VMO 2003 Problem 4

Find the largest positive integer n such that the following equations have integer solutions in x, $y_1$, $y_2$, ... , $y_n$: (x + 1)$^2$ + $y_1^2$ = (x + 2)$^2$ + $y_2^2$ = ... = (x + n)$^2$ + $y_n^2$.