Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 5.0 AIME, harder Prove it Philippines

Problem:
What is the largest integer $k$ such that $k+1$ divides
$k^{2020}+2 k^{2019}+3 k^{2018}+\cdots+2020 k+2021 ?$

Solution

Solution:
The remainder when the polynomial $x^{2020}+2 x^{2019}+3 x^{2018}+\cdots+2020 x+2021$ is divided by $x+1$ is
$(-1)^{2020}+2(-1)^{2019}+3(-1)^{2018}+\cdots+2020(-1)+2021=1010(-1)+2021=1011$
Therefore, $k+1$ divides $k^{2020}+2 k^{2019}+3 k^{2018}+\cdots+2020 k+2021$ precisely when $k+1$ divides $1011$ . The largest $k$ for which this is true is $1010$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.