}

$A$ can be at most 27 years old; for the largest cross sum possible under the given conditions is $1+8+9+9=27$. He was thus born after 1924. Let his birth year be $1900+10 a+b$ with $a, b$ integers and $2 \leq a \leq 5 ; 0 \leq b \leq 9$. His age on January 1, 1953, is therefore (according to the conditions) $1+9+a+b$ years.

Thus, if he was born on January 1 (Case 1):

$$\begin{aligned} 1+9+a+b & =1953-(1900+10 a+b) \\ 43 & =11 a+2 b \end{aligned}$$

This equation, considering the conditions for $a$ and $b$, is only satisfied by $a=3$ and $b=5$. Therefore, A was born on January 1, 1935, and is 18 years old.

However, he could also have been born on another day (Case 2):

$$\begin{aligned} 1+9+a+b & =1952-(1900+10 a+b) \\ 42 & =11 a+2 b \end{aligned}$$

This equation is not satisfied by any pair $(a, b)$ under the conditions of the problem. The solution given for Case 1 is thus the only one.

Solutions of the I. Round 1966 adopted from [5]

\subsection*{7.8.2 II. Round 1966, Class 10}