Find for which positive integers $n$ there exists real number $x$ such that:
\[\sum_{i=1}^{n}\left \lfloor ix \right \rfloor= n\]
Where $\left \lfloor x \right \rfloor=$greatest integer which is less or equal to $x$
Bangladesh TST(Team selection test) 2011 Exam 4 Problem 1
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You spin my head right round right round,
When you go down, when you go down down......(-$from$ "$THE$ $UGLY$ $TRUTH$" )
When you go down, when you go down down......(-$from$ "$THE$ $UGLY$ $TRUTH$" )
Re: Bangladesh TST(Team selection test) 2011 Exam 4 Problem
Is it all $n$ which are divisible by $3$ ?
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Re: Bangladesh TST(Team selection test) 2011 Exam 4 Problem
$1$-ও তো হয়। $x=1.5$ নেন।mahathir wrote:Is it all $n$ which are divisible by $3$ ?
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Re: Bangladesh TST(Team selection test) 2011 Exam 4 Problem
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Use $L^AT_EX$, It makes our work a lot easier!
Nur Muhammad Shafiullah | Mahi
Use $L^AT_EX$, It makes our work a lot easier!
Nur Muhammad Shafiullah | Mahi
- Phlembac Adib Hasan
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Re: Bangladesh TST(Team selection test) 2011 Exam 4 Problem
Any $x$ in the interval $\dfrac 1 k>x\ge \dfrac {2}{2k+1}$ satisfies the equation if $n=3k$. If $n=3k+1$, take any $x$ from the interval $\dfrac 2 {2k+1}>x\ge \dfrac 1 {k+1}$. And if $3\mid n-2$, it is not that hard to prove there is no such $x$. Just keep in mind $2a\ge 2\Longrightarrow a\ge 1$.