<s>[INST] Let $a,b,c >0$ Prove the inequality $\displaystyle{\frac{a}{a+1}+\frac{b}{b+1}+\frac{c}{c+1}+\frac{1}{a+b+c+1} \geq 1}$ Let $a,b,c >0$ 
Prove the inequality $\displaystyle{\frac{a}{a+1}+\frac{b}{b+1}+\frac{c}{c+1}+\frac{1}{a+b+c+1} \geq 1}$
I dont even know where to begin.
Only interested in hints (not solution)
 [/INST] We have $a+1\leq a+b+c+1$, $b+1\leq a+b+c+1$ and $c+1\leq a+b+c+1$ so $\frac 1{x+1}\geq \frac 1{1+a+b+c}$ where $x=a,b$ or $c$. Now multiply by $x$ to get $\frac x{x+1}\geq \frac x{1+a+b+c}$ and sum.
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