<s>[INST] Find minimum of $\frac{a+3c}{a+2b+c}+\frac{7a+6b+3c}{a+b+2c}+\frac{c-a}{2a+b+c}$ for non-negative reals 
Let $a, b, c\ge 0$, not all zero. Find the minimum of
$$N = \frac{a+3c}{a+2b+c}+\frac{7a+6b+3c}{a+b+2c}+\frac{c-a}{2a+b+c}. $$

 [/INST] Hint: Let $x,y,z$ be the denominators of the three fractions, and express $a,b,c$ in terms of $x,y,z$. Then write $N$ in terms of $x,y,z$. After some simplification we will get terms like $\frac{x}{z},\frac{x}{y},\frac{y}{x},\frac{y}{z},\cdots$Then consider inequality $\frac{x}{y}+\frac{y}{x}\geqslant2$ if $xy$ is positive and analogously $\frac{x}{z}+\frac{z}{x}\geqslant2$ and so on...
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