Consider a polynomial $P(x,y,z)$ in three variables with integer coefficients such that for any real numbers $a,b,c,$ $$P(a,b,c)=0 \Leftrightarrow a=b=c.$$
Find the largest integer $r$ such that for all such polynomials $P(x,y,z)$ and integers $m,n,$ $$m^r\mid P(n,n+m,n+2m).$$

[i]Proposed by Ma Zhao Yu
1. **Understanding the problem**: We need to find the largest integer \( r \) such that for any polynomial \( P(x,y,z) \) with integer coefficients satisfying \( P(a,b,c) = 0 \Leftrightarrow a = b = c \), the expression \( m^r \) divides \( P(n, n+m, n+2m) \) for all integers \( m \) and \( n \).

2. **Analyzing a specific polynomial**: Consider the polynomial \( P(x,y,z) = (x-y)^2 + (y-z)^2 \). This polynomial satisfies \( P(a,b,c) = 0 \Leftrightarrow a = b = c \).

3. **Evaluating the polynomial at specific points**: Evaluate \( P(x,y,z) \) at \( (x,y,z) = (n, n+m, n+2m) \):
   \[
   P(n, n+m, n+2m) = (n - (n+m))^2 + ((n+m) - (n+2m))^2 = (-m)^2 + (-m)^2 = m^2 + m^2 = 2m^2
   \]
   This shows that \( 2m^2 \) is always divisible by \( m^2 \), implying \( r \leq 2 \).

4. **Generalizing the result**: To show that \( m^2 \) always divides \( P(n, n+m, n+2m) \) for any polynomial \( P(x,y,z) \) satisfying the given condition, consider the polynomial \( Q(t) = P(n, n+t, n+2t) \) for any fixed \( n \).

5. **Properties of \( Q(t) \)**: The polynomial \( Q(t) \) has exactly one root at \( t = 0 \) because \( P(n, n, n) = 0 \). Since \( P(a,b,c) = 0 \Leftrightarrow a = b = c \), \( t = 0 \) is the only root of \( Q(t) \).

6. **Multiplicity of the root**: Since \( Q(t) \) is a polynomial with integer coefficients and \( t = 0 \) is its only root, the root must have a multiplicity of at least 2. This is because if the root had a multiplicity of 1, \( Q(t) \) would change sign around \( t = 0 \), contradicting the condition that \( P(a,b,c) = 0 \Leftrightarrow a = b = c \).

7. **Divisibility by \( t^2 \)**: Therefore, \( t^2 \) divides \( Q(t) \), implying \( m^2 \) divides \( Q(m) = P(n, n+m, n+2m) \).

8. **Conclusion**: Since \( m^2 \) always divides \( P(n, n+m, n+2m) \) for any polynomial \( P(x,y,z) \) satisfying the given condition, the largest integer \( r \) is 2.

The final answer is \( \boxed{ r = 2 } \).