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Quadratic P with divisors of P(1), P(2),... increasing

Source: XVIII Tuymaada Mathematical Olympiad (2011), Senior Level

July 29, 2011
quadraticsalgebrapolynomialAnalytic Number TheoryInteger sequenceInteger Polynomial

Problem Statement

Let P(n)P(n) be a quadratic trinomial with integer coefficients. For each positive integer nn, the number P(n)P(n) has a proper divisor dnd_n, i.e., 1<dn<P(n)1<d_n<P(n), such that the sequence d1,d2,d3,…d_1,d_2,d_3,\ldots is increasing. Prove that either P(n)P(n) is the product of two linear polynomials with integer coefficients or all the values of P(n)P(n), for positive integers nn, are divisible by the same integer m>1m>1.
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