Sums and Squares

$a + b = a^{2} + b^{2} \Longrightarrow a^{2} - a + b^{2} - b = 0 \Longrightarrow \left(a - \dfrac{1}{2}\right)^{2} + \left(b - \dfrac{1}{2}\right)^{2} = \dfrac{1}{2}$ ,

which describes a circle of radius $\dfrac{1}{\sqrt{2}}$ , i.e., of diameter $\sqrt{2}$ . The largest inscribed square will have as its diagonal the diameter of this circle, and thus its sides will have length $1$ and its area will therefore be $\boxed{1}$ .