<s>[INST] Can someone tell me what should i do next? Should I use the inequality between Arithmetic and Geometric mean? Problem 1: Let $a,b,c> 0$
$ab+3bc+2ca\leqslant18$
Prove that:
$\frac{3}{a} + \frac{2}{b}+ \frac{1}{c}\geqslant 3$.
I started on this way:
 $\frac{3}{a} + \frac{2}{b}+ \frac{1}{c}\geqslant 3$
$\frac{3bc+2ac+ab}{abc}\geqslant 3  $
$\frac{ab+3bc+2ac}{abc}\geqslant 3 \times \frac{abc}{3}$
$\frac{ab+3bc+2ca}{3}\geqslant abc$
 [/INST] By the AM-GM inequality, we have $$\left(\frac{ab+3bc+2ac}{3}\right)^3\ge 6{(abc)}^2$$
$$\Rightarrow \frac 1{abc}\ge \sqrt{\frac{3^3.6}{(ab+3bc+2ac)^3}}$$
$$\Rightarrow \frac{ab+3bc+2ac}{abc}\ge \sqrt{\frac {3^3.6}{ab+3bc+2ac}}\ge\sqrt{\frac{3^3.6}{18}}=3$$
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