smc

Looking at the constraints of $S_1$: $x+y>0$ ${\log }_{10}(1+x^2+y^2)\le 1+{\log }_{10}(x+y)$ ${\log }_{10}(1+x^2+y^2)\le {\log }_{10}10+{\log }_{10}(x+y)$ ${\log }_{10}(1+x^2+y^2)\le {\log }_{10}(10x+10y)$ $1+x^2+y^2\le 10x+10y$ $x^2-10x+y^2-10y\le -1$ $x^2-10x+25+y^2-10y+25\le 49$ $(x-5{)}^2+(y-5{)}^2\le (7{)}^2$ $S_1$ is a circle with a radius of $7$. So, the area of $S_1$ is $49\pi$. Looking at the constraints of $S_2$: $x+y>0$ ${\log }_{10}(2+x^2+y^2)\le 2+{\log }_{10}(x+y)$ ${\log }_{10}(2+x^2+y^2)\le {\log }_{10}100+{\log }_{10}(x+y)$ ${\log }_{10}(2+x^2+y^2)\le {\log }_{10}(100x+100y)$ $2+x^2+y^2\le 100x+100y$ $x^2-100x+y^2-100y\le -2$ $x^2-100x+2500+y^2-100y+2500\le 4998$ $(x-50{)}^2+(y-50{)}^2\le (7\sqrt{102}{)}^2$ $S_2$ is a circle with a radius of $7\sqrt{102}$. So, the area of $S_2$ is $4998\pi$. So the desired ratio is $\frac{4998\pi }{49\pi }=102\Rightarrow \boxed{102}$.