1. On a circle, 2021 points are marked. Kostya marks a point, then marks the next point to the right, then he marks the point to the right of the last marked point, skipping one, then the point to the right of the last marked point, skipping two, and so on. On which move will a point be marked for the second time?
Answer: on the 67th move. The question is about the smallest natural $b$ for which $(a+1)+(a+2)+\cdots+(b-1)+b$ is divisible by $2021=43 \cdot 47$. This sum is equal to $b(b+1) / 2-a(a+1) / 2=$ $(b-a)(b+a+1) / 2$ and it is not hard to see that the smallest $b=(b-a) / 2+(b+a) / 2$ is achieved with the factorization $(66-19)(66+19+1) / 2=47 \cdot 86 / 2=47 \cdot 43$. In this case, $b=66$, i.e., this happened on the 67th move.