Q:

Taking numbers away and then add the remaining to get 100

This question came up in a math competition a few weeks ago. 

My reasoning for (a) was that if we took away the 9 smallest numbers (1-9), the smallest 9 numbers that we would then be able to choose from then would be 10, 11, 12, 13, 14, 15, 16, 17, 18. The sum of these numbers is already more than 100 and therefore, it is not always possible. Is this reasoning valid?
I tried to use the same reasoning for (b) but if you add the the numbers from 9 to 16 you get exactly 100. What should I do? 
Thank you. 

A:

You can always add up to 100 if 8 are removed.  To add up to 100 you can use 4 pairs of numbers that add to 25.  There are 12 such pairs.  If you remove 8 numbers you can only eliminate at most 8 such pairs.  Thus at least 4 such pairs will always be left.  Thus 4 pairs will always add to 100. 

