1. M. Evdokimov
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A natural number is written on the board. If the last digit (in the units place) is erased, the remaining non-zero number will be divisible by 20, and if the first digit is erased, the remaining number will be divisible by 21. What is the smallest number that can be written on the board if its second digit is not equal to 0?
Answer: 1609.

Solution.

The second to last digit of the number is 0, since the number without the last digit is divisible by 20. Therefore, the number is at least four digits. Note that the number remaining after erasing the last digit cannot be 100 according to the condition. Also, this number cannot be 120 or 140, as numbers of the form $\overline{20 a}$ and $\overline{40 a}$ are not divisible by 21. For 160, there is only one example: 1609.