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x_{2n+1} = P(x_{2n}), x_{2n+2} = Q(x_{2n+1}) ,every pos. integer divides x_n,

Source: 2016 Saudi Arabia GMO TST level 4+, III p3

August 1, 2020
dividesdivisoralgebrapolynomial

Problem Statement

Find all polynomials P,Q∈Z[x]P,Q \in Z[x] such that every positive integer is a divisor of a certain nonzero term of the sequence (xn)n=0∞(x_n)_{n=0}^{\infty} given by the conditions:
x0=2016x_0 = 2016, x2n+1=P(x2n)x_{2n+1} = P(x_{2n}), x2n+2=Q(x2n+1)x_{2n+2} = Q(x_{2n+1}) for all n≥0n \ge 0
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