Product of numbers $\pm \sqrt{1} \pm \sqrt{2} \pm \dots \pm \sqrt{100}$ is a perfect square Let $A$ be the product of $2^{100}$ numbers of the form 
$$\pm \sqrt{1} \pm \sqrt{2} \pm \dots \pm \sqrt{100}$$
Show that $A$ is an integer, and moreover, a perfect square.
I found a similar problem here, but the induction doesn't seem to show that $A$ is a perfect square. And I think we can generalize the following problem: if $n$ is a perfect square then $A$ is a perfect square, too.

The same argument can be used to show that the product of the $2^{n-1}$ numbers $\sqrt{1} \pm \sqrt{2} \pm \cdots \pm \sqrt{n}$ is an integer.
Then your number is exactly the square of this product.
