1. Daria Dmitrievna is preparing a test on number theory. She promised to give each student as many problems as the number of addends they create in the numerical example

$$
a_{1}+a_{2}+\ldots+a_{n}=2021
$$

where all numbers $a_{i}$ are natural numbers, greater than 10, and are palindromes (do not change if their digits are written in reverse order). If a student does not find any such example, they will receive 2021 problems on the test. What is the minimum number of problems a student can receive? (20 points)
Solution: A student cannot receive one problem since 2021 is not a palindrome. Suppose he can receive two problems, then at least one of the numbers $a_{1}, a_{2}$ is a four-digit number. If it starts with 2, then the second digit is 0 and the number itself is 2002. In this case, the second number is 19, which is not a palindrome. If the number starts with 1, then its last digit is also 1, and the last digit of the second number must be zero, which is incorrect for palindromes. Therefore, the student could not have received two problems. An example of three problems exists, for instance, $1111+888+22=2021$.

Answer: 3