Which number is greater ?

99^999 + 1 99^1000 +1
A – B = ———— – ———–
99^1000 + 1 99^1001 + 1

=> (99^1001 + 1)*(99^999 +1) – (99^1000 + 1 )*(99^1000 +1)
——————————————————–
(99^1000 + 1 ) * (99^1001 + 1)

Now here we just need to see if the value of the numerator is +v or -ve

lets expand the numerator
99^2000 + 99^1001 + 99^999 + 1 – (99^2000 + 99^1000 + 99^1000 + 1)

=> 99^1001 + 99^999 – 2*99^1000
=> 99^999 ( 99^2 + 1 – 2*99)
clearly it is +ve as 99^2 is much greater than 2*99

thus A is greater than B.