ILL 1983 Problem 37

The points $A_1,A_2, \ldots , A_{1983}$ are set on the circumference of a circle and each is given one of the values $\pm 1$ . Show that if the number of points with the value $+1$ is greater than $1789$ , then at least $1207$ of the points will have the property that the partial sums that can be formed by taking the numbers from them to any other point, in either direction, are strictly positive.