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HMMT November 2022 H7

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Alice and Bob are playing in the forest. They have six sticks of length 1,2,3,4,5,61,2,3,4,5,6 inches. Somehow, they have managed to arrange these sticks, such that they form the sides of an equiangular hexagon. Compute the sum of all possible values of the area of this hexagon.

Dataset text NuminaMath olympiads. Quoted with source.

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33 \sqrt{3}

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