20) Two mathematicians, Andrea and Sara, meet one evening. Andrea says, "the sum of the digits of my age is equal to the sum of the digits of your age," and Sara replies, "but next year my sum will be four times yours," to which Andrea retorts, "yes, but in two years our sums will be equal again." Given that neither of them has yet reached 100 years old, how old is Sara?
(A) Less than 20
(E) It cannot be determined uniquely.
(D) More than 40
20) The answer is (C).

For any number $n$, the sum of the digits of $n+1$ is not equal to the sum of the digits of $n$ increased by 1 if and only if the last digit of $n$ is 9. In this case, the sum of the digits of $n+1$ is equal to the sum of the digits of $n$ decreased by 8. Let $x$ be the sum of the digits of Andrea's age (and Sara's age). We then have:

$$
\begin{aligned}
4(x-8) & =x+1, \text { hence } \\
3 x & =33, \\
x & =11
\end{aligned}
$$

Therefore, Andrea's age is 29 years, and Sara's age is 38 years.