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Bangladesh TST(Team selection test) 2011 Exam 4 Problem 1
Posted: Thu Dec 15, 2011 7:34 pm
by sourav das
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 $x$
Re: Bangladesh TST(Team selection test) 2011 Exam 4 Problem
Posted: Tue Nov 06, 2012 1:03 am
by mahathir
Is it all $n$ which are divisible by $3$ ?
Re: Bangladesh TST(Team selection test) 2011 Exam 4 Problem
Posted: Tue Nov 06, 2012 10:41 am
by Phlembac Adib Hasan
mahathir wrote:Is it all $n$ which are divisible by $3$ ?
$1$-ও তো হয়। $x=1.5$ নেন।
Re: Bangladesh TST(Team selection test) 2011 Exam 4 Problem
Posted: Tue Nov 06, 2012 12:20 pm
by *Mahi*
Re: Bangladesh TST(Team selection test) 2011 Exam 4 Problem
Posted: Tue Apr 09, 2013 10:26 am
by Phlembac Adib Hasan
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 $2a\ge 2\Longrightarrow a\ge 1$.
