Problem 3. On a circle, 999 numbers are written, each of which is 1 or -1, and not all numbers are equal. All products of ten consecutive numbers written around the circle are calculated, and the obtained products are summed.

a) (3 points) What is the smallest possible sum?

b) (3 points) And what is the largest possible sum?

Answer. a) -997. b) 995.
Solution. a) Estimation. If two adjacent products are equal, then the first number of the left decade is equal to the last number of the right one, i.e., the numbers 10 positions apart are equal. Since 10 and 999 are coprime, with a step of 10, all recorded numbers will be traversed. However, among them, there are different numbers, which means that among the products, there are also different ones. This means that at least one product is equal to 1.

Here is an example with one product equal to 1 and 998 products equal to -1: if the sequential number of the number ends in 9, we write -1, and the other numbers are equal to 1. Then the only positive product is from the 999th number to the 9th.

b) Example. If we write two -1s next to each other, and the other numbers are 1, we will have exactly two negative products.

Estimation. The product of all products is equal to the 10th power of the product of all numbers, i.e., it is equal to 1. Therefore, an even number of products are equal to -1, i.e., at least 2.