2023 OIM Problems/Problem 6

Problem

Let $P$ be a polynomial of degree greater than or equal to 4 with integer coefficients. An integer $x$ is called $P$-representable if there exist integers $a$ and $b$ such that $x = P(a) - P(b)$. Prove that, if for all $N \ge 0$, more than half of the integers in the set ${0, 1, \cdots, N}$ are $P$-representable, then all even integers are $P$-representable or all odd integers are $P$-representable.

Solution

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See also

https://sites.google.com/associacaodaobm.org/oim-brasil-2023/pruebas