Question 78
Find the greatest positive integer $n$ such that the system of equations
$$
(x+1)^2 + y_1^2 = (x+2)^2 + y_2^2 = \dots = (x+k)^2 + y_k^2 = \dots = (x+n)^2 + y_n^2
$$
has integral solution $(x, y_1, y_2, \dots, y_n)$.
$$
(x+1)^2 + y_1^2 = (x+2)^2 + y_2^2 = \dots = (x+k)^2 + y_k^2 = \dots = (x+n)^2 + y_n^2
$$
has integral solution $(x, y_1, y_2, \dots, y_n)$.