Given two positive numbers $b,\,c$. Prove $\left ( \frac{3}{b}- 1 \right )(3- b)^{2}+ \left ( \frac{b}{c}- 1 \right )(b- c)^{2}+ (c- 1)^{3}\geqq 0$ .

Question

Given two positive numbers $b,\,c$. Prove $\left ( \dfrac{3}{b}- 1 \right )(3- b)^{2}+ \left ( \dfrac{b}{c}- 1 \right )(b- c)^{2}+ (c- 1)^{3}\geqq 0$ . My problem is given a solution by user dragonheart6 (AoPS), I tried a way without using computer decompostion! He is so into using sos with the form $\dfrac{1}{q^{2}}\sum\limits_{i= 1}^{m} f_{i}g_{i}^{2}$, I will post my solution for this soon! $$354816000(b+ 1)^{2}bc.\,\left [ \left ( \frac{3}{b}- 1 \right )(3- b)^{2}+ \left ( \frac{b}{c}- 1 \right )(b- c)^{2}+ (c- 1)^{3} \right ]=$$ $$= 55440\,b^{2}(80\,bc^{2}- 97\,b^{2}- 80\,bc- 40\,c^{2}+ 200\,c)^{2}+ 1836000(16\,b^{3}- 32\,b^{2}c+ 9\,bc^{2}- 8\,bc)^{2}$$ $$+ 39424000\,c(b^{2}+ 3\,bc- 3\,b- 12)^{2}+ 38500(-\,48\,b^{2}c+ 151\,bc^{2}+ 96\,b^{2}- 288\,bc)^{2}+$$ $$+ 2772000\,bc(16\,bc- 23\,b- 16\,c+ 24)^{2}+ 1182720\,b(11\,b^{2}- 10\,bc+ 15\,c^{2}- 35\,c)^{2}+$$ $$+ 184800 bc(-\,59\,b+ 120)^{2}+ 231 b(151\,b^{2}- 1760\,bc+ 3200\,c)^{2}+ 7884800 c(-\,8\,b^{2}+ 15\,b+ 9)^{2}$$ $$+ 19219200\,b^{3}c+ 3379200\,c(7\,b^{2}- 31)^{2}+ 126\,b(553\,b^{2}- 880\,bc)^{2}+$$ $$+16896000\,c+ 18711000\,b^{2}c^{4}+ 7700(48\,b^{2}c- 85\,bc^{2})^{2}+ 1088955\,b^{5}\geqq 0$$

Answer 1

Let $1=a$.

Thus, we need to prove that $$\frac{(3a-b)^3}{b}+\frac{(b-c)^3}{c}+\frac{(c-a)^3}{a}\geq0,$$ which is true because $$\frac{(3a-b)^3}{b}+\frac{(b-c)^3}{c}+\frac{(c-a)^3}{a}>\frac{(2a-b)^3}{b}+\frac{(b-c)^3}{c}+\frac{(c-a)^3}{a}\geq0,$$ where a proof of the last inequality see here: https://artofproblemsolving.com/community/c6h1863562p12626347