Question 103
Let $S$ be a set containing $11$ distinct positive integers, each not exceeding $60$. Prove that there exist two non-empty subsets $A$ and $B$ of $S$, such that $A \cap B = \emptyset$, $|A| = |B|$, and $\sum_{a \in A} a = \sum_{b \in B} b$.
For example, consider the set $\{1, 2, 3, \ldots, 11\}$. The subsets $\{1, 4\}$ and $\{2, 3\}$ satisfy the conditions, whereas subsets $\{1, 2\}$ and $\{3\}$ do not, since they have different numbers of elements. Similarly, subsets $\{1, 2, 3, 4\}$ and $\{5, 6, 7, 8\}$ do not satisfy the condition because their sums differ. (Batbayasgalan Balkhuu)
For example, consider the set $\{1, 2, 3, \ldots, 11\}$. The subsets $\{1, 4\}$ and $\{2, 3\}$ satisfy the conditions, whereas subsets $\{1, 2\}$ and $\{3\}$ do not, since they have different numbers of elements. Similarly, subsets $\{1, 2, 3, 4\}$ and $\{5, 6, 7, 8\}$ do not satisfy the condition because their sums differ. (Batbayasgalan Balkhuu)