TITLE: How prove this $x_{1}+2x_{2}+\cdots+1990x_{1990}\neq 0$
QUESTION [7 upvotes]: Question:

let $x_{i}=1$ or $-1$,$i=1,2,\cdots,1990$, show that
  $$x_{1}+2x_{2}+\cdots+1990x_{1990}\neq 0$$

this problem it seem is easy,But I think is not easy. 
I think note
$$1+2+3+\cdots+1990\equiv \pmod { 1990}?$$

REPLY [2 votes]: Consider the equation modulo 2. Regardless of whether $x_i = 1,or -1$, $x_i\equiv 1\pmod{2}$.
Thus $\sum_{i=1}^{1990}{ix_i}\equiv 1+0+1+\cdots +0\pmod{2}$, where there are $\dfrac{1990}{2}=995$ $1$'s in the summation. 
We conclude that $\sum_{i=1}^{1990}{ix_i}\equiv 1\pmod{2}$, so it definitely cannot be equal to $0$.