49th Mathematical Olympiad in Ukraine

Since $0\le (1-x^2)\le 1$, $0\le (1-y^2)\le 1$, we get that $$ \sqrt{1-x^2} + \sqrt{1-y^2} \geq (1-x^2) + (1-y^2), \text{ and so the equality holds if } $$\{\begin{array}{l}\sqrt{1-x^2}=1-x^2\\ \sqrt{1-y^2}=1-y^2\end{array}$$ \Rightarrow $$\{\begin{array}{l}{x}^2\in \{0;1\}\\ y^2\in \{0;1\}\end{array}$$. \text{ Thus we get the points from the answer.}