Number theory · Divisibility · Residues mod 3 · Cyclic identities
Integers \(x\), \(y\), \(z\) satisfy
\[ x^{2}z + y^{2}x + z^{2}y = x^{2}y + y^{2}z + z^{2}x + x + y + z . \]
Prove that \(27 \mid x + y + z\).
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Serbian Regional Competition 2008, high school grade I, category A, problem 4. Organized by the Mathematical Society of Serbia (DMS). Source