Problem 7.1. Solve the riddle

$$
C, B A+A, A A=B, A
$$

(Different letters represent different digits, the same letters represent the same digits.)
Answer: $A=5, B=9, C=3$.

Solution. It is obvious that $A \neq 0$ (otherwise, for example, $C=B$).

The hundredths place can disappear only if the digits in the hundredths place sum to a number ending in 0. This is only possible if $A=5$. Then the puzzle can be rewritten as follows:

$$
\begin{gathered}
C, B 5+5,55=B, 5 \\
C, B 5+0,05=B-5
\end{gathered}
$$

In the last equation, the right side is an integer, and it can only be obtained by summing non-integer numbers on the left if $B=9$. Then the last equation can be rewritten as follows:

$$
C, 95+0,05=9-5
$$

From this, it is easy to see that $C=3$.

Remark. The problem could also be solved by multiplying both sides of the original equation by 100.