3. Philip's grandfather was born in this century. Philip noticed that the sum of the digits of the birth year of his grandfather is equal to the sum of the digits of the number of years his grandfather had in 1985. In which year was Philip's grandfather born?
Solution. Let Philip's grandfather be born in the year $19 x y$ for $0 \leq y \leq 9$ and $0 \leq x \leq 9$. The digits of the number of years of the grandfather's age in 1985 are: $8-x$ and $5-y$ or $7-x$ and $15-y$. Then

$$
1+9+x+y=8-x+5-y \text { or } 1+9+x+y=7-x+15-y \text {, }
$$

i.e.

$$
2(x+y)=3 \text { or } 2(x+y)=12
$$

Since 3 is not divisible by 2, we get that $x+y=6$. If $x \geq 1$, then in 1985 the grandfather is less than 71 years old, and the sum of the digits of any natural number less than 71 is less than $16=1+9+x+y$. Therefore, $x=0$ and $y=6$, i.e., the grandfather was born in 1906.