Consider the following problem:
Given $M, N$ positive integers and $\{(m,n)\}$ be a collections of pairs of non-negative real numbers satisfying with $m \leq M, n \leq N$ and $m+n = k$, where $k$ is a positive integer, consider the claim:
There are $A, B$ real numbers such that $AB < 0$ and
$$AM + BN < 0,$$ $$Am + Bn > 0,$$ forall $(m,n)$,
if, and only if, there are no pairs $(m_1,n_1), (m_2,n_2)$ with $$m_1 \leq \frac{Mk}{M+N} \leq m_2.$$
Now I am interested on the following natural generalization:
Given $(M_1,...,M_r)$ positive integer numbers, a collection $\{(t_1,...,t_r)\}$ real numbers, such that for each $(m_1,...,m_r)$ sequence of real non-negative numbers with $m_i \leq M_i$ and $m_1 + ... + m_r = k$, for a positive integer $k$,
$$t_1m_1 + .... + t_rm_r > 0$$ $$t_1M_1 + ... + t_rM_r < 0.$$
I was expecting to obtain any relation related to the previous for "two dimensional case". Any hints??? I tried to reduce to the previous case but I failed.
Thanks in advance.