IMO Shortlist 1989 problem 9
Dodao/la:
arhiva2. travnja 2012. ![\forall n > 0, n \in \mathbb{Z},](/media/m/e/4/9/e49185060d39eab91a7d8a8423e84680.png)
there exists uniquely determined integers
![a_n, b_n, c_n \in \mathbb{Z}](/media/m/c/5/c/c5cd5ce9ef5e59ac83b75daddd2b74f4.png)
such
Prove that
![c_n = 0](/media/m/1/2/3/123fd6ecb4e8ad99f3f13c5e5f7de898.png)
implies
%V0
$\forall n > 0, n \in \mathbb{Z},$ there exists uniquely determined integers $a_n, b_n, c_n \in \mathbb{Z}$ such
$$\left(1 + 4 \cdot \sqrt[3]{2} - 4 \cdot \sqrt[3]{4} \right)^n = a_n + b_n \cdot \sqrt[3]{2} + c_n \cdot \sqrt[3]{4}.$$
Prove that $c_n = 0$ implies $n = 0.$
Izvor: Međunarodna matematička olimpijada, shortlist 1989