "

f (b) = b

" q_579 A.64 3152146 question "

f (a) = a

" q_580 A.64 3152146 question "

f (b) = b

" q_581 A.64 3152146 question "

f

" q_582 A.64 3152146 question "

[a, b]

" q_583 A.64 3152146 question "

f (a) & g t; a

" q_584 A.64 3152146 question "

f (b) & l t; b

" q_585 A.64 3152146 question "

f (a) & g t; a

" q_586 A.64 3152146 question "

f (b) & l t; b

" q_587 A.64 3152146 question "

x, y \in [a, b]

" q_588 A.64 3152146 question "

f (a) = x

" q_589 A.64 3152146 question "

f (b) = y

" q_590 A.64 3152146 question "

f [a, b] = [x, y]

" q_591 A.64 3152146 question "

[x, y] \subset [a, b]

" q_592 A.64 3152146 question "

[x, y]

" q_593 A.64 3152146 question "

c \in [a, b]

" q_594 A.64 3152146 question "

x

" q_595 A.64 3152146 question "

c

" q_596 A.64 3152146 question "

y

" q_597 A.64 3152146 question "

f (c) = c

" q_598 A.65 3155771 title "

e^{2 λ t} λ^{2} \leq \frac{1}{e^{2} t^{2}}

" q_599 A.65 3155771 title "

λ, t \geq 0

" q_600 A.65 3155771 question "

e^{2 λ t} λ^{2} \leq \frac{1}{e^{2} t^{2}}

" q_601 A.65 3155771 question "

λ, t \geq 0

" q_602 A.65 3155771 question "

\ln

" q_603 A.65 3155771 question "

(1)

" q_604 A.65 3155771 question "

t λ \leq e^{t λ - 1} .

" q_605 A.65 3155771 question "

x \leq e^{x - 1}

" q_606 A.65 3155771 question "

x \geq 0

" q_607 A.66 3168147 title "

x, h \in R^{d}

" q_608 A.66 3168147 title "

A \in R^{d \times d}

" q_609 A.66 3168147 title "

{(x^{T} A h)}^{T} = h^{T} A^{T} x

" q_610 A.66 3168147 question "

x, h \in R^{d}

" q_611 A.66 3168147 question "

A \in R^{d \times d}

" q_612 A.66 3168147 question "

{(x^{T} A h)}^{T} = h^{T} A^{T} x

" q_613 A.67 3175847 question "

k \times k

" q_614 A.67 3175847 question "

k \times l

" q_615 A.67 3175847 question "

l \times l

" q_616 A.67 3175847 question "

det [\begin{matrix} A & a m p; B \\ O & a m p; C \end{matrix}] = det (A) det (C)

" q_617 A.68 3178511 title "

a^{n} + 1

" q_618 A.68 3178511 title "

a + 1

" q_619 A.68 3178511 title "

n

" q_620 A.68 3178511 question "

a^{n} + 1

" q_621 A.68 3178511 question "

a + 1

" q_622 A.68 3178511 question "

n

" q_623 A.68 3178511 question "

a

" q_624 A.68 3178511 question "

1

" q_625 A.68 3178511 question "

n \in N

" q_626 A.68 3178511 question "

n

" q_627 A.68 3178511 question "

n

" q_628 A.68 3178511 question "

2 k + 1

" q_629 A.68 3178511 question "

a^{2 (k + 1) + 1} + 1

" q_630 A.68 3178511 question "

= a^{2 k + 3} + 1

" q_631 A.68 3178511 question "

= a^{3} a^{2 k} + 1

" q_632 A.68 3178511 question "

= (a^{3} + 1) a^{2 k} - a^{2 k} + 1

" q_633 A.68 3178511 question "

a^{2 k}

" q_634 A.68 3178511 question "

a^{n} + 1

" q_635 A.68 3178511 question "

a + 1

" q_636 A.68 3178511 question "

n

" q_637 A.69 3185659 question "

(\frac{s}{s}) + (\frac{s + 1}{s}) + ... + (\frac{n}{s}) = (\frac{n + 1}{s + 1})

" q_638 A.69 3185659 question "

n \geq s

" q_639 A.69 3185659 question "

0 \leq s \leq n

" q_640 A.70 3188898 title "

\sum_{j = 0}^{N - 1} \cos (\frac{(2 j + 1) π}{2 N}) = 0

" q_641 A.70 3188898 question "

l \in Z

" q_642 A.70 3188898 question "

N \in N

" q_643 A.70 3188898 question "

(1)Equation 11=∑=j0-N1cos(⁢l⁢(+⁢2j1)π⁢2N)0

" q_644 A.70 3188898 question "

N

" q_645 A.71 3189075 title "

1^{2} + 2^{2} + ... . n^{2} = \frac{n (n + 1) (2 n + 1)}{6}

" q_646 A.71 3189075 question "

1^{2} + 2^{2} + ... . n^{2} = \frac{n (n + 1) (2 n + 1)}{6}

" q_647 A.71 3189075 question "

L H S = 1^{2}

" q_648 A.71 3189075 question "

R H S = \frac{(1 + 1) (2 + 1)}{6} = \frac{23}{6} = 1

" q_649 A.71 3189075 question "

L H S_{p} = 1^{2} + 2^{2} + ... + p^{2}

" q_650 A.71 3189075 question "

R H S_{p} = \frac{p (p + 1) (2 p + 1)}{6}

" q_651 A.71 3189075 question "

L H S_{p + 1} = 1^{2} + 2^{2} + ... . p^{2} + {(p + 1)}^{2}

" q_652 A.71 3189075 question "

R H S_{p + 1} = \frac{(p + 1) ((p + 1) + 1) (2 (p + 1) + 1)}{6}

" q_653 A.71 3189075 question "

R H S_{p + 1} = R H S_{p} + {(p + 1)}^{2}

" q_654 A.71 3189075 question "

R H S_{p + 1} = \frac{p (p + 1) (2 p + 1)}{6} + {(p + 1)}^{2}

" q_655 A.71 3189075 question "

R H S_{p + 1} = \frac{(p + 1) ((p + 1) + 1) (2 (p + 1) + 1)}{6}

" q_656 A.71 3189075 question "

\frac{p (p + 1) (2 p + 1)}{6} + {(p + 1)}^{2}

" q_657 A.72 3189733 title "

X = Y

" q_658 A.72 3189733 title "

X \in Y

" q_659 A.72 3189733 question "

X = Y

" q_660 A.72 3189733 question "

X \in Y

" q_661 A.73 3191751 title "

{(\frac{n}{0})}^{2} + {(\frac{n}{1})}^{2} + ... + {(\frac{n}{n})}^{2} = (\frac{2 n}{n})

" q_662 A.73 3191751 question "

n

" q_663 A.73 3191751 question "

{(\frac{n}{0})}^{2} + {(\frac{n}{1})}^{2} + ... + {(\frac{n}{n})}^{2} = (\frac{2 n}{n})

" q_664 A.73 3191751 question "

P

" q_665 A.73 3191751 question "

{(1 + x)}^{n} {(1 + x)}^{n} = {(1 + x)}^{2 n}

" q_666 A.73 3191751 question "

x^{n}

" q_667 A.73 3191751 question "

(\frac{n}{0}) 1 + (\frac{n}{1}) x + ... + (\frac{n}{r}) x^{r} + ... + (\frac{n}{n}) x^{n}

" q_668 A.73 3191751 question "

x^{n}

" q_669 A.73 3191751 question "

x^{i}

" q_670 A.73 3191751 question "

x^{n - i}

" q_671 A.73 3191751 question "

x^{n}

" q_672 A.73 3191751 question "

(\frac{n}{0}) (\frac{n}{n}) + (\frac{n}{1}) (\frac{n}{n - 1}) + (\frac{n}{2}) (\frac{n}{n - 2}) + ... (\frac{n}{n}) (\frac{n}{0})

" q_673 A.73 3191751 question "

(\frac{n}{n - r}) = (\frac{n}{r})

" q_674 A.73 3191751 question "

{(\frac{n}{0})}^{2} + {(\frac{n}{1})}^{2} + ... + {(\frac{n}{n})}^{2}

" q_675 A.73 3191751 question "

x^{n}

" q_676 A.73 3191751 question "

P

" q_677 A.73 3191751 question "

(\frac{2 n}{n})

" q_678 A.73 3191751 question "

x^{n}

" q_679 A.73 3191751 question "

(\frac{2 n}{n})

" q_680 A.73 3191751 question "

{(\frac{n}{0})}^{2} + {(\frac{n}{1})}^{2} + ... + {(\frac{n}{n})}^{2} = (\frac{2 n}{n})

" q_681 A.73 3191751 question "

x^{n}

" q_682 A.73 3191751 question "

a

" q_683 A.73 3191751 question "

x^{n}

" q_684 A.73 3191751 question "

b

" q_685 A.73 3191751 question "

x^{n}

" q_686 A.74 3196153 title "

f : (0, \infty) \to R

" q_687 A.74 3196153 title "

f (x) = x + \frac{1}{x}

" q_688 A.74 3196153 title "

[2, \infty)

" q_689 A.74 3196153 question "

f : (0, \infty) \to R

" q_690 A.74 3196153 question "

f (x) = x + \frac{1}{x}

" q_691 A.74 3196153 question "

[2, \infty)

" q_692 A.74 3196153 question "

x = 1

" q_693 A.74 3196153 question "

f (1) = 2

" q_694 A.74 3196153 question "

[2, \infty)

" q_695 A.75 3200738 title "

m

" q_696 A.75 3200738 title "

{lim}_{u \to \infty} \frac{u^{m}}{e^{u}} = 0

" q_697 A.75 3200738 question "

m

" q_698 A.75 3200738 question "

{lim}_{u \to \infty} \frac{u^{m}}{e^{u}} = 0

" q_699 A.75 3200738 question "

e^{u}

" q_700 A.75 3200738 question "

& g t;

" q_701 A.75 3200738 question "

\frac{u^{m + 1}}{(m + 1)!}

" q_702 A.76 3220911 title "

Z

" q_703 A.76 3220911 question "

a + b Z = {z \in Z ∣ z = a + b k for some k \in Z}

" q_704 A.76 3220911 question "

a \in Z

" q_705 A.76 3220911 question "

b \in Z - {0}

" q_706 A.76 3220911 question "

{a_{i} + b_{i} Z ∣ i \in N}

" q_707 A.76 3220911 question "

⋃_{i \in N} (a_{i} + b_{i} Z) = Z

" q_708 A.76 3220911 question "

I \subset N

" q_709 A.76 3220911 question "

⋃_{i \in I} (a_{i} + b_{i} Z) = Z

" q_710 A.76 3220911 question "

{p_{k}} = {2, 3, 5, ...}

" q_711 A.76 3220911 question "

(⋃_{k \in N} (0 + p_{k} Z)) \cup (1 + l_{1} Z) \cup (1 + l_{2} Z) = Z

" q_712 A.76 3220911 question "

l_{1}, l_{2}

" q_713 A.76 3220911 question "

l_{1} = l_{2} = 5

" q_714 A.76 3220911 question "

{0 + p_{k_{j}}}

" q_715 A.76 3220911 question "

j = 1, ..., n

" q_716 A.76 3220911 question "

(⋃_{1 \leq j \leq n} (0 + p_{k_{j}} Z)) \cup (1 + 5 Z) \cup (1 + 5 Z)

" q_717 A.76 3220911 question "

p

" q_718 A.76 3220911 question "

p & g t; \max ({p_{k_{1}}, ..., p_{k_{n}}})

" q_719 A.76 3220911 question "

p \in̸ ⋃_{1 \leq j \leq n} (0 + p_{k_{j}} Z)

" q_720 A.76 3220911 question "

p \in̸ (1 + 5 Z) \cup (1 + 5 Z)

" q_721 A.76 3220911 question "

5 ∤ p - 1

" q_722 A.76 3220911 question "

5 ∤ p + 1

" q_723 A.76 3220911 question "

p

" q_724 A.76 3220911 question "

7

" q_725 A.76 3220911 question "

7

" q_726 A.77 3233314 title "

(1) (1) = 1

" q_727 A.77 3233314 question "

(1) (1) = 1

" q_728 A.77 3233314 question "

11 = 1

" q_729 A.77 3233314 question "

1 x = x

" q_739 A.79 3244171 question "

| \frac{e^{i x u} - 1}{u} | \leq | x |

" q_740 A.79 3244171 question "

u ≠ 0

" q_741 A.79 3244171 question "

x

" q_742 A.80 3114428 question "

N

" q_743 A.80 3114428 question "

\emptyset, {1}, {2}, {1, 2}, {3}, {1, 3}, {2, 3}, {1, 2, 3}, {4}, ...

" q_744 A.80 3114428 question "

{1}

" q_745 A.80 3114428 question "

{2}

" q_746 A.80 3114428 question "

n

" q_747 A.80 3114428 question "

[n] = {1, 2, ..., n}

" q_748 A.80 3114428 question "

n

" q_749 A.80 3114428 question "

n

" q_750 A.80 3114428 question "

[n]

" q_751 A.80 3114428 question "

2^{n}

" q_752 A.81 3124479 question "

M

" q_753 A.81 3124479 question "

n \in N

" q_754 A.81 3124479 question "

{1, \dots, n}

" q_755 A.81 3124479 question "

M

" q_756 A.81 3124479 question "

M ≠ \emptyset

" q_757 A.81 3124479 question "

x \in M

" q_758 A.81 3124479 question "

f (1)

" q_759 A.81 3124479 question "

f (1) := x

" q_760 A.81 3124479 question "

f

" q_761 A.81 3124479 question "

{1}

" q_762 A.81 3124479 question "

M

" q_763 A.81 3124479 question "

f

" q_764 A.81 3124479 question "

{1, \dots, n}

" q_765 A.81 3124479 question "

M

" q_766 A.81 3124479 question "

M

" q_767 A.81 3124479 question "

M - {f (1), \dots, f (n)}

" q_768 A.81 3124479 question "

x \in M - {f (1), \dots, f (n)}

" q_769 A.81 3124479 question "

g (1), \dots, g (n + 1)

" q_770 A.81 3124479 question "

g (1) := f (1), \dots, g (n) := f (n)

" q_771 A.81 3124479 question "

g (n + 1) := x

" q_772 A.81 3124479 question "

g

" q_773 A.81 3124479 question "

{1, \dots, n + 1}

" q_774 A.81 3124479 question "

M

" q_775 A.81 3124479 question "

n_{1}

" q_776 A.81 3124479 question "

h (n_{1})

" q_777 A.81 3124479 question "

g

" q_778 A.81 3124479 question "

{1, \dots, n_{1}}

" q_779 A.81 3124479 question "

M

" q_780 A.81 3124479 question "

g (n_{1})

" q_781 A.81 3124479 question "

h (n_{1})

" q_782 A.81 3124479 question "

(1, g (1)), \dots, (n_{1}, g (n_{1}))

" q_783 A.81 3124479 question "

h (n_{2})

" q_784 A.81 3124479 question "

n_{2} \leq n_{1}

" q_785 A.81 3124479 question "

g (n_{2})

" q_786 A.81 3124479 question "

g (n_{2})

" q_787 A.81 3124479 question "

h (n_{2})

" q_788 A.81 3124479 question "

h (n_{3})

" q_789 A.81 3124479 question "

n_{3} & g t; n_{1}

" q_790 A.81 3124479 question "

(n_{1} + 1, g (n_{1} + 1)), \dots, (n_{3}, g (n_{3}))

" q_791 A.81 3124479 question "

g (n_{3})

" q_792 A.81 3124479 question "

h (n_{3})

" q_793 A.81 3124479 question "

h (n)

" q_794 A.81 3124479 question "

n \in N

" q_795 A.81 3124479 question "

h : N \to M

" q_796 A.81 3124479 question "

h : N \to M

" q_797 A.81 3124479 question "

h : N \to M

" q_798 A.82 3149215 question "

f (x)

" q_799 A.82 3149215 question "

[0, 2 π]

" q_800 A.82 3149215 question "

A

" q_801 A.82 3149215 question "

f (x)

" q_802 A.82 3149215 question "

[0, 2 π]

" q_803 A.82 3149215 question "

A = \int_{0}^{2 π} f (x) d x

" q_804 A.82 3149215 question "

g (x)

" q_805 A.82 3149215 question "

f (x) = g (x) \cos (x)

" q_806 A.82 3149215 question "

f (x)

" q_807 A.82 3149215 question "

A = \int_{0}^{2 π} g (x) \cos (x) d x

" q_808 A.82 3149215 question "

t = \sin (x)

" q_809 A.82 3149215 question "

d t = \cos (x) d x

" q_810 A.82 3149215 question "

x = \arcsin (t)

" q_811 A.82 3149215 question "

t = \sin (x)

" q_812 A.82 3149215 question "

0

" q_813 A.82 3149215 question "

\sin (0) = 0

" q_814 A.82 3149215 question "

2 π

" q_815 A.82 3149215 question "

\sin (2 π) = 0

" q_816 A.82 3149215 question "

A = \int_{0}^{0} g (\arcsin (t)) d t

" q_817 A.82 3149215 question "

0

" q_818 A.82 3149215 question "

A = 0

" q_819 A.83 3068257 question "

1, 1 + \frac{1}{2}, 1 + \frac{1}{2} + \frac{1}{3}, 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4}, ...

" q_820 A.83 3068257 question "

M

" q_821 A.83 3068257 question "

a_{n} \leq M

" q_822 A.83 3068257 question "

n

" q_823 A.84 3071031 title "

4, x

" q_824 A.84 3071031 title "

Z [x]

" q_825 A.84 3071031 question "

I = & l t; p, x & g t;

" q_826 A.84 3071031 question "

Z [x]

" q_827 A.84 3071031 question "

I = & l t; p, x & g t;

" q_828 A.84 3071031 question "

Z [x]

" q_829 A.84 3071031 question "

4, x

" q_830 A.84 3071031 question "

Z [x]

" q_831 A.85 3074005 title "

N

" q_832 A.85 3074005 question "

0

" q_833 A.85 3074005 question "

0

" q_834 A.85 3074005 question "

1

" q_835 A.85 3074005 question "

(1)

" q_836 A.85 3074005 question "

1 / 2

" q_837 A.85 3074005 question "

1 / 2

" q_838 A.85 3074005 question "

N

" q_839 A.85 3074005 question "

N

" q_840 A.85 3074005 question "

p_{n} = \frac{1}{2} p_{n - 1}

" q_841 A.85 3074005 question "

p_{n}

" q_842 A.85 3074005 question "

n

" q_843 A.85 3074005 question "

N

" q_844 A.85 3074005 question "

p_{N} = 1

" q_845 A.85 3074005 question "

p_{N - 1} = 2

" q_846 A.85 3074005 question "

p_{0} = 2^{N}

" q_847 A.85 3074005 question "

1

" q_848 A.86 3080329 title "

\sum_{k = 0}^{n} k (\begin{matrix} n \\ k \end{matrix}) = O (2^{n \log_{3} n}) qvar_F o r m u l a # q_{8} 49 : =∑=k0n⋅k(nk)⁢O(2⁢nlog3n)qvar_Formula#q850:⁢∑=k0n⋅k(nk)≤∑=k0n⋅k(n⌊n2⌋)≤⋅⋅n(+n1)2(n⌊n2⌋)≤⁢n!(+n1)≤⁢nnn≤n+n1Formula#q851:⁢:∈∀nN≥⁢(∑=i1nai)(∑=i1n1ai)n2Formula#q852:⁢aiFormula#q853:⁢:∈∀iN∈aiR+Formula#q854:⁢:∈∀nN≥⁢(∑=i1nai)(∑=i1n1ai)n2Formula#q855:⁢=n1Formula#q856:⁢=n2Formula#q857:⁢=n3Formula#q858:⁢∈a,bR+Formula#q859:⁢=⁢ab1Formula#q860:⁢≥+ab2Formula#q861:=n3

" q_862 A.87 3081320 question "

a, b, c \in R^{+}

" q_863 A.87 3081320 question "

(a + b + c) (\frac{1}{a} + \frac{1}{b} + \frac{1}{c}) = 1 + \frac{a}{b} + \frac{a}{c} + \frac{b}{a} + 1 + \frac{b}{c} + \frac{c}{a} + \frac{c}{b} + 1

" q_864 A.87 3081320 question "

= 3 + (\frac{a}{b} + \frac{b}{a}) + (\frac{a}{c} + \frac{c}{a}) + (\frac{b}{c} + \frac{c}{b})

" q_865 A.87 3081320 question "

(\frac{a}{b} + \frac{b}{a}) \geq 2

" q_866 A.87 3081320 question "

(\frac{a}{c} + \frac{c}{a}) \geq 2

" q_867 A.87 3081320 question "

(\frac{b}{c} + \frac{c}{b}) \geq 2

" q_868 A.87 3081320 question "

3 + (\frac{a}{b} + \frac{b}{a}) + (\frac{a}{c} + \frac{c}{a}) + (\frac{b}{c} + \frac{c}{b}) \geq 3 + 2 + 2 + 2 = 9 = 3^{2} ■

" q_869 A.87 3081320 question "

n

" q_870 A.87 3081320 question "

P (n) : : (\sum_{i = 1}^{n} a_{i}) (\sum_{i = 1}^{n} \frac{1}{a_{i}}) \geq n^{2}

" q_871 A.87 3081320 question "

(\sum_{i = 1}^{1} a_{i}) (\sum_{i = 1}^{1} \frac{1}{a_{i}}) = a_{1} \frac{1}{a_{1}} = 1 = 1^{2}

" q_872 A.87 3081320 question "

P (1)

" q_873 A.87 3081320 question "

P (n)

" q_874 A.87 3081320 question "

(\sum_{i = 1}^{n + 1} a_{i}) (\sum_{i = 1}^{n + 1} \frac{1}{a_{i}}) = [(\sum_{i = 1}^{n} a_{i}) + a_{n + 1}] [(\sum_{i = 1}^{n} \frac{1}{a_{i}}) + \frac{1}{a_{n + 1}}]

" q_875 A.87 3081320 question "

= (\sum_{i = 1}^{n} a_{i}) [(\sum_{i = 1}^{n} \frac{1}{a_{i}}) + \frac{1}{a_{n + 1}}] + a_{n + 1} [(\sum_{i = 1}^{n} \frac{1}{a_{i}}) + \frac{1}{a_{n + 1}}]

" q_876 A.87 3081320 question "

= (\sum_{i = 1}^{n} a_{i}) (\sum_{i = 1}^{n} \frac{1}{a_{i}}) + (\sum_{i = 1}^{n} a_{i}) \frac{1}{a_{n + 1}} + a_{n + 1} (\sum_{i = 1}^{n} \frac{1}{a_{i}}) + a_{n + 1} \frac{1}{a_{n + 1}}

" q_877 A.87 3081320 question "

= (\sum_{i = 1}^{n} a_{i}) (\sum_{i = 1}^{n} \frac{1}{a_{i}}) + (\sum_{i = 1}^{n} a_{i}) \frac{1}{a_{n + 1}} + a_{n + 1} (\sum_{i = 1}^{n} \frac{1}{a_{i}}) + 1

" q_878 A.87 3081320 question "

{\geq_{⏟}}_{I H} n^{2} + (\sum_{i = 1}^{n} a_{i}) \frac{1}{a_{n + 1}} + a_{n + 1} (\sum_{i = 1}^{n} \frac{1}{a_{i}}) + 1

" q_879 A.87 3081320 question "

(\sum_{i = 1}^{n} a_{i}) \frac{1}{a_{n + 1}} + a_{n + 1} (\sum_{i = 1}^{n} \frac{1}{a_{i}})

" q_880 A.88 3095026 title "

x^{4} + 10 x^{2} + 1

" q_881 A.88 3095026 title "

Z [x]

" q_882 A.88 3095026 question "

x^{4} + 10 x^{2} + 1

" q_883 A.88 3095026 question "

Z [x]

" q_884 A.89 3140361 question "

A^{2} + B^{2} = C^{2} + D^{2}

" q_885 A.89 3140361 question "

A, B, C, D

" q_886 A.90 3143980 question "

A

" q_887 A.90 3143980 question "

n \times n

" q_888 A.90 3143980 question "

A

" q_889 A.90 3143980 question "

n \times n

" q_890 A.90 3143980 question "

A^{1}

" q_891 A.90 3143980 question "

A^{1} A = I_{n} \land A A^{1} = I_{n} (1)

" q_892 A.90 3143980 question "

I_{n}

" q_893 A.90 3143980 question "

n \times n

" q_894 A.90 3143980 question "

A^{1} A = I_{n}

" q_895 A.90 3143980 question "

A A^{1} ≠ I_{n}

" q_896 A.91 3175187 question "

R \to R

" q_897 A.91 3175187 question "

2

" q_898 A.91 3175187 question "

x^{2}

" q_899 A.91 3175187 question "

0

" q_900 A.91 3175187 question "

3

" q_901 A.92 3176291 title "

F_{q} [X, Y]

" q_902 A.92 3176291 question "

K

" q_903 A.92 3176291 question "

K [X, Y]

" q_904 A.92 3176291 question "

F_{q} [X, Y]

" q_905 A.92 3176291 question "

I = (P (X, Y))

" q_906 A.92 3176291 question "

P

" q_907 A.92 3176291 question "

P

" q_908 A.92 3176291 question "

X

" q_909 A.92 3176291 question "

Y

" q_910 A.92 3176291 question "

Q (X)

" q_911 A.92 3176291 question "

F_{q} [X, Y] / (Q (X)) ≅ (F_{q} [X] / (Q (X))) [Y]

" q_912 A.92 3176291 question "

P

" q_913 A.92 3176291 question "

P (X, Y) = \sum_{i = 0}^{n} P_{i} (X) Y^{i}

" q_914 A.92 3176291 question "

d \geq 1

" q_915 A.92 3176291 question "

P_{i}

" q_916 A.92 3176291 question "

X

" q_917 A.92 3176291 question "

P_{n}

" q_918 A.92 3176291 question "

F_{q}

" q_919 A.92 3176291 question "

P_{n}

" q_920 A.92 3176291 question "

\prod_{α \in F_{q}} (X - α)

" q_921 A.92 3176291 question "

α \in F_{q}

" q_922 A.92 3176291 question "

P_{n} (α) ≠ 0

" q_923 A.92 3176291 question "

I := (P, X - α)

" q_924 A.92 3176291 question "

(P)

" q_925 A.92 3176291 question "

(P)

" q_926 A.92 3176291 question "

I

" q_927 A.92 3176291 question "

F_{q} [X, Y]

" q_928 A.92 3176291 question "

1 \in I

" q_929 A.92 3176291 question "

X = α

" q_930 A.92 3176291 question "

n = \deg_{Y} (P) = 0

" q_931 A.92 3176291 question "

P (X, Y) = (X^{p - 1} - 1) X Y - 1

" q_932 A.92 3176291 question "

P (X, Y) = (X^{p - 1} - 1) X Y - X - 1

" q_933 A.92 3176291 question "

F_{p} [X, Y]

" q_934 A.92 3176291 question "

p

" q_935 A.93 3178957 title "

A B =

" q_936 A.93 3178957 title "

B A

" q_937 A.93 3178957 question "

A, B

" q_938 A.93 3178957 question "

R

" q_939 A.93 3178957 question "

n x n

" q_940 A.93 3178957 question "

d e t (x I - A B) = d e t (x I - B A)

" q_941 A.93 3178957 question "

A

" q_942 A.93 3178957 question "

B

" q_943 A.94 3185231 title "

2^{א_{0}}

" q_944 A.94 3185231 question "

2^{א_{0}}

" q_945 A.94 3185231 question "

P

" q_946 A.94 3185231 question "

P

" q_947 A.94 3185231 question "

T

" q_948 A.94 3185231 question "

T

" q_949 A.95 3201967 title "

\sin (x)

" q_950 A.95 3201967 question "

\sin (x)

" q_951 A.95 3201967 question "

b

" q_952 A.95 3201967 question "

∠ B = x

" q_953 A.95 3201967 question "

1

" q_954 A.95 3201967 question "

{lim}_{x \to 0} \frac{\sin (x)}{x} = 1

" q_955 A.95 3201967 question "

\sin (x)

" q_956 A.95 3201967 question "

x

" q_957 A.95 3201967 question "

\sin (x & l t); x & l t; \tan (x)

" q_958 A.95 3201967 question "

\sin (x & l t); x & l t; \sin (x) + 1 - \cos (x)

" q_959 A.95 3201967 question "

\sin (x)

" q_960 A.95 3201967 question "

x

" q_961 A.95 3201967 question "

F

" q_962 A.95 3201967 question "

G

" q_963 A.95 3201967 question "

R^{2}

" q_964 A.95 3201967 question "

F \subset G

" q_965 A.95 3201967 question "

| \partial F | & l t; | \partial G |

" q_966 A.95 3201967 question "

| \partial F |

" q_967 A.95 3201967 question "

F

" q_968 A.95 3201967 question "

\int_{s}^{t} \sqrt{1 + y^{'} {(u)}^{2}} d u

" q_969 A.95 3201967 question "

y^{''}

" q_970 A.95 3201967 question "

y (s) = y (t)

" q_971 A.95 3201967 question "

y^{'}

" q_972 A.95 3201967 question "

y = \sqrt{1 - x^{2}}

" q_973 A.95 3201967 question "

1 \leq x \leq 1

" q_974 A.95 3201967 question "

\sin (x)

" q_975 A.96 3209203 title "

\sum_{i} a_{i}

" q_976 A.96 3209203 title "

a_{i}

" q_977 A.96 3209203 title "

\sum_{i} \frac{a_{i}}{a_{i} + a_{i + 1} + a_{i + 2} + \dots}

" q_978 A.96 3209203 question "

\sum_{i} a_{i}

" q_979 A.96 3209203 question "

a_{i}

" q_980 A.96 3209203 question "

s_{n} = \sum_{i = n}^{\infty} a_{i}

" q_981 A.96 3209203 question "

\sum_{i} \frac{a_{i}}{s_{i}}

" q_982 A.96 3209203 question "

a_{i} = r^{i}

" q_983 A.96 3209203 question "

a_{i} = 1 / i^{2}

" q_984 A.97 3218680 title "

R

" q_985 A.97 3218680 title "

Q

" q_986 A.97 3218680 question "

C

" q_987 A.97 3218680 question "

R

" q_988 A.97 3218680 question "

2

" q_989 A.97 3218680 question "

C = s p a n {1, i}

" q_990 A.97 3218680 question "

R

" q_991 A.97 3218680 question "

Q

" q_992 A.97 3218680 question "

n

" q_993 A.97 3218680 question "

m := n + 1

" q_994 A.97 3218680 question "

x_{1}, ... x_{m}

" q_995 A.97 3218680 question "

0

" q_996 A.97 3218680 question "

\frac{a_{1}}{b_{1}} x_{1} + ... + \frac{a_{m}}{b_{m}} x_{m} = 0

" q_997 A.97 3218680 question "

l c m (b_{1}, ..., b_{m})

" q_998 A.97 3218680 question "

m

" q_999 A.97 3218680 question "

0

" q_1000 A.98 3220928 title "

R : S^{1} \to S^{1}

" q_1001 A.98 3220928 title "

{R^{n} ([x])}

" q_1002 A.98 3220928 title "

S^{1}

" q_1003 A.98 3220928 question "

α

" q_1004 A.98 3220928 question "

R : S^{1} \to S^{1}

" q_1005 A.98 3220928 question "

[x] \to [x + α]

" q_1006 A.98 3220928 question "

[x] \in S^{1}

" q_1007 A.98 3220928 question "

{R^{n} ([x])}

" q_1008 A.98 3220928 question "

S^{1}

" q_1009 A.98 3220928 question "

R

" q_1010 A.98 3220928 question "

R (e^{2 π i x}) = e^{2 π i (x + α)} .

" q_1011 A.98 3220928 question "

R^{n} (x)

" q_1012 A.98 3220928 question "

n \in Z

" q_1013 A.98 3220928 question "

R^{n} (e^{2 π i x}) = e^{2 π i (x + n α)}

" q_1014 A.98 3220928 question "

e^{2 π i x} \in S^{1}

" q_1015 A.98 3220928 question "

[y] = e^{2 π i y} \in S^{1}

" q_1016 A.98 3220928 question "

V

" q_1017 A.98 3220928 question "

[y]

" q_1018 A.98 3220928 question "

[z_{y}] = e^{2 π i z}

" q_1019 A.98 3220928 question "

[z_{y}] = R^{n} ([x])

" q_1020 A.98 3220928 question "

n \in Z

" q_1021 A.98 3220928 question "

e^{2 π i z} = e^{2 π i (x + α n)} \Rightarrow 2 π i z = 2 π i (x + α n) + 2 i k π for some k \in Z \Rightarrow z = x + α n + k .

" q_1022 A.99 3229938 question "

f : R \to R

" q_1023 A.99 3229938 question "

Q

" q_1024 A.100 3230174 question "

E = (0, 1]

" q_1025 A.100 3230174 question "

E \subseteq R

" q_1026 A.100 3230174 question "

R