<s>[INST] Is there a value of $t$ above which these inequalities aren't jointly satisfiable Let $\{a_{1}, ...., a_{8}\} \in [0, 1]$ be such that $\sum \limits_{0 < i \leq 8} a_{i} = 1$  and let $t \in (\frac{1}{2}, 1]$. Is there a value of $t$ above which the following inequalities aren't jointly satisfiable? I know that they are all satisfiable for any value of $t \leq \frac{2}{3}$. 
(1) $\frac{a_{1} + a_{2}}{a_{1} + a_{2} + a_{3} + a_{4}} \geq t$
(2) $\frac{a_{1} + a_{5}}{a_{1} + a_{2} + a_{5} + a_{6}} \geq t$
(3) $\frac{a_{4} + a_{8}}{a_{2} + a_{4} + a_{6} + a_{8}} \geq t$
(4) $\frac{a_{7} + a_{8}}{a_{3} + a_{4} + a_{7} + a_{8}} \geq t$
(5) $\frac{a_{1} + a_{3}}{a_{1} + a_{2} + a_{3} + a_{4}} \leq 1 - t$
(6) $\frac{a_{6} + a_{8}}{a_{2} + a_{4} + a_{6} + a_{8}} \leq 1 - t$
 [/INST] Let $a_1=a_4=\frac{1}{2}$ and everything else equal $0$ (all fractions make sense for these choices since $a_1$ and $a_4$ appear in all denominators). Choose $t=1$. From $(1)$ we get $\frac{1}{2}\geq 1$ and from $(2)$ we get $1\geq 1$, so they are not jointly satisfiable.
(Forgive me if I understood the problem wrong).
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