1. Find the smallest 10-digit number, the sum of whose digits is not less than that of any smaller number.
Answer: 1899999999.

Solution. Among 9-digit numbers, the largest sum of digits is for the number 999999999, which is 81. Since the desired 10-digit number is greater than 999999999, we need to find the smallest number with a sum of digits no less than 81. The sum of the last eight digits of the desired number is no more than $9 \cdot 8=72$, so the sum of the first two digits is no less than $81-72=9$. Since the desired number is the smallest, its first digit is 1, and the second digit is 8. Then the sum of the remaining eight digits is no less than 72, which means the remaining digits are nines.

Criteria. Correct answer without explanation - 1 point. Proven that the sum of the digits of the desired number is no less than $81-3$ points. Complete solution - 7 points.