geometry!!!!
posted by bravo .
Let A be a subset of T={1,2,…,100} subject to the condition that if 2 distinct elements a and b are in A, then a+b is not in A. What is the maximum value of A, which is the number of elements in A?

Looks like 51 to me.
If 49 is in A, 50,51 cannot be in A
If 48 is in A, 50,51,52 cannot be in A.
if n<50 is in A, we must exclude 51n elements from A.
So, including 50,51,...100 in A, we get 51 elements.