BdMO Online Forum

Let $S(x)$ be the sum of the digits of the positive integer $x$ in its decimal
representation.
(a) Prove that for every positive integer $x$,
$\frac{S(x)}{S(2x)}\leq 5$ Can this bound be
improved?
(b) Prove that $\frac{S(x)}{S(3x)}$is not bounded.

Posting only the hints again. Do not open them all at once. Do it serially .This is because it is easier to get the later hints when you know the earlier ones.
Warning: My solution is too much abstract :S I think I should see the official solution as well.
First and basic: Then: Desperate:

what is meant by can this bound be improved ?
does it mean whether it can be shown or not that there can be a greater value than $5$ ?
if it is then

nafistiham wrote:what is meant by can this bound be improved ?
does it mean whether it can be shown or not that there can be a greater value than $5$ ?
if it is then

as it is the decimal system it cant be greater than $5$.so, the relation cant be improved

It is actually "a smaller value than $5$".

but, $\frac {5}{1}=5$ where, $x=5$ and $2x=10$ so it cant be improved, right ?

Yes, that's right.