Ukrainian Mathematical Competitions

It is clear that the largest number you can get if you use only the actions of multiplication and addition. We will show that there is actually no sense to put any characters and is the largest nine-digit number $123456789$.

First remove one "+" sign, that is write $\overline{XY}$ instead of expression $X + Y$, as a result, the expression will obviously increase, because

$$\overline{XY} \ge 10X + Y > X + Y.$$

Now the expression contains only the multiplication signs. Then, we will substitute $X \cdot Y$ with $\overline{XY}$, and then we will show that the new expression will increase. Let $Y$ have $k$ digits, i.e. $10^{k-1} < Y < 10^k$, then
$$\overline{XY} \ge 10^k X + Y > X \cdot Y \Leftrightarrow (10^k - Y)X + Y > 0.$$