HMMT 十一月 2009 · 冲刺赛 · 第 27 题

英文解析

27. [ 14 ] ABCD is a regular tetrahedron of volume 1. Maria glues regular tetrahedra A BCD , AB CD ,
    ′ ′ ′ ′ ′ ′
    ABC D , and ABCD to the faces of ABCD . What is the volume of the tetrahedron A B C D ?
    Answer: Consider the tetrahedron with vertices at W = (1 , 0 , 0), X = (0 , 1 , 0), Y = (0 , 0 , 1),125
    1 1 127
    and Z = (1 , 1 , 1). This tetrahedron is similar to ABCD . It has center O = ( , , ). We can
    2 2 2
    ′ ′ ′ ′ ′ ′ ′ ′ ′
    construct a tetrahedron W X Y Z in the same way that A B C D was constructed by letting W
    ′ 1 1 1 ′
    be the reflection of W across XY Z and so forth. Then we see that Z = ( − , − , − ), so OZ has
    3 3 3
    √ √ 5
    5 1 5
    ′ ′ ′ ′
    length 3, whereas OZ has length 3. We thus see that W X Y Z has a side length = that 6
    6 2 31
    ′ ′ ′ ′2
    of W XY Z , so by similarity the same is true of A B C D and ABCD . In particular, the volume of
    ( )
    5 1253
    ′ ′ ′ ′
    A B C D is that of ABCD , so it is .
    3 27
    2 ANNUAL HARVARD-MIT NOVEMBER TOURNAMENT, 7 NOVEMBER 2009 — GUTS ROUNDnd