Question

If number of integral coordinates $(x,y)$ which lie inside to the circle $x^2+y^2=25$ is $n,$ then $\left[\frac{n}{9}\right]$ is
( $[.]$ represents the greatest integer function. )

Answer 1

$x,y\in Z$ and $x^2+y^2<25$ for point to be internal.
If $x=0,y^2<25$
So, possible integral coordinates are
$\left\{\right(0,0),(0,\pm 1),\dots \dots ,(0,\pm 4\left)\right\}=9$

If $x=\pm 1,x^2+y^2<24$
So, possible integral coordinates are
$\left\{\right(\pm 1,0),(\pm 1,\pm 1),(\pm 1,\pm 2),(\pm 1,\pm 3),(\pm 1,\pm 4\left)\right\}=18$

If $x=\pm 2,y^2<21$
So, possible integral coordinates are
$\left\{\right(\pm 2,0),(\pm 2,\pm 1),(\pm 2,\pm 2),(\pm 2,\pm 3),(\pm 2,\pm 4\left)\right\}=18$

If $x=\pm 3,y^2<16$
So, possible integral coordinates are
$\left\{\right(\pm 3,0),(\pm 3,\pm 1),(\pm 3,\pm 2),(\pm 3,\pm 3\left)\right\}=14$

If $x=\pm 4,y^2<9$
So, possible integral coordinates are
$\left\{\right(\pm 4,0),(\pm 4,\pm 1),(\pm 4,\pm 2\left)\right\}=10$

$\Rightarrow n=9+18+18+14+10=69$
$\therefore \left[\frac{n}{9}\right]=7$