How to prove a set of inequalties in not satisfiable?

Question

For the set of inequalities

$$\begin{cases} 10 a - b - c \ge d\\ 5 b - a - c \ge d\\ 2 c - a - b \ge d\\ d \ge a + b + c\end{cases}$$

how can I show these cannot all be satisfied for $a, b, c, d$ all positive integers?

Answer 1

Let $a+b+c=t$. Then $11a-t\ge d\ge t$, so that $a\ge \frac{2t}{11}$. Similarly, $b\ge \frac{2t}{6}$ and $c\ge \frac{2t}{3}$. Summing these up, we get: $$t=a+b+c\ge\frac{2t}{11}+\frac{2t}{6}+\frac{2t}{3}>t$$ which is impossible.