Mock Geometry AIME 2011 Problems/Problem 1

Problem

Let $ABCD$ be a unit square, and let $AB_1C_1D_1$ be its image after a $30$ degree rotation about point $A.$ The area of the region consisting of all points inside at least one of $ABCD$ and $AB_1C_1D_1$ can be expressed in the form $\frac{a-\sqrt{b}} {c},$ where $a,b,c$ are positive integers, and $b$ shares no perfect square common factor with $c$ . Find $a+b+c.$

Solution

$[asy] draw(unitsquare); draw((0,1)--(1/2, (2-sqrt(3))/2)--((1+sqrt(3))/2,(3-sqrt(3))/2)--(sqrt(3)/2,3/2)--cycle); [/asy]$

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Mock Geometry AIME 2011 Problems/Problem 1

Problem

Solution