Japan Junior Mathematical Olympiad

Let $a$ and $b$ be the ten's digit and the one's digit of $x$, respectively. Since we are concerned with the maximum value of $xy$ and since $a, b, y$ are distinct by assumption, it is clear that it suffices to consider the situation where $a, b, y$ are chosen from $7, 8, 9$. If the numbers for $a$ and $b$ are chosen, then clearly if the number for $a$ is bigger than the number for $b$, the product $xy$ becomes larger. Thus it is enough to consider the three cases: $87 \times 9$, $97 \times 8$ and $98 \times 7$. Of these three numbers the largest is $87 \times 9 = 783$, which is the desired answer.