IMO Shortlist 2015 problem A6


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30. kolovoza 2018.
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Let n be a fixed integer with n \ge 2. We say that two polynomials P and Q with real coefficients are block-similar if for each i \in \{1, 2, \ldots, n\} the sequences


P(2015i), P(2015i - 1), \ldots, P(2015i - 2014) \quad \mbox{and}
Q(2015i), Q(2015i - 1), \ldots, Q(2015i - 2014)

are permutations of each other.

(a) Prove that there exist distinct block-similar polynomials of degree n + 1.
(b) Prove that there do not exist distinct block-similar polynomials of degree n.

(Canada)

Izvor: https://www.imo-official.org/problems/IMO2015SL.pdf