Question 21
(RUS) IMO3. Let $x_1,x_2,\ldots,x_n$ be real numbers satisfying the conditions
$$|x_1+x_2+\cdots+x_n|=1\quad\text{and}\quad |x_i|\le\frac{n+1}{2}\quad\text{for }i=1,2,\ldots,n.$$
Show that there exists a permutation $y_1,\ldots,y_n$ of the sequence $x_1,\ldots,x_n$ such that
$$|y_1+2y_2+\cdots+ny_n|\le\frac{n+1}{2}.$$
$$|x_1+x_2+\cdots+x_n|=1\quad\text{and}\quad |x_i|\le\frac{n+1}{2}\quad\text{for }i=1,2,\ldots,n.$$
Show that there exists a permutation $y_1,\ldots,y_n$ of the sequence $x_1,\ldots,x_n$ such that
$$|y_1+2y_2+\cdots+ny_n|\le\frac{n+1}{2}.$$