## Task 1 - 100911

When asked about his age, Mr. $X$ said:

"The sum of the digits of the number of years I have lived is exactly one third of this number. The square of the sum of the digits of the number of years I have lived is exactly three times the number of years I have lived."

Can Mr. $X$'s statements be true? If so, how old is Mr. $X$? (In full years of life)
a) If the statements of Mr. $X$ are correct, the square of the mentioned cross sum is exactly three times the number of years of life, and this in turn is three times the mentioned cross sum. Therefore, the square of this cross sum is exactly nine times the cross sum itself. It follows that 9 is the cross sum and thus 27 years is the age of Mr. $X$.

b) If 27 years is the age of Mr. $X$, then 9 is the cross sum of the number of his years of life, which is one third of this number. Furthermore, then the square 81 of the cross sum is exactly three times the number of years of life of Mr. $X$. Therefore, the statements of Mr. $X$ are correct.

From b) it follows that the statements of Mr. $X$ can be correct. From this and from a) it follows: Mr. $X$ is 27 years old.