Understanding the Set X and the Problem

The problem asks for the maximum possible number of elements in a subset S of a given set X, such that the sum of no two distinct elements in S is 100.

First, let's understand the set X. The set X is defined as $X = \{x \mid x = 2 + 4k, \text{ where } k = 0, 1, 2, 3,\dots, 24\}$.

Let's list the elements of X by substituting the values of k:

• For $k=0$, $x = 2 + 4(0) = 2$.
• For $k=1$, $x = 2 + 4(1) = 6$.
• For $k=2$, $x = 2 + 4(2) = 10$.
• ...
• For $k=12$, $x = 2 + 4(12) = 2 + 48 = 50$.
• ...
• For $k=24$, $x = 2 + 4(24) = 2 + 96 = 98$.

So, the set X is $\{2, 6, 10, 14, \dots, 50, \dots, 94, 98\}$. This is an arithmetic progression with the first term 2 and common difference 4. The number of terms in X is $24 - 0 + 1 = 25$. So, $|X|=25$.

Identifying Pairs that Sum to 100

The condition for the subset S is that the sum of no two distinct elements of S is 100. This means if $a \in S$ and $b \in S$ and $a \neq b$, then $a+b \neq 100$.

Let's find pairs of distinct elements $(a, b)$ in X such that $a+b=100$. Let $a = 2+4k_1$ and $b = 2+4k_2$, where $k_1, k_2 \in \{0, 1, \dots, 24\}$ and $k_1 \neq k_2$.

The sum is $a+b = (2+4k_1) + (2+4k_2) = 4 + 4k_1 + 4k_2 = 4(1 + k_1 + k_2)$.

We want this sum to be 100:

$4(1 + k_1 + k_2) = 100$

$1 + k_1 + k_2 = 25$

$k_1 + k_2 = 24$

We need to find distinct pairs of integers $(k_1, k_2)$ from $\{0, 1, \dots, 24\}$ that sum to 24. Let's list these pairs, assuming $k_1 < k_2$ to avoid duplicates and ensure distinct elements from X:

There are 12 such pairs of distinct elements in X that sum to 100.

What about an element that sums to 100 with itself? Let $a \in X$ such that $a+a = 100$. This means $2a = 100$, so $a=50$. We check if $50 \in X$. $50 = 2+4k \implies 48 = 4k \implies k=12$. Since $k=12$ is in the range $\{0, 1, \dots, 24\}$, the element 50 is in X.

Constructing the Maximum Subset S

The condition is that the sum of no two distinct elements of S is 100.

Consider the 12 pairs we found: (2, 98), (6, 94), ..., (46, 54). If S contains both elements from any of these pairs, the condition is violated. Therefore, from each of these 12 pairs, S can contain at most one element.

To maximize the number of elements in S, we should take exactly one element from each of these 12 pairs. This gives us 12 elements. For example, we could take the set of the smaller elements from each pair: $\{2, 6, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46\}$. This set has 12 elements, and no two distinct elements sum to 100.

Now consider the element 50. This element is not part of any pair of distinct elements in X that sum to 100. If we include 50 in S, does it violate the condition? If $50 \in S$, we need to check if $50+x=100$ for any other $x \in S$ where $x \neq 50$. The only element in X that sums to 100 with 50 is 50 itself ($50+50=100$). Since the condition is on distinct elements, including 50 in S does not create a violation with any other distinct element in S. $50+50=100$ is a sum of two elements that are the same, which is typically not covered by "sum of no two distinct elements".

Therefore, we can include 50 in our set S, along with one element from each of the 12 pairs.

The maximum possible number of elements in S is the sum of the number of elements chosen from the pairs and the number of isolated elements (like 50).

Maximum elements = (Number of pairs summing to 100) $\times$ (1 element per pair) + (Number of elements summing to 100 only with themselves)

Maximum elements = $12 \times 1 + 1$ (for the element 50)

Maximum elements = $12 + 1 = 13$.

A possible maximum subset S could be $\{2, 6, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50\}$. Any two distinct elements in this set do not sum to 100.

Maximum Possible Number of Elements in S

Based on our analysis of pairs summing to 100 and the element 50, the maximum number of elements allowed in S is 13.

Revision Table: Key Concepts

Additional Information: Related Subset Problems

This type of problem is a classic example of finding the maximum size of a subset with a specific constraint, often related to sums or differences. These problems are related to topics in combinatorics and number theory.

The general approach involves:

1. Define the set clearly.
2. Identify elements or pairs of elements that violate the subset condition if both are included.
3. Group elements into these "forbidden" pairs or sets.
4. Count elements that do not belong to any forbidden pair (e.g., the element 50 in this case).
5. Construct the maximum subset by taking at most one element from each forbidden pair and including all "safe" elements.

This method helps partition the original set into groups where including more than a certain number of elements from a group would violate the subset condition, thereby allowing us to count the maximum possible size of the desired subset S.