5Question: Given three vertices of a cube at $(0,0,0)$, $(1,0,0)$, and $(0,1,0)$, find the coordinates of the fourth vertex in the same face, assuming all coordinates are integers.

February 22, 2026

5Question: Given three vertices of a cube at $(0,0,0)$, $(1,0,0)$, and $(0,1,0)$, find the coordinates of the fourth vertex in the same face, assuming all coordinates are integers.

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Solution: The given points lie on the base face of the cube in the $xy$-plane. The fourth vertex of the square face must complete the rectangle. The missing vertex is $(1,1,0)$, as it forms equal sides and right angles with the given points.
\boxed{(1,1,0)}
Question: For a polynomial modeling the trajectory of a seed dispersal algorithm, $z^6 + z^4 + z^2 + 1 = 0$, find the maximum imaginary part of a root expressed as $\cos \theta$, where $\theta$ is an acute angle.

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