1.The largest number $\overline{a b c d}$, written in base 10, represents the birth year of "Mr. Problem", which satisfies the relationship: $\overline{(c+1) d}=5 \cdot \overline{a(b-1)}$. Determine his age.
1.The relation from the statement can also be written as:

$10 \cdot(c+1)+d=5 \cdot(10 a+b-1)(*)$.

It follows that $d \vdots 5$, so $d=0$ or $d=5$.

Case I: $d=0$.

Equation $(*)$ becomes $10 \cdot (c+1)=5 \cdot(10 a+b-1)$ or

$2 \cdot(c+1)=10 a+b-1$. From the statement, it is deduced that $2 \cdot(c+1) \leq 18$, and $a=\overline{1,9}$, so $a=1$. We get $2 \cdot(c+1)=b+9$. From the last relation, it is deduced that the digit $b$ is odd, and since the number $\overline{a b c d}$ is the largest, but such that $b+9 \leq 18$, we get $b=9$. It follows that $c=8$, and the birth year, for this case, is 1980.

Case II: $d=5$.

Equation $(*)$ becomes $10 \cdot (c+1)+5=5 \cdot(10 a+b-1)$ or

$2 \cdot(c+1)+1=10 a+b-1$. Otherwise, $2 \cdot(c+1)=10 a+b-2$. From property 2, it is deduced that $2 \cdot(c+1) \leq 18$, and $a=\overline{1,9}$, so $a=1$. We get $2 \cdot(c+1)=b+8$. From the last relation, it is deduced that the digit $b$ is even, and since the number $\overline{a b c d}$ is the largest, but such that $b+8 \leq 18, b=\overline{0,9}$, we get $b=8$. It follows that $c=7$, and the birth year, for this case, is 1875.

Comparing the numbers found in the cases listed above, it is observed that the largest of them is 1980, which represents the birth year of "Mr. Problem".

The age of "Mr. Problem" is: 2016 - 1980 = 36 years.