Let $A$ be the set $A = \{ 1,2, \ldots, n\}$. Determine the maximum number of elements of a subset $B\subset A$ such that for all elements $x,y$ from $B$, $x+y$ cannot be divisible by $x-y$. 

[i]Mircea Lascu, Dorel Mihet[/i]
1. **Partitioning the Set \( A \):**
   We start by partitioning the set \( A = \{1, 2, \ldots, n\} \) into subsets of three consecutive integers. Specifically, we form the subsets:
   \[
   \{1, 2, 3\}, \{4, 5, 6\}, \{7, 8, 9\}, \ldots
   \]
   This partitioning continues until we cover all elements of \( A \). The number of such subsets is given by:
   \[
   m = \left\lceil \frac{n}{3} \right\rceil
   \]
   where \( \left\lceil x \right\rceil \) denotes the ceiling function, which rounds \( x \) up to the nearest integer.

2. **Condition Analysis:**
   We need to ensure that for any two elements \( x \) and \( y \) in subset \( B \), the condition \( x + y \) is not divisible by \( x - y \). 

3. **Maximum Number of Elements in \( B \):**
   If \( B \) contains more than \( m \) elements, then by the pigeonhole principle, at least two elements of \( B \) must lie in the same subset of the partition \( \mathcal{P} \). We need to show that this would violate the given condition.

4. **Violation of Condition:**
   Consider any subset \( \{a, a+1, a+2\} \) from the partition \( \mathcal{P} \). If we pick any two elements from this subset, say \( x \) and \( y \), we have:
   \[
   x, y \in \{a, a+1, a+2\}
   \]
   Without loss of generality, assume \( x < y \). Then \( x - y \) can be \( -1 \) or \( -2 \). For \( x + y \) to be divisible by \( x - y \), we need:
   \[
   x + y \equiv 0 \pmod{x - y}
   \]
   This is impossible for any pair \( (x, y) \) within the same subset \( \{a, a+1, a+2\} \) because the differences \( x - y \) are too small to divide \( x + y \).

5. **Constructing \( B \):**
   To construct a subset \( B \) with exactly \( m \) elements, we can choose one element from each subset of the partition \( \mathcal{P} \). For example:
   \[
   B = \{1, 4, 7, \ldots\} \cap A
   \]
   This ensures that \( B \) has exactly \( m \) elements and satisfies the condition that \( x + y \) is not divisible by \( x - y \) for any \( x, y \in B \).

6. **Conclusion:**
   Therefore, the maximum number of elements in subset \( B \) such that \( x + y \) is not divisible by \( x - y \) for any \( x, y \in B \) is:
   \[
   \boxed{\left\lceil \frac{n}{3} \right\rceil}
   \]