1. There were 10 chatterboxes sitting in a circle. At first, one of them told one joke, the next one clockwise told two jokes, the next one told three, and so on around the circle, until one of them told 100 jokes at once. At this point, the chatterboxes got tired, and the next one clockwise told 99 jokes, the next one told 98, and so on around the circle, until one of them told just one joke, and everyone dispersed. How many jokes did each of these 10 chatterboxes tell in total?
Answer: 1000 jokes.

Solution. Let's number the chatterboxes from 1 to 10 clockwise, starting with the one who told the first joke. Then for any pair of chatterboxes with numbers $k$ and $k+1 (1 \leq k \leq 9)$, the $(k+1)$-th chatterbox initially tells one more joke per round than the $k$-th, for 10 rounds. After the chatterboxes get tired, the $(k+1)$-th chatterbox tells one less joke than the $k$-th, also for 10 rounds. Thus, the first and second told the same number of jokes, the second and third - the same number, and so on. Therefore, all of them told the same number of jokes - each one-tenth of the total. The total number of jokes told is $1+2+\cdots+99+100+99+\cdots+2+1=(1+99)+(2+98)+\cdots+(99+1)+100=100 \cdot 100=10000$. Each of the 10 chatterboxes told $\frac{10000}{10}=1000$ jokes.

Comment. It is proven that each chatterbox told one-tenth of the total number of jokes - 4 points. The total number of jokes told is found - 3 points. For potentially useful ideas and approaches in the absence of a solution - 2-3 points.