MathDB

Let

a_i, b_i

be positive real numbers, i\equal{}1,2,\ldots,n,

n\geq 2

, such that

a_i<b_i

, for all

i

, and also b_1\plus{}b_2\plus{}\cdots \plus{} b_n < 1 \plus{} a_1\plus{}\cdots \plus{} a_n. Prove that there exists a

c\in\mathbb R

such that for all i\equal{}1,2,\ldots,n, and

k\in\mathbb Z

we have (a_i\plus{}c\plus{}k)(b_i\plus{}c\plus{}k) > 0.

Problem Statement