Number theory · Polynomials · Integer coefficients · Divisibility · Existence questions
Does there exist a polynomial \(P\) with integer coefficients for which
\[ \textbf{a)}\quad P(7) = 8 \ \text{ and } \ P(15) = 12; \qquad\qquad \textbf{b)}\quad P(8) = 7 \ \text{ and } \ P(12) = 15\,? \]
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Serbian Regional Competition (Okruzno takmicenje) 2004, high school grade I, category A, problem 4. Organized by the Mathematical Society of Serbia (DMS). Source