For positive integers $a$ and $b$, let $M(a,b)$ denote their greatest common divisor. Determine all pairs of positive integers $(m,n)$ such that for any two positive integers $x$ and $y$ such that $x\mid m$ and $y\mid n$, 
$$M(x+y,mn)>1.$$

[i]Proposed by Ivan Novak[/i]
1. **Define the problem and notation:**
   Let \( M(a, b) \) denote the greatest common divisor (gcd) of two positive integers \( a \) and \( b \). We need to determine all pairs of positive integers \( (m, n) \) such that for any two positive integers \( x \) and \( y \) with \( x \mid m \) and \( y \mid n \), the condition \( M(x + y, mn) > 1 \) holds.

2. **Introduce the product \( P \):**
   Let \( P \) be the product of all prime numbers that divide \( m \) but do not divide \( n \). Therefore, \( P \mid m \) and \( \gcd(P, n) = 1 \).

3. **Set specific values for \( x \) and \( y \):**
   Set \( x = P \) and \( y = n \). We need to check the condition:
   \[
   \gcd(P + n, mn) > 1
   \]

4. **Analyze the gcd condition:**
   Since \( \gcd(P, n) = 1 \), we have:
   \[
   \gcd(P + n, n) = \gcd(P, n) = 1
   \]
   This implies that \( P + n \) is coprime with \( n \).

5. **Consider the gcd with \( m \):**
   We need to show that:
   \[
   \gcd(P + n, m) > 1
   \]
   Suppose there exists a prime \( p \) that divides both \( m \) and \( P + n \).

6. **Case analysis on \( p \):**
   - **Case 1:** If \( p \) divides \( n \) as well, then \( p \) does not divide \( P \) by definition. This leads to a contradiction because \( p \) cannot divide \( P + n \) if it does not divide \( P \).
   - **Case 2:** If \( p \) does not divide \( n \), then \( p \) must divide \( P \) by definition. This again leads to a contradiction because \( p \) cannot divide \( P + n \) if it divides \( P \) but not \( n \).

7. **Conclusion:**
   Since both cases lead to contradictions, there cannot be any prime \( p \) that divides both \( m \) and \( P + n \). Therefore, the only way for \( \gcd(P + n, m) > 1 \) to hold is if \( m \) and \( n \) share at least one common prime factor.

8. **Final condition:**
   For the condition \( \gcd(x + y, mn) > 1 \) to hold for any \( x \mid m \) and \( y \mid n \), \( m \) and \( n \) must share at least one common prime factor.

The final answer is \( \boxed{ (m, n) } \) such that \( \gcd(m, n) > 1 \).