Question 18
Find a positive integer $N$ and $a_{1}, a_{2}, \ldots, a_{N}$, where $a_{k}=1$ or $a_{k}=-1$ for each $k=1,2, \ldots, N$, such that
$$
a_{1} \cdot 1^{3}+a_{2} \cdot 2^{3}+a_{3} \cdot 3^{3}+\cdots+a_{N} \cdot N^{3}=20162016,
$$
or show that this is impossible.
$$
a_{1} \cdot 1^{3}+a_{2} \cdot 2^{3}+a_{3} \cdot 3^{3}+\cdots+a_{N} \cdot N^{3}=20162016,
$$
or show that this is impossible.