7.5. On December 31, 2011, Yevgeny Alexandrovich's age coincided with the sum of the digits of his birth year. How old was Yevgeny Alexandrovich on December 31, 2014? Prove the uniqueness of the answer.

Solution. The maximum sum of the digits of the birth year can be $1+9+9+9=28$, and the minimum - 2. Therefore, Y.A. was born no earlier than 1983 and no later than 2009. Note that if we change only the last digit of the birth year, the sum of the digits will increase, while the age will decrease (and vice versa) by the same amount. Therefore, in each decade, there can be no more than one suitable year. It remains to check the possible decades. If the birth year falls in the 2000s, we get the equation $2+0+0+x=11-x$. Thus, $2x=9$, which has no solution in integers. If the birth year falls in the 1980s, we get the equation $1+9+8+x=31-x$ or $2x=13$, which also has no solution in integers. Finally, for the 1990s, we get the equation $1+9+9+x=21-x$. Solving it, we get the only solution: $x=1$. Therefore, Y.A. was born in 1991. Thus, in 2014, he was 23 years old.
Answer: 23 years.

## Criteria:

7 points - complete solution;

4 points - equations are correctly set up;

3 points - the earliest and latest year of birth are estimated;

2 points - noted that in each decade there is no more than one suitable year.