On the coordinate plane, the function $$f ( x ) = \begin{cases} - x + 10 & ( x < 10 ) \\ ( x - 10 ) ^ { 2 } & ( x \geq 10 ) \end{cases}$$ and for a natural number $n$, there is a circle $O _ { n }$ centered at $( n , f ( n ) )$ with radius 3. Let $A _ { n }$ be the number of all points with integer coordinates that are inside circle $O _ { n }$ and below the graph of the function $y = f ( x )$, and let $B _ { n }$ be the number of all points with integer coordinates that are inside circle $O _ { n }$ and above the graph of the function $y = f ( x )$. What is the value of $\sum _ { n = 1 } ^ { 20 } \left( A _ { n } - B _ { n } \right)$? [4 points] (1) 19 (2) 21 (3) 23 (4) 25 (5) 27
On the coordinate plane, the function
$$f ( x ) = \begin{cases} - x + 10 & ( x < 10 ) \\ ( x - 10 ) ^ { 2 } & ( x \geq 10 ) \end{cases}$$
and for a natural number $n$, there is a circle $O _ { n }$ centered at $( n , f ( n ) )$ with radius 3. Let $A _ { n }$ be the number of all points with integer coordinates that are inside circle $O _ { n }$ and below the graph of the function $y = f ( x )$, and let $B _ { n }$ be the number of all points with integer coordinates that are inside circle $O _ { n }$ and above the graph of the function $y = f ( x )$. What is the value of $\sum _ { n = 1 } ^ { 20 } \left( A _ { n } - B _ { n } \right)$? [4 points]\\
(1) 19\\
(2) 21\\
(3) 23\\
(4) 25\\
(5) 27