User:Luosiji/SandboxFrom Wikipedia, the free encyclopedia 20 x 2 + 2 y 2 − 5 = 0 {\displaystyle 20x^{2}+2y^{2}-5=0\!} 36 x 2 + 68 x y + 36 y 2 − 136 x − 144 y + 139 = 0 {\displaystyle 36x^{2}+68xy+36y^{2}-136x-144y+139=0\!} y − | x | 12 = − 4 5 ( | x | + y 12 − 9 4 ) 2 3 + 2 {\displaystyle y-{\frac {|x|}{12}}=-{\frac {4}{5}}{\sqrt[{3}]{\left(\left|x\right|+{\frac {y}{12}}-{\frac {9}{4}}\right)^{2}}}+2\!} y − | x | 6 = 1 2 ( | | x | + y 6 − 9 4 | − 1 ) 2 3 − 6 5 {\displaystyle y-{\frac {|x|}{6}}={\frac {1}{2}}{\sqrt[{3}]{\left(\left|\left|x\right|+{\frac {y}{6}}-{\frac {9}{4}}\right|-1\right)^{2}}}-{\frac {6}{5}}\!} x 2 3 ± 9 − x 2 {\displaystyle {\sqrt[{3}]{x^{2}}}\pm {\sqrt {9-x^{2}}}\!} ∑ i = 0 n ∑ j = 1 m ( m j ) i m − j = ( n + 1 ) m {\displaystyle \sum _{i=0}^{n}\sum _{j=1}^{m}{m \choose j}{i}^{m-j}={\left(n+1\right)}^{m}} ∑ i = 0 n ( i + 1 ) m − i m = ( n + 1 ) m {\displaystyle \sum _{i=0}^{n}{\left(i+1\right)}^{m}-{i}^{m}={\left(n+1\right)}^{m}} E [ T ] ≤ ⌊ log n m ⌋ + 1 + m − 1 m n n − 1 . {\displaystyle E[T]\leq \lfloor \log _{n}m\rfloor +1+{\frac {m-1}{m}}{\frac {n}{n-1}}.}
20 x 2 + 2 y 2 − 5 = 0 {\displaystyle 20x^{2}+2y^{2}-5=0\!} 36 x 2 + 68 x y + 36 y 2 − 136 x − 144 y + 139 = 0 {\displaystyle 36x^{2}+68xy+36y^{2}-136x-144y+139=0\!} y − | x | 12 = − 4 5 ( | x | + y 12 − 9 4 ) 2 3 + 2 {\displaystyle y-{\frac {|x|}{12}}=-{\frac {4}{5}}{\sqrt[{3}]{\left(\left|x\right|+{\frac {y}{12}}-{\frac {9}{4}}\right)^{2}}}+2\!} y − | x | 6 = 1 2 ( | | x | + y 6 − 9 4 | − 1 ) 2 3 − 6 5 {\displaystyle y-{\frac {|x|}{6}}={\frac {1}{2}}{\sqrt[{3}]{\left(\left|\left|x\right|+{\frac {y}{6}}-{\frac {9}{4}}\right|-1\right)^{2}}}-{\frac {6}{5}}\!} x 2 3 ± 9 − x 2 {\displaystyle {\sqrt[{3}]{x^{2}}}\pm {\sqrt {9-x^{2}}}\!} ∑ i = 0 n ∑ j = 1 m ( m j ) i m − j = ( n + 1 ) m {\displaystyle \sum _{i=0}^{n}\sum _{j=1}^{m}{m \choose j}{i}^{m-j}={\left(n+1\right)}^{m}} ∑ i = 0 n ( i + 1 ) m − i m = ( n + 1 ) m {\displaystyle \sum _{i=0}^{n}{\left(i+1\right)}^{m}-{i}^{m}={\left(n+1\right)}^{m}} E [ T ] ≤ ⌊ log n m ⌋ + 1 + m − 1 m n n − 1 . {\displaystyle E[T]\leq \lfloor \log _{n}m\rfloor +1+{\frac {m-1}{m}}{\frac {n}{n-1}}.}