We use the digits $1,2, \ldots, 9$ once each to form two integers (e.g., 7419 and 82635). What two integers formed in this way have the greatest product? Prove your answer.
The answer is $9642 \times 87531$.

Place value is an important feature of this problem, but it's awkward to write about since the leading digits matter most, yet we do not know how many digits each number will have. As a workaround, let us prepend "0." to the two integers we are forming, making them into decimals. For example, 7419 and 82635 would become 0.7419 and 0.82635 . Regardless of how many digits our two integers have, the effect is to divide their product by $10^{9}$. Thus, whichever decimals formed in this way have the largest product will correspond to the integers that have the largest product in the original formulation of the problem. The benefit of this transformation is that the leading digits now have a definite place value of tenths, etc.

It is clear that the digits of each decimal should be in descending order (otherwise we can increase that number by rearranging them). Less obviously, a larger digit should never be assigned a lower place value in one decimal than the place value assigned to a smaller digit in the other decimal. For example, we should not form the decimals $0.98 \ldots$ and $0.7 \ldots$, where the larger 8 is assigned a place value of hundredths while the smaller 7 is assigned a place value of tenths. Proof: Let our two decimals be $x$ and $y$, let digit $d$ have place value $10^{-a}$ in decimal $x$, and let digit $e$ have place value $10^{-b}$ in decimal $y$, where $d>e$ and $a>b$. Then swapping these two digits increases the product of the two decimals by $(d-e)\left(10^{-b} y-10^{-a} x\right)=(d-e)\left(10^{-a}\right)\left(10^{a-b} y-x\right)$. Since $0.1 \leq x, y10^{-1} x$, and therefore the increase is positive.

As a result of the foregoing, we see that the tenths digits of our two decimals must be 9 and 8 (in some order), the hundredths digits must be 7 and 6 (in some order), the thousandths must be 5 and 4 , the ten-thousandths 3 and 2 , and the hundredthousandths 1 (and 0). Now the sum of the two decimals is fixed, so their product
is maximized by making the two numbers as close to each other as possible. This is achieved by 0.9642 and 0.87531 , and we are done.