1. Find the smallest 10-digit number, the sum of whose digits is greater than that of any smaller number.
Answer: 1999999999.

Solution. Among 9-digit numbers, the largest sum of digits is for the number 999999 999, which is 81. Since the sought 10-digit number is greater than 999999 999, we need to find the smallest number with a sum of digits no less than 82. If the first digit of this number is 1, then the sum of the remaining digits must be no less than 81, which means the other nine digits are nines.

Criteria. Correct answer without explanation - 1 point. Proven that the sum of digits of the sought number is greater than 81 - 3 points. Complete solution - 7 points.