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Complex Number

Source: 1997 National High School Mathematics League, Exam Two, Problem 2

March 4, 2020
complex numbers

Problem Statement

For real numbers x0,x1,⋯ ,xnx_0,x_1,\cdots,x_n, there exists real numbers y0,y1,⋯ ,yny_0,y_1,\cdots,y_n, satisfying that z02=z12+z22+⋯+zn2z_0^2=z_1^2+z_2^2+\cdots+z_n^2, where zk=xk+iyk(k=0,1,⋯ ,n)z_k=x_k+\text{i}y_{k}(k=0,1,\cdots,n). Find all such (x0,x1,⋯ ,xn)(x_0,x_1,\cdots,x_n).
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