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in Olympiad by (65.1k points)

show that for all real numbers x, y, z such that x + y + z = 0 and xy + yz + zx = - 3, the expression x3y + y3z + z3x is constcnt.

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Consider the equation whose roots are, y, z:

(t - x)(t - y)(t - z) = 0

This gives t3 - 3t - λ = 0 where λ = xyz. since x,y,z are root of this equation, we have

x3 - 3x - λ = 0, y3 -  3y - = 0, z3 - 3z - λ = 0

Multiplying the first by y,the second by z and the third by x, we obtain 

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