2017 AIME II Problems/Problem 15
Problem
Tetrahedron has , , and . For any point in space, suppose . The least possible value of can be expressed as , where and are positive integers, and is not divisible by the square of any prime. Find .
Official Solution (MAA)
Let and be midpoints of and . The given conditions imply that and , and therefore and . It follows that and both lie on the common perpendicular bisector of and , and thus line is that common perpendicular bisector. Points and are symmetric to and with respect to line . If is a point in space and is the point symmetric to with respect to line , then and , so .
Let be the intersection of and . Then , from which it follows that . It remains to minimize as moves along .
Allow to rotate about to point in the plane on the side of opposite . Because is a right angle, . It then follows that , and equality occurs when is the intersection of and . Thus . Because is the median of , the Length of Median Formula shows that and . By the Pythagorean Theorem .
Because and are right angles, It follows that . The requested sum is .
Solution 2
Set , , . Let be the point which minimizes .
Let and denote the midpoints of and . From and , we have , an hence is a perpendicular bisector of both segments and . Then if is any point inside tetrahedron , its orthogonal projection onto line will have smaller -value; hence we conclude that must lie on . Similarly, must lie on the line joining the midpoints of and .
Let be the centroid of triangle ; then (by vectors). If we define , , similarly, we get and so on. But from symmetry we have , hence .
Now we use the fact that an isosceles tetrahedron has circumradius .
Here so . Therefore, the answer is .
Solution 3
Isosceles tetrahedron or Disphenoid (https://en.wikipedia.org/wiki/Disphenoid) can be inscribed in a parallelepiped whose facial diagonals are the pares of equal edges of the tetrahedron where This parallelepiped is right-angled, therefore it is circumscribed and has equal diagonals. The center O of the circumscribed sphere (coincide with the centroid) has equal distance from each vertex. Tetrachedrons and are congruent, so point of symmetry O is point of minimum , where is the circumradius of parallelepiped.
vladimir.shelomovskii@gmail.com, vvsss (Reconstruction)
See Also
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