The point ( [ P + 1 ] , [ P ] ) (where, [ x ] is the greatest...

$[ P + 1 ] = [ P ] + 1$, Let $[ P ] = n$, then $n$ is integer

$∴ ( [ P + 1 ] , [ P ] ) = ( n + 1, n )$ lie inside the region of circles

${S}_{1} = x^{2} + y^{2} - 2x - 15 = 0, {C}_{1} = (1,0) , {r}_{1} = 4$

and ${S}_{2} = x^{2} + y^{2} - 2x - 7 = 0, {C}_{2} = (1,0) , {r}_{2} = 2\sqrt{2}$

Both circles are concentric.

$∴ ( n + 1{)}^{2} + n^{2} - 2 ( n + 1 ) - 7 > 0$

and $( n + 1{)}^{2} + n^{2} - 2 ( n + 1 ) - 15 < 0 ⇒ 4 < n^{2} < 8$

Which is not possible for any integer.