On August 25, 2001 Juan turned as many years as the result of adding the digits of the year of his birth.
In what year was Juan born?
We can say that Juan was born in the year 19x and since being born in 2000 in 2001 he could not turn 2 + 0 + 0 + 0 = 2 years.
Therefore, according to the statement, we have that the age at 2001 will be 10 + x + y
We match both expressions and we have to:
2001 - 19xy = 10 + x + y; 1991 = 19xy + x + y
1900 + 91 = 1900 + 10x + y + x + y; 11x + 2y = 91
Equality is met for x = 7 and y = 7
So Juan born on August 25, 1977.