For any positive integer $n{}$ define $a_n=\{n/s(n)\}$ where $s(\cdot)$ denotes the sum of the digits and $\{\cdot\}$ denotes the fractional part. Prove that there exist infinitely many positive integers $n$ such that $a_n=1/2.$ Determine the smallest positive integer $n$ such that $a_n=1/6.$ Marius Burtea