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, 2019

Number theory Difficulty 6.0 National Olympiad Prove it Hungary

Show that for every odd integer $N>5$ there exist vectors $\mathbf{u},\mathbf{v},\mathbf{w}$ in (three-dimensional) space which are pairwise perpendicular, not parallel with any of the coordinate axes, have integer coordinates, and satisfy $|\mathbf{u}|=|\mathbf{v}|=|\mathbf{w}|=N$ .

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