2021 AIME I Problems/Problem 6
Segments and are edges of a cube and is a diagonal through the center of the cube. Point satisfies , , , and . Find
First scale down the whole cube by . Let point have coordinates , point have coordinates , and be the side length. Then we have the equations These simplify into Adding the first three equations together, we get . Subtracting this from the fourth equation, we get , so . This means . However, we scaled down everything by so our answer is .
Solution 2 (Solution 1 with Slight Simplification)
Once the equations for the distance between point P and the vertices of the cube have been written, we can add the first, second, and third to receive, Subtracting the fourth equation gives Since point , and since we scaled the answer is .
Let be the vertex of the cube such that is a square. By the British Flag Theorem, we can easily we can show that and Hence, by adding the two equations together, we get . Substituting in the values we know, we get .
Thus, we can solve for , which ends up being .
For all points in space, define the function by . Then is linear; let be the center of . Then since is linear, where denotes the side length of the cube. Thus
Video Solution by Punxsutawney Phil
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