$$(x+1)^2 + y_1^2 = (x+2)^2 + y_2^2 = \cdots = (x+n)^2 + y_n^2$$
has integer solutions $(x,y_1,y_2,\ldots,y_n)$.
What is the largest positive integer $x$ that satisfies the equation?
Hint: start with $n=2$, where you have only one equation. Put the $y$'s on one side, the other terms on the other, and factor the difference of two squares.
Once you get to $n=3$ solutions become quite unlikely. The $y$'s have to be carefully chosen. You will get one solution set for each pair of $y$'s, then have to take the intersection to see if there are any $x$'s at all.