Question 145
a) Determine all 4-tuples $(x_0, x_1, x_2, x_3)$ of pairwise distinct integers such that each $x_k$ is coprime to $x_{k+1}$ (indices are reduced modulo 4), and the cyclic sum $x_0/x_1 + x_1/x_2 + x_2/x_3 + x_3/x_0$ is an integer.
b) Show that there are infinitely many 5-tuples $(x_0, x_1, x_2, x_3, x_4)$ of pairwise distinct integers such that each $x_k$ is coprime to $x_{k+1}$ (indices are reduced modulo 5), and the cyclic sum $x_0/x_1 + x_1/x_2 + x_2/x_3 + x_3/x_4 + x_4/x_0$ is an integer.
b) Show that there are infinitely many 5-tuples $(x_0, x_1, x_2, x_3, x_4)$ of pairwise distinct integers such that each $x_k$ is coprime to $x_{k+1}$ (indices are reduced modulo 5), and the cyclic sum $x_0/x_1 + x_1/x_2 + x_2/x_3 + x_3/x_4 + x_4/x_0$ is an integer.