MathDB

The given sequences are

(x_1, x_2, \ldots, x_n)

(y_1, y_2, \ldots, y_n)

with positive terms. Prove that there exists a permutation

p

of the set

\{1, 2, \ldots, n\}

such that for every real

t

the sequence

(x_{p(1)}+ty_{p(1)}, x_{p(2)}+ty_{p(2)}, \ldots, x_{p(n)}+ty_{p(n) })

has the following property: there is a number

k

such that

1 \leq k \leq n

and all non-zero terms of the sequence with indices less than

k

are of the same sign and all non-zero terms of the sequence with indices not less than

k

are the same sign.

Problem Statement