Algebra · Polynomials · Integer coefficients · Integer roots · Divisibility · Parity
Let \(P(x)\) be a polynomial with integer coefficients such that, for every positive integer \(n\), dividing \(P(P(n))\) by \(n\) leaves remainder \(n - 1\). Prove that \(P(x)\) has no integer root.
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Serbian National Competition (Drzavno takmicenje) 2012, high school grade I, category A, problem 3. Organized by the Mathematical Society of Serbia (DMS). Source