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Omni-MATH / For a positive integer n we denote by s(n) the sum of the digits of n. Let P(x)=x^(n)+a_(n−1)x^(n−1)+⋯+a₁x+a₀ be a polynomial, where n≥2 and a_(i) is a positive integer for all …

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problem

For a positive integer nn we denote by s(n)s(n) the sum of the digits of nn. Let P(x)=xn+an−1xn−1+⋯+a1x+a0P(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0 be a polynomial, where n≥2n \geqslant 2 and aia_i is a positive integer for all 0≤i≤n−10 \leqslant i \leqslant n-1. Could it be the case that, for all positive integers kk, s(k)s(k) and s(P(k))s(P(k)) have the same parity?
Plain-text mathematical notation (without MathML)
For a positive integer n we denote by s(n) the sum of the digits of n. Let P(x)=x^(n)+a_(n−1)x^(n−1)+⋯+a₁x+a₀ be a polynomial, where n≥2 and a_(i) is a positive integer for all 0≤i≤n−1. Could it be the case that, for all positive integers k, s(k) and s(P(k)) have the same parity?
Original LaTeX notation
For a positive integer $n$ we denote by $s(n)$ the sum of the digits of $n$. Let $P(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ be a polynomial, where $n \geqslant 2$ and $a_i$ is a positive integer for all $0 \leqslant i \leqslant n-1$. Could it be the case that, for all positive integers $k$, $s(k)$ and $s(P(k))$ have the same parity?

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Source and history

task_id
ca5d8b74-d919-5b83-b3ee-44d6e0733e9c
task_key
test--ca5d8b74-d919-5b83-b3ee-44d6e0733e9c
task_revision_id
1
config / split
/ test

Official source

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