5. In a Cartesian coordinate system, points with both coordinates being integers are called integer points, and the set of all integer points is denoted as $X$. Find the largest real number $\alpha$, such that there exists a function $f: X \rightarrow \mathbf{N}^{*}$, satisfying: (1) There exists a positive real number $c$, such that for each positive integer $n$, the number of integer points $A$ satisfying $f(A)=n$ is greater than $c n^{\alpha}$; (2) For any integer point $\triangle A B C$ (i.e., $A, B, C$ are all integer points), $f(A)>|B C|, f(B)>|A C|$, $f(C)>|A B|$ at least one of these conditions holds.