Let X = {x | x = 2 + 4k, where k = 0, 1, 2, 3,...24}. Let S be a subset of X such that the sum of no two elements of S is 100. What is the maximum possible number of elements in S ?  

Calculation:

The set X is given by

{2, 6, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50, 54, 58, 62, 66, 70, 74, 78, 82, 86, 90, 94, 98}.

We want to find the maximum size of a subset S of X such that no two elements sum to 100.

The pairs in X that sum to 100 are

(2, 98), (6, 94), (10, 90), (14, 86), (18, 82), (22, 78), (26, 74), (30, 70), (34, 66), (38, 62), (42, 58), (46, 54), (50, 50){note: 50 appears only once in X }

Therefore, 

To maximize the number of elements in S while ensuring no two elements sum to 100:

• Choose one element from each of the 12 pairs (but not both)
• Additionally, include the element 50

The maximum possible number of elements in S = 13

∴ The maximum possible number of elements in S be 13.