Competition problem — Algebra — Maths Olympiad Prep

Library / /10 of 155

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. $\square$

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement and solution reproduced as published; topic and difficulty added by this site.