If $a,b,c$ are positive integers,and $\cfrac{a}{b}+\cfrac{b}{c}+\cfrac{c}{a}$ and $\cfrac{b}{a}+\cfrac{c}{b}+\cfrac{a}{c}$ are integers then $a=b=c$
Solution 1:
If they are not all equal, there must be some prime $p$ such $a = p^i A$, $b = p^j B$, $c = p^k C$ where $A,B,C$ are each coprime to $p$ and $i,j,k$ are nonnegative integers that are not all equal. Without loss of generality $i \le j \le k$ (otherwise permute $a$, $b$, $c$ appropriately) and $i < k$.
Then $ b/a + c/b + a/c$ has $p^{k-i}$ in its denominator, and can't be an integer.