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No rational solutions to a system of equations

Source: Bundeswettbewerb Mathematik 2020, Round 2 - Problem 2

November 17, 2020
number theorynumber theory unsolvedrational numberalgebrasystem of equations

Problem Statement

Prove that there are no rational numbers x,y,zx,y,z with x+y+z=0x+y+z=0 and x2+y2+z2=100x^2+y^2+z^2=100.
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