Philippine Mathematical Olympiad 2015

All four digit numbers has digital sum between $1,2,3,.....,36$ .

Every digital sum can be mapped to a different four-digit numbers.But there exists a smallest number $n$ for every $s_n$

For $s_n=1$ smallest $n$ is $1000$

For $s_n=2$ smallest $n$ is $1001$

For $s_n=3$ smallest $n$ is $1011$

and continues...

The $\frac{n}{s_n}$ will continue to decrease upto $s_n=17$ and again continue to increase after $n \geq 20$ .So minimum lies within $18,19$ .

For $s_n=18$ smallest $n=1089$ then $n/s_n=60.5$

For $s_n=19$ smallest $n=1088$ then $n/s_n=57.84$

After some more values it can be seen that this will be the smallest.