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2021 WSMO Team Round Problems/Problem 6

Problem

Suppose that regular dodecagon $ABCDEFGHIJKL$ has side length $5.$ The area of the shaded region can be expressed as $a+b\sqrt{c},$ where $c$ is not divisible by the square of any prime. Find $a+b+c$.

[asy] size(150); filldraw(polygon(12),grey); filldraw(rotate(75)*(dir(60)--dir(150)--dir(240)--dir(330)--cycle),white); for(int i=30; i<=360; i+=30){ dot(rotate(75)*dir(i)); } label("$C$",dir(45),NE); label("$B$",dir(75),N); label("$A$",dir(105),N); label("$L$",dir(135),NW); label("$K$",dir(165),W); label("$J$",dir(195),W); label("$I$",dir(225),SW); label("$H$",dir(255),S); label("$G$",dir(285),S); label("$F$",dir(315),SE); label("$E$",dir(345),E); label("$D$",dir(375),E); [/asy]

Proposed by mahaler

Solution

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