Good positive integers

For discussing Olympiad Level Number Theory problems
tanmoy
Posts: 305
Joined: Fri Oct 18, 2013 11:56 pm
Location: Rangpur,Bangladesh

Good positive integers

Unread post by tanmoy » Sun Sep 27, 2015 7:16 pm

For a positive real number $c>0$,call a positive integer $c$-$good$ if for all positive integer $m<n$,$\frac {m} {n}$ can be represented as
$\frac{m}{n}=\frac{a_{0}}{b_{0}}+...+\frac{a_{k}} {b_{k}}$
for some non-negative integers $k<\frac{n} {c}$ and $2b_{i}\leq n$ and $0\leq a_{i}<b_{j}$ and $0\leq j\leq k$.Show that,for any real $c$,there are an infinite number of $c$-$good$ positive integers.
"Questions we can't answer are far better than answers we can't question"

User avatar
Masum
Posts: 592
Joined: Tue Dec 07, 2010 1:12 pm
Location: Dhaka,Bangladesh

Re: Good positive integers

Unread post by Masum » Sat Oct 10, 2015 12:20 am

Aha. That problem of mine! It's quite easy actually. But it can be made a bit hard with more restrictions. Have you solved it?
One one thing is neutral in the universe, that is $0$.

Post Reply