Question

Let $f:N\to N$ be a function such that (i) $x-f\left(x\right)=19\left[\frac{x}{19}\right]-90\left[\frac{f\left(x\right)}{90}\right]\forall xϵN$, where [] denotes the greatest integer function (ii) 1900< f(1990)< 2000. Let sum of all the possible values of f(1990) be k.Find sum of digits of $k$ .

Answer 1

Since, 1900< f(1990)< 2000
$\Rightarrow$ $\frac{1900}{90}<\frac{f\left(1990\right)}{90}<\frac{2000}{90}\Rightarrow 21.1<\frac{f\left(1990\right)}{90}<22.2$
$\therefore$ $\left[\frac{f\left(1990\right)}{90}\right]=21,22$ (i)
Given, $x-f\left(x\right)=19\left[\frac{x}{19}\right]-90\left[\frac{f\left(x\right)}{90}\right]$
Now, there are two cases which arise.
Case I: $\left[\frac{f\left(1990\right)}{90}\right]=21$
We have, $1990-f\left(1990\right)=19\left[\frac{1990}{19}\right]-90\left[\frac{1990}{90}\right]$
1990-f(1990)=19(104)-90(21)
$\Rightarrow$ f(1990)=1904 (ii)
Case II: $\left[\frac{f\left(1990\right)}{90}\right]=22$
We have, $1990-f\left(1990\right)=19\left[\frac{1990}{19}\right]-90\left[\frac{1990}{90}\right]$
$\Rightarrow$ f(1990)=1994 (iii)
From Eqs. (ii) and (iii), we have f(1990)=1904 or 1994.