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Math (Algebra)

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If the sum of 3 non-zero distinct real numbers a, b, and c is 2, and the two sets {a,b,c} and {1/a,1/b,1/c} are the same, what is the value of a^2+b^2+c^2?

Note: Two sets are the same if there is a one-to-one correspondence between their elements. For example, the sets {1,2,3} and {3,2,1} are the same. Neither of them are the same as {1,2,1}.

  • Math (Algebra) -

    assuming a<=b<=c
    a = 1/c
    b = 1/b
    c = 1/a
    a+b+c=2
    We see that b=±1, so a+c=1 or 3

    If b=1,
    a+ 1/a = 1
    a^2 - a + 1 = 0
    2a = 1±√3 i
    Nope

    If b = -1,
    a + 1/a = 3
    a^2 - 3a + 1 = 0
    a = (3±√5)/2
    c = 2/(3±√5) = (3∓√5)/2

    a^2+b^2+c^2 = (14+6√5)/4 + 1 + (14-6√5)/4 = 8

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