Number theory · Integer valued polynomials · Parity · Divisibility · Difference quotients
Does there exist a polynomial \(P(x)\) whose coefficients are not all integers, such that
\[ P(0)=0 \qquad\text{and}\qquad \frac{P(a)-P(b)}{a-b}\ \text{ is an integer for every pair of distinct integers } a, b\ ? \]
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Serbian National Competition (Drzavno takmicenje) 2015, high school grade I, category A, problem 4. Organized by the Mathematical Society of Serbia (DMS). Source