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Complex numbers!

Source: Iran 3rd round 2013 - Algebra Exam - Problem 3

September 11, 2013
complex numbersalgebra proposedalgebra

Problem Statement

For every positive integer n≥2n \geq 2, Prove that there is no n−n-tuple of distinct complex numbers (x1,x2,…,xn)(x_1,x_2,\dots,x_n) such that for each 1≤k≤n1 \leq k \leq n following equality holds. ∏1≤i≤ni≠k(xk−xi)=∏1≤i≤ni≠k(xk+xi)\prod_{\underset{i \neq k}{1 \leq i \leq n}}^{ } (x_k - x_i) = \prod_{\underset{i \neq k}{1 \leq i \leq n}}^{ } (x_k + x_i) (20 points)
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