Question 71
Let $(x_{1}, y_{1}),(x_{2}, y_{2}),(x_{3}, y_{3}),(x_{4}, y_{4})$, and $(x_{5}, y_{5})$ be the vertices of a regular pentagon centered at $(0,0)$. Compute the product of all positive integers $k$ such that the equality
$$
x_{1}^{k}+x_{2}^{k}+x_{3}^{k}+x_{4}^{k}+x_{5}^{k}=y_{1}^{k}+y_{2}^{k}+y_{3}^{k}+y_{4}^{k}+y_{5}^{k}
$$
must hold for all possible choices of the pentagon.
Proposed by: Daniel Zhu
$$
x_{1}^{k}+x_{2}^{k}+x_{3}^{k}+x_{4}^{k}+x_{5}^{k}=y_{1}^{k}+y_{2}^{k}+y_{3}^{k}+y_{4}^{k}+y_{5}^{k}
$$
must hold for all possible choices of the pentagon.
Proposed by: Daniel Zhu