## AIME II 2010 #9

American Mathematics Contest & American Invitational Mathematics Examination. See http://www.unl.edu/amc for more information
Note:USAMO/IMO-level questions need to go in the Olympiad section.

### AIME II 2010 #9

Let ABCDEF be a regular hexagon. Let be the midpoints of sides , respectively. The segments 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 where and are relatively prime positive integers. Find .
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stupidityismygam
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### Re: AIME II 2010 #9

stupidityismygam wrote:Let ABCDEF be a regular hexagon. Let be the midpoints of sides , respectively. The segments 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 where and are relatively prime positive integers. Find .

The length of AH, from Law of Cosines, is

The apothem of the inner, smaller triangle is drawn of the hexagons' center to AH.

From similar triangles, .

This means that the apothem length is

For an apothem of , the length of the side of the smaller hexagon is

Now the ratio of the area is the ratio of the squares of the sides of the hexagons:

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