Jana and David are practicing adding decimal numbers by each writing one number, and then adding these two numbers together. Their last problem resulted in 11.11. David's number had the same number of digits before and after the decimal point, as did Jana's. David's number was written with all different digits, while Jana's number had exactly two identical digits.

Determine the largest possible number that David could have written.

(M. Petrová)

Hint. How many digits before/after the decimal point could the numbers being added have?
For the sum to be of the given form, none of the numbers could have more than two digits before or after the decimal point. Thus, both Jan's and David's numbers were of the type **,** or $*, *$. Since the resulting number had a non-zero digit in the hundredths place, at least one of the numbers had to have a non-zero digit in the hundredths place—and therefore also in the tens place. If the other number also had the same properties, the result would be greater than 11.11. Therefore, the second number was of the type $*, *$, and the entire problem can be written as follows:

![](https://cdn.mathpix.com/cropped/2024_04_17_adee0188cf1f8dcaeca5g-1.jpg?height=166&width=169&top_left_y=1162&top_left_x=949)

Since only the first number has non-zero digits in the tens and hundredths places, both of these digits must be 1. The entire problem then looks like this:

![](https://cdn.mathpix.com/cropped/2024_04_17_adee0188cf1f8dcaeca5g-1.jpg?height=172&width=167&top_left_y=1521&top_left_x=952)

Since David's number was written with different digits, the first number is Jan's and the second is David's. We are interested in the largest number David could have written, or the smallest number Jan could have written:

- The smallest conceivable Jan's number is 10.01, but this does not meet the condition that exactly two digits are the same.
- The next conceivable Jan's number is 10.11, but this does not meet the same condition.
- The next conceivable Jan's number is 10.21, in which case David's number would be 0.9, and this option meets all the conditions in the problem.

The largest number David could have written was 0.9.