1. [3] Several natural numbers are written in a row with a sum of 20. No number and no sum of several consecutive numbers equals 3. Could there be more than 10 numbers written?

A. Shapovalov
Answer: could.

Solution. An example with 11 numbers: $1,1,4,1,1,4,1,1,4,1,1$. It can be proven that more than 11 numbers could not have been written down. See also the more general problem 5 for higher grades.