Sámson writes the number 123456789 on a piece of paper. Then he can insert multiplication signs between any two adjacent digits, at different places, or not at all. The digits between the multiplication signs, when read together, form a number, resulting in an expression that is a product of numbers, for example, 1234$\cdot$56$\cdot$789. What is the maximum possible value of the resulting number?
Solution. Let's choose the position for the first multiplication sign arbitrarily, thereby dividing the original number into $A$ and $B$. Let $k$ be the length of the number $B$. The value of the original number is $A \cdot 10^{k} + B$, while the value of the product formed by inserting the multiplication sign is $A \cdot B$. Since the number $B$ consists of $k$ digits, we have $10^{k} > B$. Therefore,

$$
A \cdot 10^{k} + B > A \cdot B + B \geq A \cdot B
$$

We can apply this line of reasoning for every subsequent insertion of a multiplication sign.

From this, we can conclude that each insertion of a multiplication sign decreases the value of the expression. Therefore, the largest number is obtained when no multiplication sign is inserted.

Thus, the largest possible number is 123456789.