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Omni-MATH / Suppose that x,y, and z are complex numbers of equal magnitude that satisfy x+y+z=−(√(3))/(2)−i√(5) and xyz=√(3)+i√(5). If x=x₁+ix₂,y=y₁+iy₂, and z=z₁+iz₂ for real x₁,x₂,y₁,y₂,z₁, …

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problem

Suppose that x,yx, y, and zz are complex numbers of equal magnitude that satisfy x+y+z=−32−i5x+y+z=-\frac{\sqrt{3}}{2}-i \sqrt{5} and xyz=3+i5.x y z=\sqrt{3}+i \sqrt{5}. If x=x1+ix2,y=y1+iy2x=x_{1}+i x_{2}, y=y_{1}+i y_{2}, and z=z1+iz2z=z_{1}+i z_{2} for real x1,x2,y1,y2,z1x_{1}, x_{2}, y_{1}, y_{2}, z_{1}, and z2z_{2}, then (x1x2+y1y2+z1z2)2\left(x_{1} x_{2}+y_{1} y_{2}+z_{1} z_{2}\right)^{2} can be written as ab\frac{a}{b} for relatively prime positive integers aa and bb. Compute 100a+b100 a+b.
Plain-text mathematical notation (without MathML)
Suppose that x,y, and z are complex numbers of equal magnitude that satisfy x+y+z=−(√(3))/(2)−i√(5) and xyz=√(3)+i√(5). If x=x₁+ix₂,y=y₁+iy₂, and z=z₁+iz₂ for real x₁,x₂,y₁,y₂,z₁, and z₂, then (x₁x₂+y₁y₂+z₁z₂)² can be written as (a)/(b) for relatively prime positive integers a and b. Compute 100a+b.
Original LaTeX notation
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 $$x y z=\sqrt{3}+i \sqrt{5}.$$ If $x=x_{1}+i x_{2}, y=y_{1}+i y_{2}$, and $z=z_{1}+i z_{2}$ for real $x_{1}, x_{2}, y_{1}, y_{2}, z_{1}$, and $z_{2}$, then $$\left(x_{1} x_{2}+y_{1} y_{2}+z_{1} z_{2}\right)^{2}$$ can be written as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $100 a+b$.

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Source and history

task_id
920485a6-009a-54c9-bbf5-d76affa9917a
task_key
test--920485a6-009a-54c9-bbf5-d76affa9917a
task_revision_id
1
config / split
/ test

Official source

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