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Part of 2003 Polish MO Finals

Problems(1)

x^2 + y^2 + z^2 is divisible by x + y + z - Poland 2003

Source:

11/1/2010
A prime number pp and integers x,y,zx, y, z with 0<x<y<z<p0 < x < y < z < p are given. Show that if the numbers x3,y3,z3x^3, y^3, z^3 give the same remainder when divided by pp, then x2+y2+z2x^2 + y^2 + z^2 is divisible by x+y+z.x + y + z.
modular arithmeticnumber theorynumber theory unsolved