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2023 Iran MO 2nd round P5

Source: 2023 Iran MO 2nd round

May 17, 2023
algebrapolynomial

Problem Statement

5. We call (Pn)n∈N(P_n)_{n\in \mathbb{N}} an arithmetic sequence with common difference Q(x)Q(x) if ∀n:Pn+1=Pn+Q\forall n: P_{n+1} = P_n + Q \newline We have an arithmetic sequence with a common difference Q(x)Q(x) and the first term P(x)P(x) such that P,QP,Q are monic polynomials with integer coefficients and don't share an integer root. Each term of the sequence has at least one integer root. Prove that: \newline a) P(x)P(x) is divisible by Q(x)Q(x) \newline b) deg(P(x)Q(x))=1\text{deg}(\frac{P(x)}{Q(x)}) = 1
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