1 + 109
2 + 108
3 + 107
4 + 106
(...)
106 + 4
107 + 3
108 + 2
109 + 1
There are 109 such pairs - unless you don't want to count the combinations of the same numbers but in different order (e.g. 50 + 60 and 60 + 50) - then there are 55 such pairs (up to 55 + 55).
You can also count 0 - sometimes we treat it as a whole number, sometimes we don't - then add two pairs (0 + 110 and 110 + 0) to the 109 pairs result, or 1 pair for 55 pairs result.