Question 9
Let $ABCDEF$ be a regular hexagon. Let $G,$ $H,$ $I,$ $J,$ $K,$ and $L$ be the midpoints of sides $AB,$ $BC,$ $CD,$ $DE,$ $EF,$ and $AF,$ respectively. The segments $AH,$ $BI,$ $CJ,$ $DK,$ $EL,$ and $FG$ bound a smaller regular hexagon. Let the ratio of the area of the smaller hexagon to the area of $ABCDEF$ be expressed as a fraction $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$