CN-SMO 2006 Problem 7

(1) Find the number of positive integer solutions $(m,n,r)$ of the indeterminate equation $mn+nr+mr=2(m+n+r)$ .(2) Given an integer $k (k>1)$ , prove that indeterminate equation $mn+nr+mr=k(m+n+r)$ has at least $3k+1$ positive integer solutions $(m,n,r)$ .