HMMT 二月 2005 · 冲刺赛 · 第 19 题

HMMT February 2005 — Guts Round — Problem 19

[8] Regular tetrahedron ABCD is projected onto a plane sending A , B , C , and D ′ ′ ′ ′ ′ ′ ′ ′ to A , B , C , and D respectively. Suppose A B C D is a convex quadrilateral with ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ A B = B C ’ and C D = D A , and suppose that the area of A B C D = 4. Given these conditions, the set of possible lengths of AB consists of all real numbers in the interval [ a, b ). Compute b .

解析

Regular tetrahedron ABCD is projected onto a plane sending A , B , C , and D to ′ ′ ′ ′ ′ ′ ′ ′ A , B , C , and D respectively. Suppose A B C D is a convex quadrilateral with ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ A B = A D and C B = C D , and suppose that the area of A B C D = 4. Given these conditions, the set of possible lengths of AB consists of all real numbers in the interval [ a, b ). Compute b . √ 4 Solution: 2 6 ′ ′ ′ ′ The value of b occurs when the quadrilateral A B C D degenerates to an isosceles triangle. This occurs when the altitude from A to BCD is parallel to the plane. Let s = AB . Then the altitude from A intersects the center E of face BCD . Since √ √ 2 s s s 6 ′ ′ 2 √ EB = , it follows that A C = AE = s − = . Then since BD is parallel to 3 3 3 √ √ 2 1 s 6 ′ ′ ′ ′ ′ ′ 2 the plane, B D = s . Then the area of A B C D is 4 = · , implying s = 4 6, or 2 3 √ 4 s = 2 6.