imc

The equation of the circle is $(x-5{)}^2+(y-5{)}^2=100$. Plugging in the given conditions we have $(x-5{)}^2+(-x-5{)}^2\le 100$. Expanding gives: $x^2-10x+25+x^2+10x+25\le 100$, which simplifies to $x^2\le 25$ and therefore $x\le 5$ and $x\ge -5$. So $x$ ranges from $-5$ to $5$, for a total of $\boxed{11}$ integer values. Note by Williamgolly: Alternatively, draw out the circle and see that these points must be on the line $y=-x$.