2001 AIME II Problems/Problem 15
Let , , and be three adjacent square faces of a cube, for which , and let be the eighth vertex of the cube. Let , , and , be the points on , , and , respectively, so that . A solid is obtained by drilling a tunnel through the cube. The sides of the tunnel are planes parallel to , and containing the edges, , , and . The surface area of , including the walls of the tunnel, is , where , , and are positive integers and is not divisible by the square of any prime. Find .
Set the coordinate system so that vertex , where the drilling starts, is at . Using a little visualization (involving some similar triangles, because we have parallel lines) shows that the tunnel meets the bottom face (the xy plane one) in the line segments joining to , and to , and similarly for the other three faces meeting at the origin (by symmetry). So one face of the tunnel is the polygon with vertices (in that order), , and the other two faces of the tunnel are congruent to this shape.
Observe that this shape is made up of two congruent trapezoids each with height and bases and . Together they make up an area of . The total area of the tunnel is then . Around the corner we're missing an area of , the same goes for the corner opposite . So the outside area is . Thus the the total surface area is , and the answer is .
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