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Let z_r (1 <= r <= 4) be complex numbers with |z_r| = sqrt(r + 1), and suppose |30 z1 + 20 z2 + 15 z3 + 12 z4| = k * |z1 z2 z3 + z2 z3 z4 + z3 z4 z1 + z4 z1 z2|. Find k.

1. |z1 z2 z3 z4|
2. |z1 z2 z3|
3. |z2 z3 z4|
4. |z3 z4 z1|

Correct answer: |z1 z2 z3 z4|

Solution

Using |z_r|² = r+1, the coefficient of z_r is 60/(r+1) = 60/|z_r|², so the left expression equals 60|conj(z1)+...| pattern that factors to give k = |z1 z2 z3 z4|.

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