Problem 7. Find all positive integer solutions of

$$
\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}+\frac{1}{a+b+c-2}=1
$$
Solution. If $a, b, c \geq 2$, then the expression is $\leq 1$, with equality if and only if $a=$ $b=c=2$. So we now assume that one of them is 1, let's say without loss of generality $a=1$. If $b, c \geq 3$ it is again found that the expression is less than 1, so the only cases to analyze now are $b=1$ and $b=2$, which do not yield more solutions.