maximum sum We can notice invariant, that whenever we switch 1 to -1 or -1 to 1 sum changes its value by 10. Because there is odd count of numbers in the circle. one of sums of -1 or 1 must be odd. so at least ten products has different value than the others- so all products cannot be +1 or -1(so sum +999 is not possible) Now its enough to prove that we can gain sum +989 this we can handle easily by setting all numbers 1 and but one -1 (and we get the maximum sum)

MatusKo wrote: maximum sum We can notice invariant, that whenever we switch 1 to -1 or -1 to 1 sum changes its value by 10. This is not true. If you change the sign of one number, it just changes the sign of the 10 products containing it. The change of + to - and - to + may be balanced. My solution:- I think we can get the sum 995 The sequence formed by the repitition of the block +1,-1,-1,-1,-1 and in this sequence the last block is +1,-1,-1,-1. In this sequence the sum is 995. It is the maximum because we will surely get two -1 products.