HMMT 二月 2013 · 冲刺赛 · 第 25 题

英文解析

25. [ 17 ] The sequence ( z ) of complex numbers satisfies the following properties:
    • z and z are not real.n
    1 2
    • z = z z for all integers n ≥ 1.2
    n +2 nn +1
    n +3 z
    • is real for all integers n ≥ 1.
    z 2
    ∣ ∣ ∣ ∣n
    ∣ ∣ ∣ ∣
    z z
    3 4
    ∣ ∣ ∣ ∣
    • = = 2.
    ∣ ∣ ∣ ∣
    z z
    4 5
    Find the product of all possible values of z .
    iθ1
    Answer: 65536 All complex numbers can be expressed as r (cos θ + i sin θ ) = re . Let z benr e .niθn
    2 5 2
    z z z z z πkn +3 n +1 nn +2 n +1 n = = = z is real for all n ≥ 1, so θ = for all n ≥ 2, where k is an 5
    n nn +1
    2 2 2
    z z z 5
    n n nπkinteger. θ + 2 θ = θ , so we may write θ = with k an integer.1
    1 2 3 1 1
    r r r r 1 125
    3 4 3
    2 2 24 = ⇒ r = = r r , so r = 1. = 2 ⇒ r = , r = r r ⇒ r = , and r = r r ⇒ r = 4.
    5 3 3 4 4 2 2 3 1 1
    4 3 2
    r r r r 2 2
    4 5 3 4
    10 10
    Therefore, the possible values of z are the nonreal roots of the equation x − 4 = 0, and the product
    4101
    of the eight possible values is = 4 = 65536. For these values of z , it is not difficult to construct 8
    1421
    a sequence which works, by choosing z nonreal so that | z | = .
    2 2 2