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Question Number 213663 by efronzo1 last updated on 13/Nov/24

$$\mathrm{Given}\mathrm{a},\mathrm{b},\mathrm{c}\mathrm{is}\mathrm{natural}\mathrm{numbers}$$$$\mathrm{such}\mathrm{that}(\mathrm{a}-\mathrm{b})(\mathrm{b}-\mathrm{c})(\mathrm{c}-\mathrm{a})=\mathrm{a}+\mathrm{b}+\mathrm{c}.$$$$\mathrm{find}\min \mathrm{value}\mathrm{of}\mathrm{a}+\mathrm{b}+\mathrm{c}$$

Answered by Ghisom last updated on 13/Nov/24

$$\mathrm{let}a<b<c\left({}_{c<a<b\mathrm{possible}}^{\mathrm{also}b<c<a\mathrm{or}}\right)$$$$b=a+p\wedge c=b+q$$$$p(p+q)q=3a+2p+q$$$$a=\frac{p^{2}q+{pq}^2-2p-q}{3}$$$$\Rightarrow p=3m\wedge q=3n$$$$a=9m(m+n)n-2m-n$$$$b=9m(m+n)n+m-n$$$$c=9m(m+n)n+m+2n$$$$a+b+c=27m(m+n)n$$$$\mathrm{if}0\in \mathbb{N}$$$$\mathrm{minimum}\mathrm{at}m=n=a=b=c=a+b+c=0$$$$\mathrm{if}0\notin \mathbb{N}$$$$\mathrm{minimum}\mathrm{at}m=n=1$$$$a=15$$$$b=18$$$$c=21$$$$a+b+c=54$$

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