Problem 3

Let

x_1,x_2,\ldots,x_n

be real numbers satisfying

|x_1+x_2+\cdots+x_n|=1

and

|x_i|\le(n+1)/2

for every

i

. Show that there is a permutation

y_1,\ldots,y_n

of the

x_i

such that

|y_1+2y_2+\cdots+ny_n|\le(n+1)/2

Step 1 of 4: Normalize the total sum

Previous step Next step

In plain words

Normalize the total sum

x_1+\cdots+x_n=1

Detailed analysis

If the total sum is $-1$ , replace every $x_i$ by $-x_i$ ; this preserves the hypotheses and changes the final weighted sum by a sign. Thus assume the total sum is $1$ .