Skip to main content
MathOverflow

Return to Question

deleted 1 character in body
Source Link
math110
math110
  • 4.3k
  • 18
  • 46

The following question has postipost mathsatck :Nice problem, perhaps from a question of expectation.The problem seemed so elementary, but I had tried to solve it for a long time and eventually failed.I 'd like to get your help or help out here.Thanks!

Show that: for any real numbers $x_{1},x_{2},\cdots,x_{n}$, there exist $k\in\{1,2,\cdots,n\}$ such that $$\dfrac{1}{n}\sum_{i=1}^{n}\left(\{kx_{i}\}-\dfrac{1}{2}\right)^2>\dfrac{1}{12}-\dfrac{1}{6n},$$ where $\{x\}=x-\lfloor x\rfloor$.

The following question has posti mathsatck :Nice problem, perhaps from a question of expectation.The problem seemed so elementary, but I had tried to solve it for a long time and eventually failed.I 'd like to get your help or help out here.Thanks!

Show that: for any real numbers $x_{1},x_{2},\cdots,x_{n}$, there exist $k\in\{1,2,\cdots,n\}$ such that $$\dfrac{1}{n}\sum_{i=1}^{n}\left(\{kx_{i}\}-\dfrac{1}{2}\right)^2>\dfrac{1}{12}-\dfrac{1}{6n},$$ where $\{x\}=x-\lfloor x\rfloor$.

The following question has post mathsatck :Nice problem, perhaps from a question of expectation.The problem seemed so elementary, but I had tried to solve it for a long time and eventually failed.I 'd like to get your help or help out here.Thanks!

Show that: for any real numbers $x_{1},x_{2},\cdots,x_{n}$, there exist $k\in\{1,2,\cdots,n\}$ such that $$\dfrac{1}{n}\sum_{i=1}^{n}\left(\{kx_{i}\}-\dfrac{1}{2}\right)^2>\dfrac{1}{12}-\dfrac{1}{6n},$$ where $\{x\}=x-\lfloor x\rfloor$.

edited tags
Link
GH from MO
GH from MO
  • 105.2k
  • 8
  • 292
  • 398
Source Link
math110
math110
  • 4.3k
  • 18
  • 46

Show that there exist $k\in\{1,2,\cdots,n\}$ such that $\frac{1}{n}\sum_{i=1}^{n}\left(\{kx_{i}\}-\frac{1}{2}\right)^2>\frac{1}{12}-\frac{1}{6n}$

The following question has posti mathsatck :Nice problem, perhaps from a question of expectation.The problem seemed so elementary, but I had tried to solve it for a long time and eventually failed.I 'd like to get your help or help out here.Thanks!

Show that: for any real numbers $x_{1},x_{2},\cdots,x_{n}$, there exist $k\in\{1,2,\cdots,n\}$ such that $$\dfrac{1}{n}\sum_{i=1}^{n}\left(\{kx_{i}\}-\dfrac{1}{2}\right)^2>\dfrac{1}{12}-\dfrac{1}{6n},$$ where $\{x\}=x-\lfloor x\rfloor$.