A.K.

On a circle, 999 numbers are arranged, each being 1 or -1, and not all numbers are the same. We will take the product of every 10 consecutive numbers and sum them up.

a) What is the smallest sum that can be obtained?

b) And what is the largest?
a) Evaluation. If two adjacent products are equal, then the first number of the left is equal to the last number of the right, meaning the numbers 10 places apart are equal. Since 10 and 999 are coprime, stepping by 10 will cover all numbers. But among the numbers, there are different ones, so among the products, there are different ones as well. Therefore, there is at least one product equal to 1. An example where exactly one product is equal to 1 and the other 998 are equal to -1: if the number ends in 9, place -1, otherwise place 1. Then the only positive product is from the 999th to the 9th place.

b) Example: two minus ones next to each other, the rest of the numbers are ones. The negative products will only be those where one minus one is included, but the other is not; and there are exactly two such cases.

Evaluation. The product of all 999 products is the 10th power of the product of all numbers, which is 1. Therefore, among the products, there is an even number of minus ones, meaning there are at least two.

## Answer

a) -997;
b) 995.

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