MathDB

Suppose that

x

y

, and

z

are complex numbers of equal magnitude that satisfy

x+y+z = -\frac{\sqrt{3}}{2}-i\sqrt{5}

and

xyz=\sqrt{3} + i\sqrt{5}.

x=x_1+ix_2, y=y_1+iy_2,

and

z=z_1+iz_2

for real

x_1,x_2,y_1,y_2,z_1

and

z_2

then

(x_1x_2+y_1y_2+z_1z_2)^2

can be written as

\tfrac{a}{b}

for relatively prime positive integers

a

and

b

. Compute

100a+b.

Problem Statement