Japan Mathematical Olympiad

Let us write $p=abc$, $q=def$ and $r=ghi$. We first show that it is possible to make the maximum of the three numbers $p,q,r$ to be no more than $72$. Indeed, if we consider the following, we see that this is possible: $$1\times 8\times 9=72,2\times 5\times 7=70,3\times 4\times 6=72.$$ Next, we will show that the maximum of the three numbers $p,q,r$ must be at least $72$. Since all the numbers $a,b,c,d,e,f,g,h,i$ lie in between $1$ and $9$ and are distinct, we see that the product $pqr$ of the three numbers $p,q,r$ must be a constant, which from the example above must equal $72\times 70\times 72$. In particular, we have $70^3<pqr$. Thus the maximum of $p,q,r$ must be greater than $70$. Since $71$ cannot be written as a product of integers lying in between $1$ through $9$, we see that it is impossible to have any of $p,q,r$ to be $71$. Therefore, the maximum of $p,q,r$ has to be at least $72$.