Let $n > 1$ be an integer and $X = \{1, 2, \cdots , n^2 \}$. If there exist $x, y$ such that $x^2\mid y$ in all subsets of $X$ with $k$ elements, find the least possible value of $k$.
1. **Define the problem and set up the notation:**
   Let \( n > 1 \) be an integer and \( X = \{1, 2, \ldots, n^2 \} \). We need to find the least possible value of \( k \) such that in any subset of \( X \) with \( k \) elements, there exist \( x \) and \( y \) such that \( x^2 \mid y \).

2. **Identify the maximum subset size \( t \) that does not satisfy \( x^2 \mid y \):**
   To find the least value of \( k \), we need to find the maximum value of \( t \) such that no pair \( (x, y) \) in a subset of size \( t \) satisfies \( x^2 \mid y \). Then, \( k = t + 1 \).

3. **Construct a subset of \( X \) that avoids \( x^2 \mid y \):**
   Consider the subset \( \{n+1, n+2, \ldots, (n-1)^2, n^2\} \). This subset contains \( n^2 - n \) elements. We need to show that for any \( x \) and \( y \) in this subset, \( x^2 \nmid y \).

4. **Verify that \( x^2 \nmid y \) for the constructed subset:**
   For \( 0 \leq k \leq (n-1) \), we need to show that \( (n-k) < (n-k)^2 < n^2 \). This can be verified as follows:
   \[
   n - k < (n - k)^2 < n^2
   \]
   - For \( n - k < (n - k)^2 \):
     \[
     n - k < n^2 - 2nk + k^2
     \]
     Simplifying, we get:
     \[
     0 < n^2 - 2nk + k^2 - (n - k)
     \]
     \[
     0 < n^2 - 2nk + k^2 - n + k
     \]
     Since \( n^2 - 2nk + k^2 \) is always positive for \( 0 \leq k \leq (n-1) \), the inequality holds.

5. **Conclude the maximum value of \( t \):**
   Thus, the maximum value of \( t \) is \( n^2 - n \).

6. **Determine the least value of \( k \):**
   Since \( k = t + 1 \), we have:
   \[
   k = n^2 - n + 1
   \]

The final answer is \( \boxed{n^2 - n + 1} \).