ILL 1977 Problem 7

Prove the following assertion: If $c_1,c_2,\ldots ,c_n\ (n\ge 2)$ are real numbers such that $$(n-1)(c_1^2+c_2^2+\cdots +c_n^2)=(c_1+c_2+\cdots + c_n)^2,$$ then either all these numbers are nonnegative or all these numbers are nonpositive.