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Saudi Arabian Mathematical Competitions

Algebra Difficulty 4.8 AIME Prove it Saudi Arabia

Find all integers $n$ such that there exists a polynomial $P(x)$ with integer coefficients satisfying
$P\left(\sqrt[3]{n^{2}}+\sqrt[3]{n}\right)=2016 n+20 \sqrt[3]{n^{2}}+16 \sqrt[3]{n} .$

Solution

First, we prove 2 lemmas as the problem in Level 4's Test. Then turn to the problem, since $P\left(\sqrt[3]{n^{2}}+\sqrt[3]{n}\right)=2016 n+20 \sqrt[3]{n^{2}}+16 \sqrt[3]{n}$ , then by the second lemma, we obtain $n-1 \mid 20-16=4$ .
That is, $n \in\{5,3,2,0,-1,-3\}$ , and we can easily check that these numbers satisfy the condition of the problem. $\square$

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