Proof in mathematical question

Question

Let x_1, x_2, ..., x_n be arbitrary real numbers such that the total sum x_1+...+x_n is strictly positive. Prove that there exists an index k in {1,...,n} such that each of the following n sums x_k, x_k+x_{k+1}, x_k+x_{k+1}+x_{k+2}, ..., x_k+...+x_n, x_k+...+x_n+x_1, x_k+...+x_n+x_1+x_2, ..., x_k+...+x_n+x_1+...+x_{k-1} is strictly positive. (In other words, all the sums x_k+...+x_{k+l}, l=0,1,...,n-1, with indices counted modulo n, must be positive.)

Answer 1

Let S be the set of all the numbers in the sum. Let P be the set of all the numbers in S which are strictly positive. Let N be the set of all numbers in S which are negative. S=P+N.

It follows that k must be within the indices of the elements within P because x_k>0 is one of the conditions. The sum of the elements of P must be greater than the sum of the elements of N. The minimum number of elements in P is 1, so k will be the index of the only positive element in S. Let p=x_k>0. Also p is greater than any single element of N as well as being greater than the sum of all the elements of N, therefore k meets all the required conditions.

Consider the following sum: 1-100+2-99+3-98+4-97+X>0, so X=x₉. We can write this as 10-394+X>0, so X>384. In this case, k=9 only. In this worst scenario (minimum) case, the last element makes the sum strictly positive. If X is placed elsewhere in the sum, there may be other values of k. For example: X+1-100+2-99+3-98+4-97. We know that X is greater than all the other negatives, individually or collectively, so k=1 is the unique value of k for this sum. So from this we can see that there is at least one value of k that meets all the conditions.